Termination w.r.t. Q proof of /tmp/tmpD-78Dd/tpa09.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

Q is empty.

↳ QTRS

  ↳ AAECC Innermost

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

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

Q is empty.

We have applied [19,8] to switch to innermost. The TRS R 1 is

min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
min(0, y) → 0

The TRS R 2 is

f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The signature Sigma is {f}

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

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

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

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

-¹(s(x), s(y)) → -¹(x, y)
F(s(x)) → MAX(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))
MIN(s(x), s(y)) → MIN(x, y)
F(s(x)) → -¹(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))
*¹(x, s(y)) → *¹(x, y)
F(s(x)) → +¹(s(x), s(s(s(s(0)))))
F(s(x)) → +¹(s(x), s(s(s(0))))
MAX(s(x), s(y)) → MAX(x, y)
+¹(s(x), y) → +¹(x, y)
F(s(x)) → *¹(s(x), s(x))
F(s(x)) → F(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))
*¹(x, s(y)) → +¹(x, *(x, y))
F(s(x)) → MAX(*(s(x), s(x)), +(s(x), s(s(s(0)))))

The TRS R consists of the following rules:

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

-¹(s(x), s(y)) → -¹(x, y)
F(s(x)) → MAX(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))
MIN(s(x), s(y)) → MIN(x, y)
F(s(x)) → -¹(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))
*¹(x, s(y)) → *¹(x, y)
F(s(x)) → +¹(s(x), s(s(s(s(0)))))
F(s(x)) → +¹(s(x), s(s(s(0))))
MAX(s(x), s(y)) → MAX(x, y)
+¹(s(x), y) → +¹(x, y)
F(s(x)) → *¹(s(x), s(x))
F(s(x)) → F(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))
*¹(x, s(y)) → +¹(x, *(x, y))
F(s(x)) → MAX(*(s(x), s(x)), +(s(x), s(s(s(0)))))

The TRS R consists of the following rules:

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 6 SCCs with 7 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

-¹(s(x), s(y)) → -¹(x, y)

The TRS R consists of the following rules:

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

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.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

-¹(s(x), s(y)) → -¹(x, y)

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

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

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

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

-¹(s(x), s(y)) → -¹(x, y)

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:

-¹(s(x), s(y)) → -¹(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

+¹(s(x), y) → +¹(x, y)

The TRS R consists of the following rules:

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

+¹(s(x), y) → +¹(x, y)

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

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

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

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

+¹(s(x), y) → +¹(x, y)

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

+¹(s(x), y) → +¹(x, y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

*¹(x, s(y)) → *¹(x, y)

The TRS R consists of the following rules:

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

*¹(x, s(y)) → *¹(x, y)

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

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

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

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

*¹(x, s(y)) → *¹(x, y)

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

*¹(x, s(y)) → *¹(x, y)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

MAX(s(x), s(y)) → MAX(x, y)

The TRS R consists of the following rules:

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

MAX(s(x), s(y)) → MAX(x, y)

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

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

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

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

MAX(s(x), s(y)) → MAX(x, y)

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

MAX(s(x), s(y)) → MAX(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

F(s(x)) → F(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The TRS R consists of the following rules:

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

F(s(x)) → F(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

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

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
f(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

              ↳ QDP

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

F(s(x)) → F(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(x)) → F(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(x)) → F(-(max(+(s(x), *(s(x), x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

              ↳ QDP

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

F(s(x)) → F(-(max(+(s(x), *(s(x), x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(x)) → F(-(max(+(s(x), *(s(x), x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(x)) → F(-(max(s(+(x, *(s(x), x))), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

              ↳ QDP

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

F(s(x)) → F(-(max(s(+(x, *(s(x), x))), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(x)) → F(-(max(s(+(x, *(s(x), x))), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) at position [0,0,1] we obtained the following new rules:

F(s(x)) → F(-(max(s(+(x, *(s(x), x))), s(+(x, s(s(s(0)))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

              ↳ QDP

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

F(s(x)) → F(-(max(s(+(x, *(s(x), x))), s(+(x, s(s(s(0)))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(x)) → F(-(max(s(+(x, *(s(x), x))), s(+(x, s(s(s(0)))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) at position [0,0] we obtained the following new rules:

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

              ↳ QDP

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

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) at position [0,1,0,0] we obtained the following new rules:

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(+(s(x), *(s(x), x))), +(s(x), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

              ↳ QDP

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

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(+(s(x), *(s(x), x))), +(s(x), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(+(s(x), *(s(x), x))), +(s(x), s(s(s(s(0)))))))) at position [0,1,0,0] we obtained the following new rules:

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(s(+(x, *(s(x), x)))), +(s(x), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

              ↳ QDP

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

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(s(+(x, *(s(x), x)))), +(s(x), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(s(+(x, *(s(x), x)))), +(s(x), s(s(s(s(0)))))))) at position [0,1,1] we obtained the following new rules:

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(s(+(x, *(s(x), x)))), s(+(x, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

              ↳ QDP

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

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(s(+(x, *(s(x), x)))), s(+(x, s(s(s(s(0)))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), max(s(s(+(x, *(s(x), x)))), s(+(x, s(s(s(s(0))))))))) at position [0,1] we obtained the following new rules:

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), s(max(s(+(x, *(s(x), x))), +(x, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

              ↳ QDP

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

F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), s(max(s(+(x, *(s(x), x))), +(x, s(s(s(s(0)))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(x)) → F(-(s(max(+(x, *(s(x), x)), +(x, s(s(s(0)))))), s(max(s(+(x, *(s(x), x))), +(x, s(s(s(s(0))))))))) at position [0] we obtained the following new rules:

F(s(x)) → F(-(max(+(x, *(s(x), x)), +(x, s(s(s(0))))), max(s(+(x, *(s(x), x))), +(x, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

              ↳ QDP

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

F(s(x)) → F(-(max(+(x, *(s(x), x)), +(x, s(s(s(0))))), max(s(+(x, *(s(x), x))), +(x, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

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

F(s(0)) → F(-(max(+(0, 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(+(s(x1), *(s(s(x1)), s(x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(+(0, 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(+(s(x1), *(s(s(x1)), s(x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(+(0, 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(max(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(+(s(x1), *(s(s(x1)), s(x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(+(s(x1), *(s(s(x1)), s(x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(+(s(x1), *(s(s(x1)), s(x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(+(s(x1), *(s(s(x1)), s(x1))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0))))))))) at position [0,0,0] we obtained the following new rules:

F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(+(s(x0), *(s(s(x0)), s(x0))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(+(0, *(s(0), 0)), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0))))))) at position [0,0,0] we obtained the following new rules:

F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(+(0, *(s(0), 0)), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0)))))))) at position [0,0] we obtained the following new rules:

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0,0,0,1] we obtained the following new rules:

F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(s(+(x1, *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0,0,0,1] we obtained the following new rules:

F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0))))))))) at position [0,0,0,0,1] we obtained the following new rules:

F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(s(+(x0, *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0)))))))) at position [0,0] we obtained the following new rules:

F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(*(s(0), 0), +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0))))))) at position [0,0,0] we obtained the following new rules:

F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(*(s(0), 0), s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0)))))))) at position [0,0,0] we obtained the following new rules:

F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0)))))))) at position [0,0] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0,0,0,1,0] we obtained the following new rules:

F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, 0)), +(0, s(s(s(s(0)))))))) at position [0,0] we obtained the following new rules:

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0,0,0,1] we obtained the following new rules:

F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0))))))))) at position [0,0,0,0,1] we obtained the following new rules:

F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(s(max(+(x0, *(s(s(x0)), s(x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0)))))))) at position [0,0,0,0,1] we obtained the following new rules:

F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(0, +(0, s(s(s(0))))), max(s(+(0, *(s(0), 0))), s(s(s(s(0))))))) at position [0,0] we obtained the following new rules:

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(max(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0)))))))) at position [0,0] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), +(0, s(s(s(s(0)))))))) at position [0,1,0,0] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), max(s(*(s(0), 0)), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(*(s(0), 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0,1] we obtained the following new rules:

F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(*(s(0), 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, 0)), +(0, s(s(s(s(0)))))))) at position [0,0] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(*(s(0), 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(s(+(x1, s(+(s(x1), *(s(s(x1)), x1))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0,0,0,1,0] we obtained the following new rules:

F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(*(s(0), 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0))))))))) at position [0,0,0,0,1,0] we obtained the following new rules:

F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(*(s(0), 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(s(max(+(x0, +(s(s(x0)), *(s(s(x0)), x0))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0)))))))) at position [0,0,0,0,1] we obtained the following new rules:

F(s(s(x0))) → F(-(s(max(+(x0, s(+(s(x0), *(s(s(x0)), x0)))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(*(s(0), 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(+(s(x0), *(s(s(x0)), x0)))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(+(0, s(s(s(0)))), max(s(+(0, *(s(0), 0))), s(s(s(s(0))))))) at position [0,0] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(*(s(0), 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(+(s(x0), *(s(s(x0)), x0)))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(s(0))), max(s(*(s(0), 0)), +(0, s(s(s(s(0)))))))) at position [0,1,0,0] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(s(s(0))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(+(s(x0), *(s(s(x0)), x0)))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0] we obtained the following new rules:

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, 0)), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(+(s(x0), *(s(s(x0)), x0)))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(s(0))), max(s(+(0, 0)), +(0, s(s(s(s(0)))))))) at position [0,1,0,0] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(+(s(x0), *(s(s(x0)), x0)))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(s(x1), s(s(s(0))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0,1] we obtained the following new rules:

F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(+(s(x0), *(s(s(x0)), x0)))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(s(x0), s(s(s(0))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0))))))))) at position [0,0,1] we obtained the following new rules:

F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(+(s(x0), *(s(s(x0)), x0)))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(s(max(+(x0, s(+(s(x0), *(s(s(x0)), x0)))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0)))))))) at position [0,0,0,0,1,0] we obtained the following new rules:

F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), +(s(x0), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), s(s(s(s(0)))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(s(0))), max(s(+(0, *(s(0), 0))), s(s(s(s(0))))))) at position [0,1] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), s(max(+(0, *(s(0), 0)), s(s(s(0)))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), s(max(+(0, *(s(0), 0)), s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(s(0))), max(s(0), +(0, s(s(s(s(0)))))))) at position [0,1,1] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), max(s(0), s(s(s(s(0)))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), s(max(+(0, *(s(0), 0)), s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), *(s(s(x1)), s(x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,1,0,0] we obtained the following new rules:

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, *(s(s(x1)), s(x1))))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, *(s(s(x1)), s(x1))))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), s(max(+(0, *(s(0), 0)), s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), s(+(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,0] we obtained the following new rules:

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, *(s(s(x1)), s(x1))))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), s(max(+(0, *(s(0), 0)), s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), s(+(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0))))))))) at position [0,0] we obtained the following new rules:

F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, *(s(s(x1)), s(x1))))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), max(s(0), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(s(0))), s(max(+(0, *(s(0), 0)), s(s(s(0)))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(s(0))), s(max(+(0, *(s(0), 0)), s(s(s(0))))))) at position [0] we obtained the following new rules:

F(s(0)) → F(-(s(s(0)), max(+(0, *(s(0), 0)), s(s(s(0))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(s(s(0))), max(s(0), s(s(s(s(0)))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, *(s(s(x1)), s(x1))))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), max(+(0, *(s(0), 0)), s(s(s(0))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(s(0))), max(s(0), s(s(s(s(0))))))) at position [0,1] we obtained the following new rules:

F(s(0)) → F(-(s(s(s(0))), s(max(0, s(s(s(0)))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(s(s(0))), s(max(0, s(s(s(0)))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, *(s(s(x1)), s(x1))))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), max(+(0, *(s(0), 0)), s(s(s(0))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, *(s(s(x1)), s(x1))))), +(s(x1), s(s(s(s(0)))))))) at position [0,1,0,0,0,1] we obtained the following new rules:

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1))))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(s(s(0))), s(max(0, s(s(s(0)))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1))))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), max(+(0, *(s(0), 0)), s(s(s(0))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(+(s(x1), +(s(s(x1)), *(s(s(x1)), x1)))), +(s(x1), s(s(s(s(0)))))))) at position [0,1,0,0] we obtained the following new rules:

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1))))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(s(s(0))), s(max(0, s(s(s(0)))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1))))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), max(+(0, *(s(0), 0)), s(s(s(0))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), max(s(+(s(x0), *(s(s(x0)), s(x0)))), s(+(x0, s(s(s(s(0))))))))) at position [0,1] we obtained the following new rules:

F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), s(max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), s(max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(s(s(s(0))), s(max(0, s(s(s(0)))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1))))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), max(+(0, *(s(0), 0)), s(s(s(0))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(0)), max(+(0, *(s(0), 0)), s(s(s(0)))))) at position [0,1,0] we obtained the following new rules:

F(s(0)) → F(-(s(s(0)), max(*(s(0), 0), s(s(s(0))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), s(max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(s(s(s(0))), s(max(0, s(s(s(0)))))))
F(s(0)) → F(-(s(s(0)), max(*(s(0), 0), s(s(s(0))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1))))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

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

F(s(0)) → F(-(s(s(0)), max(0, s(s(s(0))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), s(max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0)))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1))))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), max(*(s(0), 0), s(s(s(0))))))
F(s(0)) → F(-(s(s(0)), max(0, s(s(s(0))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, +(s(s(x1)), *(s(s(x1)), x1))))), +(s(x1), s(s(s(s(0)))))))) at position [0,1,0,0,0,1] we obtained the following new rules:

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(+(s(x1), *(s(s(x1)), x1)))))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(+(s(x1), *(s(s(x1)), x1)))))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), s(max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0)))))))))
F(s(0)) → F(-(s(s(0)), max(*(s(0), 0), s(s(s(0))))))
F(s(0)) → F(-(s(s(0)), max(0, s(s(s(0))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(s(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0)))))), s(max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0))))))))) at position [0] we obtained the following new rules:

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(+(s(x1), *(s(s(x1)), x1)))))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), max(*(s(0), 0), s(s(s(0))))))
F(s(0)) → F(-(s(s(0)), max(0, s(s(s(0))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(0)), max(*(s(0), 0), s(s(s(0)))))) at position [0,1,0] we obtained the following new rules:

F(s(0)) → F(-(s(s(0)), max(0, s(s(s(0))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(+(s(x1), *(s(s(x1)), x1)))))), +(s(x1), s(s(s(s(0))))))))
F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), max(0, s(s(s(0))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(0)) → F(-(s(s(0)), max(0, s(s(s(0)))))) at position [0,1] we obtained the following new rules:

F(s(0)) → F(-(s(s(0)), s(s(s(0)))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(+(s(x1), *(s(s(x1)), x1)))))), +(s(x1), s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), s(s(s(0)))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(+(s(x1), *(s(s(x1)), x1)))))), +(s(x1), s(s(s(s(0)))))))) at position [0,1,0,0,0,1,0] we obtained the following new rules:

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), +(s(x1), s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0))))))))
F(s(0)) → F(-(s(s(0)), s(s(s(0)))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(+(s(x0), *(s(s(x0)), s(x0))), +(x0, s(s(s(s(0)))))))) at position [0,1,0] we obtained the following new rules:

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, *(s(s(x0)), s(x0)))), +(x0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(s(0)), s(s(s(0)))))
F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, *(s(s(x0)), s(x0)))), +(x0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

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

F(s(0)) → F(-(s(0), s(s(0))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(0), s(s(0))))
F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, *(s(s(x0)), s(x0)))), +(x0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), +(s(x1), s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), +(s(x1), s(s(s(s(0)))))))) at position [0,1,1] we obtained the following new rules:

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), s(+(x1, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(0), s(s(0))))
F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, *(s(s(x0)), s(x0)))), +(x0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), s(+(x1, s(s(s(s(0)))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, *(s(s(x0)), s(x0)))), +(x0, s(s(s(s(0)))))))) at position [0,1,0,0,1] we obtained the following new rules:

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(x0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(0)) → F(-(s(0), s(s(0))))
F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(x0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), s(+(x1, s(s(s(s(0)))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

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

F(s(0)) → F(-(0, s(0)))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

              ↳ QDP

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

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(x0, s(s(s(s(0))))))))
F(s(0)) → F(-(0, s(0)))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), s(+(x1, s(s(s(s(0)))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                                ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(x0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), s(+(x1, s(s(s(s(0)))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), max(s(s(+(x1, s(s(+(x1, *(s(s(x1)), x1))))))), s(+(x1, s(s(s(s(0))))))))) at position [0,1] we obtained the following new rules:

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), s(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(x1, s(s(s(s(0)))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                                    ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), s(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(x1, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(x0, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, +(s(s(x0)), *(s(s(x0)), x0)))), +(x0, s(s(s(s(0)))))))) at position [0,1,0,0,1] we obtained the following new rules:

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(x0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                                        ↳ Rewriting

              ↳ QDP

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

F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), s(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(x1, s(s(s(s(0)))))))))
F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(x0, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x1))) → F(-(s(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0)))))), s(max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(x1, s(s(s(s(0))))))))) at position [0] we obtained the following new rules:

F(s(s(x1))) → F(-(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0))))), max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(x1, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                                                            ↳ Rewriting

              ↳ QDP

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

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(x0, s(s(s(s(0))))))))
F(s(s(x1))) → F(-(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0))))), max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(x1, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, s(+(s(x0), *(s(s(x0)), x0))))), +(x0, s(s(s(s(0)))))))) at position [0,1,0,0,1,0] we obtained the following new rules:

F(s(s(x0))) → F(-(max(+(x0, s(s(+(x0, *(s(s(x0)), x0))))), +(x0, s(s(s(0))))), max(s(+(x0, s(s(+(x0, *(s(s(x0)), x0)))))), +(x0, s(s(s(s(0))))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                                                                                                                              ↳ QDP

              ↳ QDP

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

F(s(s(x1))) → F(-(max(+(x1, s(s(+(x1, *(s(s(x1)), x1))))), +(x1, s(s(s(0))))), max(s(+(x1, s(s(+(x1, *(s(s(x1)), x1)))))), +(x1, s(s(s(s(0))))))))

The TRS R consists of the following rules:

*(x, s(y)) → +(x, *(x, y))
+(s(x), y) → s(+(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
+(0, y) → y
*(x, 0) → 0

The set Q consists of the following terms:

max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

MIN(s(x), s(y)) → MIN(x, y)

The TRS R consists of the following rules:

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))))

The set Q consists of the following terms:

min(0, x0)
min(x0, 0)
min(s(x0), s(x1))
max(0, x0)
max(x0, 0)
max(s(x0), s(x1))
+(0, x0)
+(s(x0), x1)
-(x0, 0)
-(s(x0), s(x1))
*(x0, 0)
*(x0, s(x1))
f(s(x0))

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