Termination w.r.t. Q proof of /tmp/tmpMfA87v/tpa5.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:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Q is empty.

↳ QTRS

  ↳ AAECC Innermost

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

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Q is empty.

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

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The TRS R 2 is

f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The signature Sigma is {f}

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

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

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

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

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

-¹(s(x), s(y)) → -¹(x, y)
F(s(x), s(y)) → -¹(x, min(x, y))
F(s(x), s(y)) → F(-(x, min(x, y)), s(twice(min(x, y))))
MIN(s(x), s(y)) → MIN(x, y)
F(s(x), s(y)) → TWICE(min(x, y))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
TWICE(s(x)) → TWICE(x)
F(s(x), s(y)) → MIN(x, y)
F(s(x), s(y)) → -¹(y, min(x, y))

The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

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

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), s(y)) → -¹(x, min(x, y))
F(s(x), s(y)) → F(-(x, min(x, y)), s(twice(min(x, y))))
MIN(s(x), s(y)) → MIN(x, y)
F(s(x), s(y)) → TWICE(min(x, y))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
TWICE(s(x)) → TWICE(x)
F(s(x), s(y)) → MIN(x, y)
F(s(x), s(y)) → -¹(y, min(x, y))

The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 4 SCCs with 4 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

TWICE(s(x)) → TWICE(x)

The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

TWICE(s(x)) → TWICE(x)

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

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

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

TWICE(s(x)) → TWICE(x)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

TWICE(s(x)) → TWICE(x)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ 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:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

MIN(s(x), s(y)) → MIN(x, y)

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

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

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

MIN(s(x), s(y)) → MIN(x, y)

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

MIN(s(x), s(y)) → MIN(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

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:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ 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:

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

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

-¹(s(x), s(y)) → -¹(x, y)

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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

F(s(x), s(y)) → F(-(x, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))

The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

F(s(x), s(y)) → F(-(x, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

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

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

f(s(x0), s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

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

F(s(x), s(y)) → F(-(x, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

F(s(s(x0)), s(s(x1))) → F(-(s(x0), s(min(x0, x1))), s(twice(min(s(x0), s(x1)))))
F(s(x0), s(0)) → F(-(x0, 0), s(twice(min(x0, 0))))
F(s(0), s(x0)) → F(-(0, 0), s(twice(min(0, x0))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

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

F(s(0), s(x0)) → F(-(0, 0), s(twice(min(0, x0))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(s(x0)), s(s(x1))) → F(-(s(x0), s(min(x0, x1))), s(twice(min(s(x0), s(x1)))))
F(s(x0), s(0)) → F(-(x0, 0), s(twice(min(x0, 0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0)), s(s(x1))) → F(-(s(x0), s(min(x0, x1))), s(twice(min(s(x0), s(x1))))) at position [0] we obtained the following new rules:

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(min(s(x0), s(x1)))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(min(s(x0), s(x1)))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(0), s(x0)) → F(-(0, 0), s(twice(min(0, x0))))
F(s(x0), s(0)) → F(-(x0, 0), s(twice(min(x0, 0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

F(s(x0), s(0)) → F(x0, s(twice(min(x0, 0))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

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

F(s(0), s(x0)) → F(-(0, 0), s(twice(min(0, x0))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(min(s(x0), s(x1)))))
F(s(x0), s(0)) → F(x0, s(twice(min(x0, 0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

F(s(0), s(x0)) → F(0, s(twice(min(0, x0))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(min(s(x0), s(x1)))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(x0), s(0)) → F(x0, s(twice(min(x0, 0))))
F(s(0), s(x0)) → F(0, s(twice(min(0, x0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(min(s(x0), s(x1)))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(x0), s(0)) → F(x0, s(twice(min(x0, 0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(min(s(x0), s(x1))))) at position [1,0,0] we obtained the following new rules:

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(s(min(x0, x1)))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(s(min(x0, x1)))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(x0), s(0)) → F(x0, s(twice(min(x0, 0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

F(s(x0), s(0)) → F(x0, s(twice(0)))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

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

F(s(x0), s(0)) → F(x0, s(twice(0)))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(s(min(x0, x1)))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(twice(s(min(x0, x1))))) at position [1,0] we obtained the following new rules:

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

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

F(s(x0), s(0)) → F(x0, s(twice(0)))
F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

F(s(x0), s(0)) → F(x0, s(0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(x0), s(0)) → F(x0, s(0))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

F(s(s(x0)), s(s(x1))) → F(-(s(x1), s(min(x0, x1))), s(twice(min(s(x0), s(x1)))))
F(s(0), s(x0)) → F(-(x0, 0), s(twice(min(0, x0))))
F(s(x0), s(0)) → F(-(0, 0), s(twice(min(x0, 0))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

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

F(s(s(x0)), s(s(x1))) → F(-(s(x1), s(min(x0, x1))), s(twice(min(s(x0), s(x1)))))
F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(x0), s(0)) → F(-(0, 0), s(twice(min(x0, 0))))
F(s(x0), s(0)) → F(x0, s(0))
F(s(0), s(x0)) → F(-(x0, 0), s(twice(min(0, x0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0)), s(s(x1))) → F(-(s(x1), s(min(x0, x1))), s(twice(min(s(x0), s(x1))))) at position [0] we obtained the following new rules:

F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(min(s(x0), s(x1)))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(x0), s(0)) → F(-(0, 0), s(twice(min(x0, 0))))
F(s(x0), s(0)) → F(x0, s(0))
F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(min(s(x0), s(x1)))))
F(s(0), s(x0)) → F(-(x0, 0), s(twice(min(0, x0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

F(s(0), s(x0)) → F(x0, s(twice(min(0, x0))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(x0), s(0)) → F(-(0, 0), s(twice(min(x0, 0))))
F(s(x0), s(0)) → F(x0, s(0))
F(s(0), s(x0)) → F(x0, s(twice(min(0, x0))))
F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(min(s(x0), s(x1)))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

F(s(x0), s(0)) → F(0, s(twice(min(x0, 0))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(x0), s(0)) → F(x0, s(0))
F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(min(s(x0), s(x1)))))
F(s(0), s(x0)) → F(x0, s(twice(min(0, x0))))
F(s(x0), s(0)) → F(0, s(twice(min(x0, 0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(x0), s(0)) → F(x0, s(0))
F(s(0), s(x0)) → F(x0, s(twice(min(0, x0))))
F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(min(s(x0), s(x1)))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(min(s(x0), s(x1))))) at position [1,0,0] we obtained the following new rules:

F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(s(min(x0, x1)))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

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

F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(s(min(x0, x1)))))
F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(x0), s(0)) → F(x0, s(0))
F(s(0), s(x0)) → F(x0, s(twice(min(0, x0))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

F(s(0), s(x0)) → F(x0, s(twice(0)))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

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

F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(s(min(x0, x1)))))
F(s(0), s(x0)) → F(x0, s(twice(0)))
F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(x0), s(0)) → F(x0, s(0))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                              ↳ UsableRulesProof

                                                                                            ↳ QDP

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

F(s(0), s(x0)) → F(x0, s(twice(0)))
F(s(x0), s(0)) → F(x0, s(0))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                              ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                                                  ↳ QReductionProof

                                                                                            ↳ QDP

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

F(s(0), s(x0)) → F(x0, s(twice(0)))
F(s(x0), s(0)) → F(x0, s(0))

The TRS R consists of the following rules:

twice(0) → 0

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(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.

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                              ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                                                  ↳ QReductionProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                            ↳ QDP

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

F(s(0), s(x0)) → F(x0, s(twice(0)))
F(s(x0), s(0)) → F(x0, s(0))

The TRS R consists of the following rules:

twice(0) → 0

The set Q consists of the following terms:

twice(0)
twice(s(x0))

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

F(s(0), s(x0)) → F(x0, s(0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                              ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                                                  ↳ QReductionProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                            ↳ QDP

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

F(s(0), s(x0)) → F(x0, s(0))
F(s(x0), s(0)) → F(x0, s(0))

The TRS R consists of the following rules:

twice(0) → 0

The set Q consists of the following terms:

twice(0)
twice(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                              ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                                                  ↳ QReductionProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                            ↳ QDP

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

F(s(0), s(x0)) → F(x0, s(0))
F(s(x0), s(0)) → F(x0, s(0))

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

twice(0)
twice(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.

twice(0)
twice(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                              ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                                                  ↳ QReductionProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                            ↳ QDP

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

F(s(x0), s(0)) → F(x0, s(0))
F(s(0), s(x0)) → F(x0, s(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(s(0), s(x0)) → F(x0, s(0)) we obtained the following new rules:

F(s(0), s(0)) → F(0, s(0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                              ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                                                  ↳ QReductionProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                            ↳ QDP

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

F(s(x0), s(0)) → F(x0, s(0))
F(s(0), s(0)) → F(0, s(0))

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                              ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                                                  ↳ QReductionProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ ForwardInstantiation

                                                                                            ↳ QDP

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

F(s(x0), s(0)) → F(x0, s(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(s(x0), s(0)) → F(x0, s(0)) we obtained the following new rules:

F(s(s(y_0)), s(0)) → F(s(y_0), s(0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                              ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                                                  ↳ QReductionProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ ForwardInstantiation

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QDPSizeChangeProof

                                                                                            ↳ QDP

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

F(s(s(y_0)), s(0)) → F(s(y_0), s(0))

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

F(s(s(y_0)), s(0)) → F(s(y_0), s(0))
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                            ↳ QDP

                                                                                              ↳ Rewriting

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

F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(s(min(x0, x1)))))
F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(twice(s(min(x0, x1))))) at position [1,0] we obtained the following new rules:

F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(s(s(twice(min(x0, x1))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                            ↳ QDP

                                                                                              ↳ Rewriting

                                                                                                ↳ QDP

                                                                                                  ↳ Instantiation

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

F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(s(s(twice(min(x0, x1))))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(s(s(x0)), s(s(x1))) → F(-(x0, min(x0, x1)), s(s(s(twice(min(x0, x1)))))) we obtained the following new rules:

F(s(s(x0)), s(s(s(y_3)))) → F(-(x0, min(x0, s(y_3))), s(s(s(twice(min(x0, s(y_3)))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                            ↳ QDP

                                                                                              ↳ Rewriting

                                                                                                ↳ QDP

                                                                                                  ↳ Instantiation

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

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

F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(s(s(twice(min(x0, x1))))))
F(s(s(x0)), s(s(s(y_3)))) → F(-(x0, min(x0, s(y_3))), s(s(s(twice(min(x0, s(y_3)))))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(s(s(x0)), s(s(x1))) → F(-(x1, min(x0, x1)), s(s(s(twice(min(x0, x1)))))) we obtained the following new rules:

F(s(s(x0)), s(s(s(y_3)))) → F(-(s(y_3), min(x0, s(y_3))), s(s(s(twice(min(x0, s(y_3)))))))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ AND

                                                                                            ↳ QDP

                                                                                            ↳ QDP

                                                                                              ↳ Rewriting

                                                                                                ↳ QDP

                                                                                                  ↳ Instantiation

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

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

F(s(s(x0)), s(s(s(y_3)))) → F(-(s(y_3), min(x0, s(y_3))), s(s(s(twice(min(x0, s(y_3)))))))
F(s(s(x0)), s(s(s(y_3)))) → F(-(x0, min(x0, s(y_3))), s(s(s(twice(min(x0, s(y_3)))))))

The TRS R consists of the following rules:

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))

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