Termination w.r.t. Q proof of /tmp/tmpnqMES8/23.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

Q is empty.

↳ QTRS

  ↳ AAECC Innermost

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

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

Q is empty.

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

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The TRS R 2 is

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)

The signature Sigma is {cond1, cond2}

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

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

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

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

COND1(true, x, y, z) → GR(y, z)
COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND2(true, x, y, z) → P(y)
COND2(false, x, y, z) → P(x)
COND2(true, x, y, z) → GR(y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, x, y, z) → GR(x, z)
GR(s(x), s(y)) → GR(x, y)

The TRS R consists of the following rules:

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

COND1(true, x, y, z) → GR(y, z)
COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND2(true, x, y, z) → P(y)
COND2(false, x, y, z) → P(x)
COND2(true, x, y, z) → GR(y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, x, y, z) → GR(x, z)
GR(s(x), s(y)) → GR(x, y)

The TRS R consists of the following rules:

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 5 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

GR(s(x), s(y)) → GR(x, y)

The TRS R consists of the following rules:

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

GR(s(x), s(y)) → GR(x, y)

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

cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(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.

cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

GR(s(x), s(y)) → GR(x, y)

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

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)

The TRS R consists of the following rules:

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(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.

cond1(true, x0, x1, x2)
cond2(true, x0, x1, x2)
cond2(false, x0, x1, x2)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

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

COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND2(false, x, y, z) → COND1(gr(x, z), p(x), y, z) at position [0] we obtained the following new rules:

COND2(false, 0, y1, x0) → COND1(false, p(0), y1, x0)
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), p(s(x0)), y1, s(x1))
COND2(false, s(x0), y1, 0) → COND1(true, p(s(x0)), y1, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

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

COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), p(s(x0)), y1, s(x1))
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND2(false, 0, y1, x0) → COND1(false, p(0), y1, x0)
COND2(false, s(x0), y1, 0) → COND1(true, p(s(x0)), y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

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

COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), p(s(x0)), y1, s(x1))
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND2(false, s(x0), y1, 0) → COND1(true, p(s(x0)), y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), p(s(x0)), y1, s(x1)) at position [1] we obtained the following new rules:

COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

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

COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, s(x0), y1, 0) → COND1(true, p(s(x0)), y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(false, s(x0), y1, 0) → COND1(true, p(s(x0)), y1, 0) at position [1] we obtained the following new rules:

COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

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

COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z)
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND2(true, x, y, z) → COND2(gr(y, z), x, p(y), z) at position [0] we obtained the following new rules:

COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, p(s(x0)), s(x1))
COND2(true, y0, 0, x0) → COND2(false, y0, p(0), x0)
COND2(true, y0, s(x0), 0) → COND2(true, y0, p(s(x0)), 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

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

COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND2(true, y0, s(x0), 0) → COND2(true, y0, p(s(x0)), 0)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, p(s(x0)), s(x1))
COND2(true, y0, 0, x0) → COND2(false, y0, p(0), x0)
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, p(s(x0)), s(x1)) at position [2] we obtained the following new rules:

COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

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

COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, y0, s(x0), 0) → COND2(true, y0, p(s(x0)), 0)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(true, y0, 0, x0) → COND2(false, y0, p(0), x0)
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(true, y0, 0, x0) → COND2(false, y0, p(0), x0) at position [2] we obtained the following new rules:

COND2(true, y0, 0, x0) → COND2(false, y0, 0, x0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

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

COND2(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND2(true, y0, s(x0), 0) → COND2(true, y0, p(s(x0)), 0)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

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

COND2(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND2(true, y0, s(x0), 0) → COND2(true, y0, p(s(x0)), 0)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

p(s(x)) → x
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(true, y0, s(x0), 0) → COND2(true, y0, p(s(x0)), 0) at position [2] we obtained the following new rules:

COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

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

COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)
COND2(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

p(s(x)) → x
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

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

COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)
COND2(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(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.

p(0)
p(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

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

COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)
COND2(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND1(true, x, y, z) → COND2(gr(y, z), x, y, z) at position [0] we obtained the following new rules:

COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND1(true, y0, s(x0), 0) → COND2(true, y0, s(x0), 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

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

COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)
COND2(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND1(true, y0, s(x0), 0) → COND2(true, y0, s(x0), 0)
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND2(true, y0, 0, x0) → COND2(false, y0, 0, x0) we obtained the following new rules:

COND2(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(true, z0, 0, 0) → COND2(false, z0, 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

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

COND2(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(true, z0, 0, 0) → COND2(false, z0, 0, 0)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, y0, s(x0), 0) → COND2(true, y0, s(x0), 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND2(false, s(x0), y1, 0) → COND1(true, x0, y1, 0) we obtained the following new rules:

COND2(false, s(x0), 0, 0) → COND1(true, x0, 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

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

COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)
COND2(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(false, s(x0), 0, 0) → COND1(true, x0, 0, 0)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(true, z0, 0, 0) → COND2(false, z0, 0, 0)
COND1(true, y0, s(x0), 0) → COND2(true, y0, s(x0), 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                    ↳ QDP

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

COND2(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(false, s(x0), 0, 0) → COND1(true, x0, 0, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0) we obtained the following new rules:

COND1(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND1(true, z0, 0, 0) → COND2(false, z0, 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                    ↳ QDP

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

COND1(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(false, s(x0), 0, 0) → COND1(true, x0, 0, 0)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND1(true, z0, 0, 0) → COND2(false, z0, 0, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                                ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                    ↳ QDP

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

COND2(false, s(x0), 0, 0) → COND1(true, x0, 0, 0)
COND1(true, z0, 0, 0) → COND2(false, z0, 0, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                                ↳ UsableRulesProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QReductionProof

                                                                                              ↳ QDP

                                                                                    ↳ QDP

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

COND2(false, s(x0), 0, 0) → COND1(true, x0, 0, 0)
COND1(true, z0, 0, 0) → COND2(false, z0, 0, 0)

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

gr(0, x0)
gr(s(x0), 0)
gr(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.

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                                ↳ UsableRulesProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QReductionProof

                                                                                                      ↳ QDP

                                                                                                        ↳ ForwardInstantiation

                                                                                              ↳ QDP

                                                                                    ↳ QDP

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

COND2(false, s(x0), 0, 0) → COND1(true, x0, 0, 0)
COND1(true, z0, 0, 0) → COND2(false, z0, 0, 0)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND1(true, z0, 0, 0) → COND2(false, z0, 0, 0) we obtained the following new rules:

COND1(true, s(y_0), 0, 0) → COND2(false, s(y_0), 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                                ↳ UsableRulesProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QReductionProof

                                                                                                      ↳ QDP

                                                                                                        ↳ ForwardInstantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ ForwardInstantiation

                                                                                              ↳ QDP

                                                                                    ↳ QDP

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

COND1(true, s(y_0), 0, 0) → COND2(false, s(y_0), 0, 0)
COND2(false, s(x0), 0, 0) → COND1(true, x0, 0, 0)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND2(false, s(x0), 0, 0) → COND1(true, x0, 0, 0) we obtained the following new rules:

COND2(false, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                                ↳ UsableRulesProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QReductionProof

                                                                                                      ↳ QDP

                                                                                                        ↳ ForwardInstantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ ForwardInstantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPSizeChangeProof

                                                                                              ↳ QDP

                                                                                    ↳ QDP

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

COND2(false, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND1(true, s(y_0), 0, 0) → COND2(false, s(y_0), 0, 0)

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

COND2(false, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
The graph contains the following edges 2 > 2, 3 >= 3, 4 >= 3, 3 >= 4, 4 >= 4
COND1(true, s(y_0), 0, 0) → COND2(false, s(y_0), 0, 0)
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 3, 3 >= 4, 4 >= 4

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                              ↳ QDP

                                                                                                ↳ ForwardInstantiation

                                                                                    ↳ QDP

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

COND1(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1))
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND2(false, s(x0), y1, s(x1)) → COND1(gr(x0, x1), x0, y1, s(x1)) we obtained the following new rules:

COND2(false, s(x0), 0, s(x2)) → COND1(gr(x0, x2), x0, 0, s(x2))
COND2(false, s(x0), s(y_2), s(x2)) → COND1(gr(x0, x2), x0, s(y_2), s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                              ↳ QDP

                                                                                                ↳ ForwardInstantiation

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                    ↳ QDP

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

COND2(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND1(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(false, s(x0), 0, s(x2)) → COND1(gr(x0, x2), x0, 0, s(x2))
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(false, s(x0), s(y_2), s(x2)) → COND1(gr(x0, x2), x0, s(y_2), s(x2))

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                              ↳ QDP

                                                                                                ↳ ForwardInstantiation

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ AND

                                                                                                        ↳ QDP

                                                                                                          ↳ ForwardInstantiation

                                                                                                        ↳ QDP

                                                                                    ↳ QDP

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

COND1(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2))
COND2(false, s(x0), 0, s(x2)) → COND1(gr(x0, x2), x0, 0, s(x2))

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND1(true, z0, 0, s(z2)) → COND2(false, z0, 0, s(z2)) we obtained the following new rules:

COND1(true, s(y_0), 0, s(x1)) → COND2(false, s(y_0), 0, s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                              ↳ QDP

                                                                                                ↳ ForwardInstantiation

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ AND

                                                                                                        ↳ QDP

                                                                                                          ↳ ForwardInstantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ QDPSizeChangeProof

                                                                                                        ↳ QDP

                                                                                    ↳ QDP

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

COND2(false, s(x0), 0, s(x2)) → COND1(gr(x0, x2), x0, 0, s(x2))
COND1(true, s(y_0), 0, s(x1)) → COND2(false, s(y_0), 0, s(x1))

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

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:

COND1(true, s(y_0), 0, s(x1)) → COND2(false, s(y_0), 0, s(x1))
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4
COND2(false, s(x0), 0, s(x2)) → COND1(gr(x0, x2), x0, 0, s(x2))
The graph contains the following edges 2 > 2, 3 >= 3, 4 >= 4

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                              ↳ QDP

                                                                                                ↳ ForwardInstantiation

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ AND

                                                                                                        ↳ QDP

                                                                                                        ↳ QDP

                                                                                                          ↳ ForwardInstantiation

                                                                                    ↳ QDP

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

COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1))
COND2(false, s(x0), s(y_2), s(x2)) → COND1(gr(x0, x2), x0, s(y_2), s(x2))

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, x0, s(x1)) we obtained the following new rules:

COND2(true, s(y_1), s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), s(y_1), s(y_2), s(x2))
COND2(true, x0, s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), x0, s(y_2), s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                              ↳ QDP

                                                                                                ↳ ForwardInstantiation

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ AND

                                                                                                        ↳ QDP

                                                                                                        ↳ QDP

                                                                                                          ↳ ForwardInstantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ ForwardInstantiation

                                                                                    ↳ QDP

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

COND2(true, s(y_1), s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), s(y_1), s(y_2), s(x2))
COND2(true, x0, s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), x0, s(y_2), s(x2))
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(false, s(x0), s(y_2), s(x2)) → COND1(gr(x0, x2), x0, s(y_2), s(x2))

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1)) we obtained the following new rules:

COND1(true, s(y_1), s(x1), s(x2)) → COND2(gr(x1, x2), s(y_1), s(x1), s(x2))
COND1(true, s(y_1), s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), s(y_1), s(s(y_2)), s(x2))
COND1(true, x0, s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), x0, s(s(y_2)), s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ AND

                                                                                              ↳ QDP

                                                                                              ↳ QDP

                                                                                                ↳ ForwardInstantiation

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ AND

                                                                                                        ↳ QDP

                                                                                                        ↳ QDP

                                                                                                          ↳ ForwardInstantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ ForwardInstantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ QDPSizeChangeProof

                                                                                    ↳ QDP

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

COND1(true, s(y_1), s(x1), s(x2)) → COND2(gr(x1, x2), s(y_1), s(x1), s(x2))
COND2(true, s(y_1), s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), s(y_1), s(y_2), s(x2))
COND1(true, s(y_1), s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), s(y_1), s(s(y_2)), s(x2))
COND2(true, x0, s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), x0, s(y_2), s(x2))
COND1(true, x0, s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), x0, s(s(y_2)), s(x2))
COND2(false, s(x0), s(y_2), s(x2)) → COND1(gr(x0, x2), x0, s(y_2), s(x2))

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

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:

COND2(false, s(x0), s(y_2), s(x2)) → COND1(gr(x0, x2), x0, s(y_2), s(x2))
The graph contains the following edges 2 > 2, 3 >= 3, 4 >= 4
COND1(true, s(y_1), s(x1), s(x2)) → COND2(gr(x1, x2), s(y_1), s(x1), s(x2))
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4
COND1(true, s(y_1), s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), s(y_1), s(s(y_2)), s(x2))
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4
COND1(true, x0, s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), x0, s(s(y_2)), s(x2))
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4
COND2(true, s(y_1), s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), s(y_1), s(y_2), s(x2))
The graph contains the following edges 2 >= 2, 3 > 3, 4 >= 4
COND2(true, x0, s(s(y_2)), s(x2)) → COND2(gr(s(y_2), x2), x0, s(y_2), s(x2))
The graph contains the following edges 2 >= 2, 3 > 3, 4 >= 4

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

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

COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)

The TRS R consists of the following rules:

gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ QReductionProof

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

COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)

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

gr(0, x0)
gr(s(x0), 0)
gr(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.

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ QReductionProof

                                                                                            ↳ QDP

                                                                                              ↳ ForwardInstantiation

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

COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND2(true, y0, s(x0), 0) → COND2(true, y0, x0, 0) we obtained the following new rules:

COND2(true, x0, s(s(y_1)), 0) → COND2(true, x0, s(y_1), 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ Instantiation

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ AND

                                                                                    ↳ QDP

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ QReductionProof

                                                                                            ↳ QDP

                                                                                              ↳ ForwardInstantiation

                                                                                                ↳ QDP

                                                                                                  ↳ QDPSizeChangeProof

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

COND2(true, x0, s(s(y_1)), 0) → COND2(true, x0, s(y_1), 0)

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

COND2(true, x0, s(s(y_1)), 0) → COND2(true, x0, s(y_1), 0)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4