Termination w.r.t. Q proof of /tmp/tmp4kjnRu/19.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(x, 0), x, y, z)
cond2(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0), x, y, z)
cond3(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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(x, 0), x, y, z)
cond2(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0), x, y, z)
cond3(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0) → 0
p(s(x)) → x

The TRS R 2 is

cond1(true, x, y, z) → cond2(gr(x, 0), x, y, z)
cond2(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0), x, y, z)
cond3(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, y, z)

The signature Sigma is {cond1, cond3, 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(x, 0), x, y, z)
cond2(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0), x, y, z)
cond3(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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)
cond3(true, x0, x1, x2)
cond3(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)
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:

COND2(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), p(x), y, z)
COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND1(true, x, y, z) → GR(x, 0)
COND3(false, x, y, z) → GR(x, z)
COND2(true, x, y, z) → OR(gr(x, z), gr(y, z))
COND2(true, x, y, z) → GR(x, z)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND3(true, x, y, z) → GR(y, z)
COND3(false, x, y, z) → OR(gr(x, z), gr(y, z))
COND2(false, x, y, z) → GR(y, 0)
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND3(true, x, y, z) → P(y)
COND3(true, x, y, z) → GR(x, z)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND3(true, x, y, z) → OR(gr(x, z), gr(y, z))
COND3(false, x, y, z) → GR(y, z)
COND2(true, x, y, z) → GR(y, z)
COND2(true, x, y, z) → P(x)
GR(s(x), s(y)) → GR(x, y)

The TRS R consists of the following rules:

cond1(true, x, y, z) → cond2(gr(x, 0), x, y, z)
cond2(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0), x, y, z)
cond3(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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)
cond3(true, x0, x1, x2)
cond3(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)
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:

COND2(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), p(x), y, z)
COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND1(true, x, y, z) → GR(x, 0)
COND3(false, x, y, z) → GR(x, z)
COND2(true, x, y, z) → OR(gr(x, z), gr(y, z))
COND2(true, x, y, z) → GR(x, z)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND3(true, x, y, z) → GR(y, z)
COND3(false, x, y, z) → OR(gr(x, z), gr(y, z))
COND2(false, x, y, z) → GR(y, 0)
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND3(true, x, y, z) → P(y)
COND3(true, x, y, z) → GR(x, z)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND3(true, x, y, z) → OR(gr(x, z), gr(y, z))
COND3(false, x, y, z) → GR(y, z)
COND2(true, x, y, z) → GR(y, z)
COND2(true, x, y, z) → P(x)
GR(s(x), s(y)) → GR(x, y)

The TRS R consists of the following rules:

cond1(true, x, y, z) → cond2(gr(x, 0), x, y, z)
cond2(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0), x, y, z)
cond3(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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)
cond3(true, x0, x1, x2)
cond3(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)
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 13 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(x, 0), x, y, z)
cond2(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0), x, y, z)
cond3(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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)
cond3(true, x0, x1, x2)
cond3(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)
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)
cond3(true, x0, x1, x2)
cond3(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)
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)
cond3(true, x0, x1, x2)
cond3(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)
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(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), p(x), y, z)
COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)

The TRS R consists of the following rules:

cond1(true, x, y, z) → cond2(gr(x, 0), x, y, z)
cond2(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) → cond3(gr(y, 0), x, y, z)
cond3(true, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) → cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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)
cond3(true, x0, x1, x2)
cond3(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)
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(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), p(x), y, z)
COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND3(true, x, y, z) → COND1(or(gr(x, z), 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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)
cond3(true, x0, x1, x2)
cond3(false, x0, x1, x2)
gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)
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)
cond3(true, x0, x1, x2)
cond3(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(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), p(x), y, z)
COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND3(true, x, y, z) → COND1(or(gr(x, z), 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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) → COND1(or(gr(x, z), gr(y, z)), p(x), y, z) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

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

COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), p(y0), s(x0), 0)
COND2(true, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), p(s(x0)), y1, 0)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), p(0), y1, x0)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), p(s(x0)), y1, 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

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

COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

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

COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND2(true, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), p(y0), s(x0), 0)
COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND2(true, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), p(s(x0)), y1, 0)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), p(s(x0)), y1, 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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) → COND1(or(gr(y0, 0), true), p(y0), s(x0), 0) at position [0] we obtained the following new rules:

COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

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

COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND2(true, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), p(s(x0)), y1, 0)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), p(s(x0)), y1, 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

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

COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

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

COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND2(true, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), p(s(x0)), y1, 0)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

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

COND2(true, 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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

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

COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

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

COND2(true, 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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

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

COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z)
COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND3(false, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, y, z) at position [0] we obtained the following new rules:

COND3(false, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, s(x0), s(x1))
COND3(false, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), y0, s(x0), 0)
COND3(false, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), s(x0), y1, 0)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND3(false, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, s(x0), s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), y0, s(x0), 0)
COND3(false, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), s(x0), y1, 0)
COND2(true, 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND3(false, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), y0, s(x0), 0) at position [0] we obtained the following new rules:

COND3(false, y0, s(x0), 0) → COND1(true, y0, s(x0), 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND3(false, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, s(x0), s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, y0, s(x0), 0) → COND1(true, y0, s(x0), 0)
COND3(false, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), s(x0), y1, 0)
COND2(true, 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND3(false, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), s(x0), y1, 0) at position [0] we obtained the following new rules:

COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND3(false, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, s(x0), s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND2(false, x, y, z) → COND3(gr(y, 0), x, y, z)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, y0, s(x0), 0) → COND1(true, y0, s(x0), 0)
COND2(true, 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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) → COND3(gr(y, 0), x, y, z) at position [0] we obtained the following new rules:

COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND3(false, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, s(x0), s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, y0, s(x0), 0) → COND1(true, y0, s(x0), 0)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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 2 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND1(true, x, y, z) → COND2(gr(x, 0), x, y, z)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, 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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

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

COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND3(true, x, y, z) → COND1(or(gr(x, z), gr(y, z)), x, p(y), z) at position [0] we obtained the following new rules:

COND3(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, p(0), x0)
COND3(true, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), s(x0), p(y1), 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, p(s(x0)), s(x1))
COND3(true, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), y0, p(s(x0)), 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), y0, p(s(x0)), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), s(x0), p(y1), 0)
COND3(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, p(0), x0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, p(s(x0)), s(x1))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), y0, p(s(x0)), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), s(x0), p(y1), 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, p(s(x0)), s(x1))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND3(true, s(x0), y1, 0) → COND1(or(true, gr(y1, 0)), s(x0), p(y1), 0) at position [0] we obtained the following new rules:

COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND3(true, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), y0, p(s(x0)), 0)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, p(s(x0)), s(x1))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, p(s(x0)), s(x1)) at position [2] we obtained the following new rules:

COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND3(true, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), y0, p(s(x0)), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND3(true, y0, s(x0), 0) → COND1(or(gr(y0, 0), true), y0, p(s(x0)), 0) at position [0] we obtained the following new rules:

COND3(true, y0, s(x0), 0) → COND1(true, y0, p(s(x0)), 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)
COND3(true, y0, s(x0), 0) → COND1(true, y0, p(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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND3(true, y0, s(x0), 0) → COND1(true, y0, p(s(x0)), 0) at position [2] we obtained the following new rules:

COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

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

COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND2(true, y0, 0, x0) → COND1(or(gr(y0, x0), false), p(y0), 0, x0) at position [1] we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

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

COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, 0, 0, y1) → COND1(or(gr(0, y1), false), 0, 0, y1)
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

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

COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND2(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), p(y0), s(x0), s(x1)) at position [1] we obtained the following new rules:

COND2(true, 0, s(y1), s(y2)) → COND1(or(gr(0, s(y2)), gr(y1, y2)), 0, s(y1), s(y2))
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(s(x0), s(y2)), gr(y1, y2)), x0, s(y1), s(y2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

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

COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(s(x0), s(y2)), gr(y1, y2)), x0, s(y1), s(y2))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(true, 0, s(y1), s(y2)) → COND1(or(gr(0, s(y2)), gr(y1, y2)), 0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

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

COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(s(x0), s(y2)), gr(y1, y2)), x0, s(y1), s(y2))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(s(x0), s(y2)), gr(y1, y2)), x0, s(y1), s(y2)) at position [0,0] we obtained the following new rules:

COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

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

COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0)
COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND2(true, y0, s(x0), 0) → COND1(true, p(y0), s(x0), 0) at position [1] we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

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

COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND2(true, 0, s(y1), 0) → COND1(true, 0, s(y1), 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

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

COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND3(false, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, y1, x0) at position [0] we obtained the following new rules:

COND3(false, 0, s(x0), s(x1)) → COND1(or(false, gr(x0, x1)), 0, s(x0), s(x1))
COND3(false, 0, s(x0), 0) → COND1(or(false, true), 0, s(x0), 0)
COND3(false, 0, 0, x0) → COND1(or(false, false), 0, 0, x0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

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

COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, 0, s(x0), s(x1)) → COND1(or(false, gr(x0, x1)), 0, s(x0), s(x1))
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, 0, s(x0), 0) → COND1(or(false, true), 0, s(x0), 0)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(false, 0, 0, x0) → COND1(or(false, false), 0, 0, x0)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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 3 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

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

COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND3(true, 0, y1, x0) → COND1(or(false, gr(y1, x0)), 0, p(y1), x0) we obtained the following new rules:

COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, p(s(z1)), z2)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

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

COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, p(s(z1)), z2)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, p(s(z1)), z2) at position [2] we obtained the following new rules:

COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

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

COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND2(false, y0, s(x0), y2) → COND3(true, y0, s(x0), y2) we obtained the following new rules:

COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

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

COND3(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), p(y1), s(x1))
COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, s(x0), y1, 0) → COND1(true, s(x0), p(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
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 2 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

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

COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

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)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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))
or(false, false)
or(true, x0)
or(x0, true)
p(0)
p(s(x0))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

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

COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)
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

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

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

COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND3(true, y0, s(x0), 0) → COND1(true, y0, x0, 0) we obtained the following new rules:

COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

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

COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND3(true, y0, s(x0), s(x1)) → COND1(or(gr(y0, s(x1)), gr(x0, x1)), y0, x0, s(x1)) we obtained the following new rules:

COND3(true, 0, s(z0), s(x2)) → COND1(or(gr(0, s(x2)), gr(z0, x2)), 0, z0, s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

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

COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, s(z0), s(x2)) → COND1(or(gr(0, s(x2)), gr(z0, x2)), 0, z0, s(x2))
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND3(true, 0, s(z0), s(x2)) → COND1(or(gr(0, s(x2)), gr(z0, x2)), 0, z0, s(x2)) at position [0,0] we obtained the following new rules:

COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

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

COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1))
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND3(false, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), s(x0), y1, s(x1)) we obtained the following new rules:

COND3(false, s(x0), 0, s(x2)) → COND1(or(gr(x0, x2), gr(0, s(x2))), s(x0), 0, s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

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

COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), 0, s(x2)) → COND1(or(gr(x0, x2), gr(0, s(x2))), s(x0), 0, s(x2))
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND3(false, s(x0), 0, s(x2)) → COND1(or(gr(x0, x2), gr(0, s(x2))), s(x0), 0, s(x2)) at position [0,1] we obtained the following new rules:

COND3(false, s(x0), 0, s(x2)) → COND1(or(gr(x0, x2), false), s(x0), 0, s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

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

COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), 0, s(x2)) → COND1(or(gr(x0, x2), false), s(x0), 0, s(x2))
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND3(false, s(x0), y1, 0) → COND1(true, s(x0), y1, 0) we obtained the following new rules:

COND3(false, s(x0), 0, 0) → COND1(true, s(x0), 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

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

COND3(false, s(x0), 0, 0) → COND1(true, s(x0), 0, 0)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), 0, s(x2)) → COND1(or(gr(x0, x2), false), s(x0), 0, s(x2))
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND2(false, y0, 0, y2) → COND3(false, y0, 0, y2) we obtained the following new rules:

COND2(false, 0, 0, z1) → COND3(false, 0, 0, z1)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

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

COND3(false, s(x0), 0, 0) → COND1(true, s(x0), 0, 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND3(false, s(x0), 0, s(x2)) → COND1(or(gr(x0, x2), false), s(x0), 0, s(x2))
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, 0, 0, z1) → COND3(false, 0, 0, z1)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

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

COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)
COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0)
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, 0, 0, z1) → COND3(false, 0, 0, z1)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule COND3(false, y0, 0, x0) → COND1(or(gr(y0, x0), false), y0, 0, x0) at position [0] we obtained the following new rules:

COND3(false, s(x0), 0, s(x1)) → COND1(or(gr(x0, x1), false), s(x0), 0, s(x1))
COND3(false, 0, 0, x0) → COND1(or(false, false), 0, 0, x0)
COND3(false, s(x0), 0, 0) → COND1(or(true, false), s(x0), 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

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

COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)
COND3(false, s(x0), 0, s(x1)) → COND1(or(gr(x0, x1), false), s(x0), 0, s(x1))
COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(false, s(x0), 0, 0) → COND1(or(true, false), s(x0), 0, 0)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(false, 0, 0, x0) → COND1(or(false, false), 0, 0, x0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(false, 0, 0, z1) → COND3(false, 0, 0, z1)
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                        ↳ QDP

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

COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                        ↳ QDP

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

COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2)
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND1(true, 0, y1, y2) → COND2(false, 0, y1, y2) we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                        ↳ QDP

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

COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2)
COND1(true, 0, s(y_0), x1) → COND2(false, 0, s(y_0), x1)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND3(true, 0, s(z1), z2) → COND1(or(false, gr(s(z1), z2)), 0, z1, z2) we obtained the following new rules:

COND3(true, 0, s(s(y_1)), x1) → COND1(or(false, gr(s(s(y_1)), x1)), 0, s(y_1), x1)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                        ↳ QDP

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

COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND3(true, 0, s(s(y_1)), x1) → COND1(or(false, gr(s(s(y_1)), x1)), 0, s(y_1), x1)
COND1(true, 0, s(y_0), x1) → COND2(false, 0, s(y_0), x1)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND2(false, 0, s(x1), z1) → COND3(true, 0, s(x1), z1) we obtained the following new rules:

COND2(false, 0, s(x0), 0) → COND3(true, 0, s(x0), 0)
COND2(false, 0, s(s(y_0)), x1) → COND3(true, 0, s(s(y_0)), x1)
COND2(false, 0, s(x0), s(y_1)) → COND3(true, 0, s(x0), s(y_1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                        ↳ QDP

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

COND2(false, 0, s(x0), 0) → COND3(true, 0, s(x0), 0)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND3(true, 0, s(s(y_1)), x1) → COND1(or(false, gr(s(s(y_1)), x1)), 0, s(y_1), x1)
COND2(false, 0, s(s(y_0)), x1) → COND3(true, 0, s(s(y_0)), x1)
COND2(false, 0, s(x0), s(y_1)) → COND3(true, 0, s(x0), s(y_1))
COND1(true, 0, s(y_0), x1) → COND2(false, 0, s(y_0), x1)
COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND3(true, 0, s(z0), 0) → COND1(true, 0, z0, 0) we obtained the following new rules:

COND3(true, 0, s(s(y_0)), 0) → COND1(true, 0, s(y_0), 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                        ↳ QDP

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

COND2(false, 0, s(x0), 0) → COND3(true, 0, s(x0), 0)
COND3(true, 0, s(s(y_0)), 0) → COND1(true, 0, s(y_0), 0)
COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2))
COND3(true, 0, s(s(y_1)), x1) → COND1(or(false, gr(s(s(y_1)), x1)), 0, s(y_1), x1)
COND2(false, 0, s(s(y_0)), x1) → COND3(true, 0, s(s(y_0)), x1)
COND1(true, 0, s(y_0), x1) → COND2(false, 0, s(y_0), x1)
COND2(false, 0, s(x0), s(y_1)) → COND3(true, 0, s(x0), s(y_1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND3(true, 0, s(z0), s(x2)) → COND1(or(false, gr(z0, x2)), 0, z0, s(x2)) we obtained the following new rules:

COND3(true, 0, s(s(y_1)), s(x1)) → COND1(or(false, gr(s(y_1), x1)), 0, s(y_1), s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ QDPSizeChangeProof

                                                                                                                                                                                                        ↳ QDP

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

COND2(false, 0, s(x0), 0) → COND3(true, 0, s(x0), 0)
COND3(true, 0, s(s(y_0)), 0) → COND1(true, 0, s(y_0), 0)
COND3(true, 0, s(s(y_1)), x1) → COND1(or(false, gr(s(s(y_1)), x1)), 0, s(y_1), x1)
COND2(false, 0, s(s(y_0)), x1) → COND3(true, 0, s(s(y_0)), x1)
COND3(true, 0, s(s(y_1)), s(x1)) → COND1(or(false, gr(s(y_1), x1)), 0, s(y_1), s(x1))
COND2(false, 0, s(x0), s(y_1)) → COND3(true, 0, s(x0), s(y_1))
COND1(true, 0, s(y_0), x1) → COND2(false, 0, s(y_0), x1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

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, 0, s(y_0), x1) → COND2(false, 0, s(y_0), x1)
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4
COND3(true, 0, s(s(y_1)), s(x1)) → COND1(or(false, gr(s(y_1), x1)), 0, s(y_1), s(x1))
The graph contains the following edges 2 >= 2, 3 > 3, 4 >= 4
COND3(true, 0, s(s(y_1)), x1) → COND1(or(false, gr(s(s(y_1)), x1)), 0, s(y_1), x1)
The graph contains the following edges 2 >= 2, 3 > 3, 4 >= 4
COND3(true, 0, s(s(y_0)), 0) → COND1(true, 0, s(y_0), 0)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 2, 3 > 3, 2 >= 4, 4 >= 4
COND2(false, 0, s(x0), s(y_1)) → COND3(true, 0, s(x0), s(y_1))
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4
COND2(false, 0, s(s(y_0)), x1) → COND3(true, 0, s(s(y_0)), x1)
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 4
COND2(false, 0, s(x0), 0) → COND3(true, 0, s(x0), 0)
The graph contains the following edges 2 >= 2, 4 >= 2, 3 >= 3, 2 >= 4, 4 >= 4

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

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

COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND1(true, s(x0), y1, y2) → COND2(true, s(x0), y1, y2) we obtained the following new rules:

COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

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

COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(true, s(x0), y1, s(x1)) → COND1(or(gr(x0, x1), gr(y1, s(x1))), x0, y1, s(x1))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

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

COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), gr(0, s(x2))), s(y_1), 0, s(x2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(s(y_2), s(x2))), s(y_1), s(y_2), s(x2))
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

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

COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), gr(0, s(x2))), s(y_1), 0, s(x2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(s(y_2), s(x2))), s(y_1), s(y_2), s(x2))
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), gr(0, s(x2))), s(y_1), 0, s(x2)) at position [0,1] we obtained the following new rules:

COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

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

COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(s(y_2), s(x2))), s(y_1), s(y_2), s(x2))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(s(y_2), s(x2))), s(y_1), s(y_2), s(x2)) at position [0,1] we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

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

COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))
COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND2(true, s(x0), y1, 0) → COND1(true, x0, y1, 0) we obtained the following new rules:

COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND2(true, s(x0), 0, y1) → COND1(or(gr(s(x0), y1), false), x0, 0, y1) we obtained the following new rules:

COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND2(true, s(s(y_1)), 0, s(y_3)) → COND1(or(gr(s(s(y_1)), s(y_3)), false), s(y_1), 0, s(y_3))
COND2(true, s(s(y_1)), 0, 0) → COND1(or(gr(s(s(y_1)), 0), false), s(y_1), 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(s(y_1)), 0, s(y_3)) → COND1(or(gr(s(s(y_1)), s(y_3)), false), s(y_1), 0, s(y_3))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(s(y_1)), 0, 0) → COND1(or(gr(s(s(y_1)), 0), false), s(y_1), 0, 0)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(true, s(s(y_1)), 0, s(y_3)) → COND1(or(gr(s(s(y_1)), s(y_3)), false), s(y_1), 0, s(y_3)) at position [0,0] we obtained the following new rules:

COND2(true, s(s(y_1)), 0, s(y_3)) → COND1(or(gr(s(y_1), y_3), false), s(y_1), 0, s(y_3))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(s(y_1)), 0, 0) → COND1(or(gr(s(s(y_1)), 0), false), s(y_1), 0, 0)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(true, s(s(y_1)), 0, 0) → COND1(or(gr(s(s(y_1)), 0), false), s(y_1), 0, 0) at position [0,0] we obtained the following new rules:

COND2(true, s(s(y_1)), 0, 0) → COND1(or(true, false), s(y_1), 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ Rewriting

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND2(true, s(s(y_1)), 0, 0) → COND1(or(true, false), s(y_1), 0, 0)
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(true, s(s(y_1)), 0, 0) → COND1(or(true, false), s(y_1), 0, 0) at position [0] we obtained the following new rules:

COND2(true, s(s(y_1)), 0, 0) → COND1(true, s(y_1), 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ ForwardInstantiation

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND2(true, s(x0), s(y1), s(y2)) → COND1(or(gr(x0, y2), gr(y1, y2)), x0, s(y1), s(y2)) we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND2(true, s(x0), s(y1), 0) → COND1(true, x0, s(y1), 0) we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND1(true, s(x0), 0, z1) → COND2(true, s(x0), 0, z1) we obtained the following new rules:

COND1(true, s(s(y_0)), 0, s(y_2)) → COND2(true, s(s(y_0)), 0, s(y_2))
COND1(true, s(s(y_0)), 0, x1) → COND2(true, s(s(y_0)), 0, x1)
COND1(true, s(s(y_0)), 0, 0) → COND2(true, s(s(y_0)), 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND1(true, s(s(y_0)), 0, x1) → COND2(true, s(s(y_0)), 0, x1)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0)
COND1(true, s(s(y_0)), 0, s(y_2)) → COND2(true, s(s(y_0)), 0, s(y_2))
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(s(y_0)), 0, 0) → COND2(true, s(s(y_0)), 0, 0)
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND1(true, s(x0), z1, 0) → COND2(true, s(x0), z1, 0) we obtained the following new rules:

COND1(true, s(s(y_0)), s(y_1), 0) → COND2(true, s(s(y_0)), s(y_1), 0)
COND1(true, s(s(y_0)), x1, 0) → COND2(true, s(s(y_0)), x1, 0)
COND1(true, s(s(y_0)), 0, 0) → COND2(true, s(s(y_0)), 0, 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND1(true, s(s(y_0)), x1, 0) → COND2(true, s(s(y_0)), x1, 0)
COND1(true, s(s(y_0)), 0, x1) → COND2(true, s(s(y_0)), 0, x1)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND1(true, s(s(y_0)), s(y_1), 0) → COND2(true, s(s(y_0)), s(y_1), 0)
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2))
COND1(true, s(s(y_0)), 0, s(y_2)) → COND2(true, s(s(y_0)), 0, s(y_2))
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))
COND1(true, s(s(y_0)), 0, 0) → COND2(true, s(s(y_0)), 0, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND1(true, s(x0), z1, s(z2)) → COND2(true, s(x0), z1, s(z2)) we obtained the following new rules:

COND1(true, s(s(y_0)), x1, s(x2)) → COND2(true, s(s(y_0)), x1, s(x2))
COND1(true, s(s(y_0)), s(y_1), s(x2)) → COND2(true, s(s(y_0)), s(y_1), s(x2))
COND1(true, s(s(y_0)), 0, s(x2)) → COND2(true, s(s(y_0)), 0, s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

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

COND1(true, s(s(y_0)), s(y_1), s(x2)) → COND2(true, s(s(y_0)), s(y_1), s(x2))
COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND1(true, s(s(y_0)), 0, x1) → COND2(true, s(s(y_0)), 0, x1)
COND1(true, s(s(y_0)), x1, 0) → COND2(true, s(s(y_0)), x1, 0)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND1(true, s(s(y_0)), x1, s(x2)) → COND2(true, s(s(y_0)), x1, s(x2))
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND1(true, s(s(y_0)), 0, s(y_2)) → COND2(true, s(s(y_0)), 0, s(y_2))
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(s(y_0)), s(y_1), 0) → COND2(true, s(s(y_0)), s(y_1), 0)
COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0)
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(s(y_0)), 0, 0) → COND2(true, s(s(y_0)), 0, 0)
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND1(true, s(x0), s(z1), 0) → COND2(true, s(x0), s(z1), 0) we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

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

COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND1(true, s(s(y_0)), s(y_1), s(x2)) → COND2(true, s(s(y_0)), s(y_1), s(x2))
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND1(true, s(s(y_0)), x1, 0) → COND2(true, s(s(y_0)), x1, 0)
COND1(true, s(s(y_0)), 0, x1) → COND2(true, s(s(y_0)), 0, x1)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND1(true, s(s(y_0)), x1, s(x2)) → COND2(true, s(s(y_0)), x1, s(x2))
COND1(true, s(s(y_0)), s(y_1), 0) → COND2(true, s(s(y_0)), s(y_1), 0)
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(s(y_0)), 0, s(y_2)) → COND2(true, s(s(y_0)), 0, s(y_2))
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2))
COND1(true, s(s(y_0)), 0, 0) → COND2(true, s(s(y_0)), 0, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule COND1(true, s(x0), s(z1), s(z2)) → COND2(true, s(x0), s(z1), s(z2)) we obtained the following new rules:

COND1(true, s(s(y_0)), s(x1), s(x2)) → COND2(true, s(s(y_0)), s(x1), s(x2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ Rewriting

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ Rewriting

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ QDPSizeChangeProof

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

COND1(true, s(s(y_0)), s(y_1), s(x2)) → COND2(true, s(s(y_0)), s(y_1), s(x2))
COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
COND2(true, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
COND1(true, s(s(y_0)), 0, x1) → COND2(true, s(s(y_0)), 0, x1)
COND1(true, s(s(y_0)), x1, 0) → COND2(true, s(s(y_0)), x1, 0)
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
COND1(true, s(s(y_0)), x1, s(x2)) → COND2(true, s(s(y_0)), x1, s(x2))
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
COND1(true, s(s(y_0)), 0, s(y_2)) → COND2(true, s(s(y_0)), 0, s(y_2))
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
COND1(true, s(s(y_0)), s(y_1), 0) → COND2(true, s(s(y_0)), s(y_1), 0)
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(y_2, x2)), s(y_1), s(y_2), s(x2))
COND1(true, s(s(y_0)), 0, 0) → COND2(true, s(s(y_0)), 0, 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
or(false, false)
or(true, x0)
or(x0, true)

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(s(y_0)), s(y_1), s(x2)) → COND2(true, s(s(y_0)), s(y_1), s(x2))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
COND1(true, s(s(y_0)), x1, s(x2)) → COND2(true, s(s(y_0)), x1, s(x2))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
COND1(true, s(s(y_0)), 0, x1) → COND2(true, s(s(y_0)), 0, x1)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
COND1(true, s(s(y_0)), 0, s(y_2)) → COND2(true, s(s(y_0)), 0, s(y_2))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
COND1(true, s(s(y_0)), x1, 0) → COND2(true, s(s(y_0)), x1, 0)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
COND1(true, s(s(y_0)), 0, 0) → COND2(true, s(s(y_0)), 0, 0)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 3, 3 >= 4, 4 >= 4
COND1(true, s(s(y_0)), s(y_1), 0) → COND2(true, s(s(y_0)), s(y_1), 0)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
COND2(true, s(s(y_1)), 0, x1) → COND1(or(gr(s(s(y_1)), x1), false), s(y_1), 0, x1)
The graph contains the following edges 2 > 2, 3 >= 3, 4 >= 4
COND2(true, s(s(y_0)), x1, 0) → COND1(true, s(y_0), x1, 0)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
COND2(true, s(s(y_0)), 0, 0) → COND1(true, s(y_0), 0, 0)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 3, 3 >= 4, 4 >= 4
COND2(true, s(s(y_0)), s(y_1), 0) → COND1(true, s(y_0), s(y_1), 0)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
COND2(true, s(s(y_1)), s(y_2), s(x2)) → COND1(or(gr(s(y_1), x2), gr(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, s(s(y_1)), x1, s(x2)) → COND1(or(gr(s(y_1), x2), gr(x1, s(x2))), s(y_1), x1, s(x2))
The graph contains the following edges 2 > 2, 3 >= 3, 4 >= 4
COND2(true, s(s(y_1)), 0, s(x2)) → COND1(or(gr(s(y_1), x2), false), s(y_1), 0, s(x2))
The graph contains the following edges 2 > 2, 3 >= 3, 4 >= 4