Termination w.r.t. Q proof of /tmp/tmpS-oyKr/21.trs

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

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

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

Q is empty.

↳ QTRS

  ↳ AAECC Innermost

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

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

Q is empty.

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

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

The TRS R 2 is

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

The signature Sigma is {cond1, cond2}

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

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

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

The set Q consists of the following terms:

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

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(false, x, y, z) → COND1(and(eq(x, y), gr(x, z)), x, y, z)
COND1(true, x, y, z) → GR(y, z)
COND2(false, x, y, z) → EQ(x, y)
COND2(true, x, y, z) → P(y)
COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, x, y, z) → AND(eq(x, y), gr(x, z))
COND2(true, x, y, z) → GR(y, z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
EQ(s(x), s(y)) → EQ(x, y)
COND2(false, x, y, z) → GR(x, 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(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), p(x), p(y), z)
cond2(false, x, y, z) → cond1(and(eq(x, y), gr(x, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
eq(0, 0) → true
eq(s(x), 0) → false
eq(0, s(x)) → false
eq(s(x), s(y)) → eq(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

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

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(false, x, y, z) → COND1(and(eq(x, y), gr(x, z)), x, y, z)
COND1(true, x, y, z) → GR(y, z)
COND2(false, x, y, z) → EQ(x, y)
COND2(true, x, y, z) → P(y)
COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, x, y, z) → AND(eq(x, y), gr(x, z))
COND2(true, x, y, z) → GR(y, z)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
EQ(s(x), s(y)) → EQ(x, y)
COND2(false, x, y, z) → GR(x, 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(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), p(x), p(y), z)
cond2(false, x, y, z) → cond1(and(eq(x, y), gr(x, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
eq(0, 0) → true
eq(s(x), 0) → false
eq(0, s(x)) → false
eq(s(x), s(y)) → eq(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 7 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

EQ(s(x), s(y)) → EQ(x, y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

EQ(s(x), s(y)) → EQ(x, y)

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

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

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

EQ(s(x), s(y)) → EQ(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:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

GR(s(x), s(y)) → GR(x, y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

GR(s(x), s(y)) → GR(x, y)

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

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

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ 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)

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

              ↳ QDP

                ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

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

COND2(false, 0, s(x0), y2) → COND1(and(false, gr(0, y2)), 0, s(x0), y2)
COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, s(x0), y1, 0) → COND1(and(eq(s(x0), y1), true), s(x0), y1, 0)
COND2(false, s(x0), 0, y2) → COND1(and(false, gr(s(x0), y2)), s(x0), 0, y2)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND2(false, 0, y1, x0) → COND1(and(eq(0, y1), false), 0, y1, x0)
COND2(false, 0, 0, y2) → COND1(and(true, gr(0, y2)), 0, 0, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

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

COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, s(x0), y1, 0) → COND1(and(eq(s(x0), y1), true), s(x0), y1, 0)
COND2(false, s(x0), 0, y2) → COND1(and(false, gr(s(x0), y2)), s(x0), 0, y2)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND2(false, 0, y1, x0) → COND1(and(eq(0, y1), false), 0, y1, x0)
COND2(false, 0, 0, y2) → COND1(and(true, gr(0, y2)), 0, 0, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

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

COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, s(x0), 0, y2) → COND1(false, s(x0), 0, y2)
COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, s(x0), y1, 0) → COND1(and(eq(s(x0), y1), true), s(x0), y1, 0)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND2(false, 0, y1, x0) → COND1(and(eq(0, y1), false), 0, y1, x0)
COND2(false, 0, 0, y2) → COND1(and(true, gr(0, y2)), 0, 0, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

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

COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, s(x0), y1, 0) → COND1(and(eq(s(x0), y1), true), s(x0), y1, 0)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND2(false, 0, y1, x0) → COND1(and(eq(0, y1), false), 0, y1, x0)
COND2(false, 0, 0, y2) → COND1(and(true, gr(0, y2)), 0, 0, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

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

COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, s(x0), y1, 0) → COND1(and(eq(s(x0), y1), true), s(x0), y1, 0)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND2(false, 0, y1, x0) → COND1(false, 0, y1, x0)
COND2(false, 0, 0, y2) → COND1(and(true, gr(0, y2)), 0, 0, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

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

COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, s(x0), y1, 0) → COND1(and(eq(s(x0), y1), true), s(x0), y1, 0)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND2(false, 0, 0, y2) → COND1(and(true, gr(0, y2)), 0, 0, y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(false, 0, 0, y2) → COND1(and(true, gr(0, y2)), 0, 0, y2) at position [0,1] we obtained the following new rules:

COND2(false, 0, 0, y2) → COND1(and(true, false), 0, 0, y2)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

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

COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, s(x0), y1, 0) → COND1(and(eq(s(x0), y1), true), s(x0), y1, 0)
COND2(false, 0, 0, y2) → COND1(and(true, false), 0, 0, y2)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

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

COND2(true, x, y, z) → COND2(gr(y, z), p(x), p(y), z)
COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2)
COND1(true, x, y, z) → COND2(gr(y, z), x, y, z)
COND2(false, s(x0), y1, 0) → COND1(and(eq(s(x0), y1), true), s(x0), y1, 0)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

COND2(false, s(x0), 0, 0) → COND1(and(false, true), s(x0), 0, 0)
COND2(false, s(x0), s(x1), 0) → COND1(and(eq(x0, x1), true), s(x0), s(x1), 0)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ 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) → COND2(true, p(y0), x0, 0)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2)
COND2(true, s(x0), 0, y1) → COND2(false, x0, 0, y1)
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(true, 0, 0, y1) → COND2(false, 0, 0, y1)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND1(true, y0, s(x0), 0) → COND2(true, y0, s(x0), 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND2(false, s(x0), s(x1), y2) → COND1(and(eq(x0, x1), gr(s(x0), y2)), s(x0), s(x1), y2) we obtained the following new rules:

COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(s(x0), s(z2))), s(x0), s(z1), s(z2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

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

COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(s(x0), s(z2))), s(x0), s(z1), s(z2))
COND2(true, y0, s(x0), 0) → COND2(true, p(y0), x0, 0)
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(true, s(x0), 0, y1) → COND2(false, x0, 0, y1)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND1(true, y0, s(x0), 0) → COND2(true, y0, s(x0), 0)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                    ↳ QDP

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

COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(s(x0), s(z2))), s(x0), s(z1), s(z2))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(true, s(x0), 0, y1) → COND2(false, x0, 0, y1)
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(s(x0), s(z2))), s(x0), s(z1), s(z2)) at position [0,1] we obtained the following new rules:

COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                    ↳ QDP

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

COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))
COND1(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), y0, s(x0), s(x1))
COND2(true, s(x0), 0, y1) → COND2(false, x0, 0, y1)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))
COND2(true, s(x0), 0, y1) → COND2(false, x0, 0, y1)
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

COND2(true, s(x0), 0, s(z2)) → COND2(false, x0, 0, s(z2))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND1(true, y0, 0, x0) → COND2(false, y0, 0, x0)
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND2(true, s(x0), 0, s(z2)) → COND2(false, x0, 0, s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND1(true, s(z0), 0, s(z2)) → COND2(false, s(z0), 0, s(z2))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))
COND2(false, s(x0), y1, s(x1)) → COND1(and(eq(s(x0), y1), gr(x0, x1)), s(x0), y1, s(x1))
COND2(true, s(x0), 0, s(z2)) → COND2(false, x0, 0, s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND1(true, s(z0), 0, s(z2)) → COND2(false, s(z0), 0, s(z2))
COND2(false, s(x0), s(y_2), s(x2)) → COND1(and(eq(s(x0), s(y_2)), gr(x0, x2)), s(x0), s(y_2), s(x2))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))
COND2(false, s(x0), 0, s(x2)) → COND1(and(eq(s(x0), 0), gr(x0, x2)), s(x0), 0, s(x2))
COND2(true, s(x0), 0, s(z2)) → COND2(false, x0, 0, s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(false, s(x0), 0, s(x2)) → COND1(and(eq(s(x0), 0), gr(x0, x2)), s(x0), 0, s(x2)) at position [0,0] we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ UsableRulesProof

                                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QReductionProof

                                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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.

eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(false, s(x0), 0, s(x2)) → COND1(and(false, gr(x0, x2)), s(x0), 0, s(x2)) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND2(false, s(x0), s(y_2), s(x2)) → COND1(and(eq(s(x0), s(y_2)), gr(x0, x2)), s(x0), s(y_2), s(x2))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND2(false, s(x0), s(y_2), s(x2)) → COND1(and(eq(s(x0), s(y_2)), gr(x0, x2)), s(x0), s(y_2), s(x2)) at position [0,0] we obtained the following new rules:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(true, y0, s(x0), s(x1)) → COND2(gr(x0, x1), p(y0), x0, s(x1))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(COND1(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(COND2(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(and(x₁, x₂)) = 0
POL(eq(x₁, x₂)) = 0
POL(false) = 1
POL(gr(x₁, x₂)) = 1
POL(p(x₁)) = x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 1

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ UsableRulesProof

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QReductionProof

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))

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)
eq(0, 0) → true
eq(s(x), 0) → false
eq(0, s(x)) → false
eq(s(x), s(y)) → eq(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QReductionProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ NonInfProof

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))

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)
eq(0, 0) → true
eq(s(x), 0) → false
eq(0, s(x)) → false
eq(s(x), s(y)) → eq(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [18] with the following steps:
Note that final constraints are written in bold face.

For Pair COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2)) the following chains were created:

We consider the chain COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2)), COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2)) which results in the following constraint:
(1)    (COND2(gr(x4, x5), s(x3), s(x4), s(x5))=COND2(false, s(x6), s(x7), s(x8)) ⇒ COND1(true, s(x3), s(x4), s(x5))≥COND2(gr(x4, x5), s(x3), s(x4), s(x5)))

We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
(2)    (gr(x4, x5)=false ⇒ COND1(true, s(x3), s(x4), s(x5))≥COND2(gr(x4, x5), s(x3), s(x4), s(x5)))

We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on gr(x4, x5)=false which results in the following new constraints:
(3)    (false=false ⇒ COND1(true, s(x3), s(0), s(x18))≥COND2(gr(0, x18), s(x3), s(0), s(x18)))

(4)    (gr(x20, x21)=false∧(∀x22:gr(x20, x21)=false ⇒ COND1(true, s(x22), s(x20), s(x21))≥COND2(gr(x20, x21), s(x22), s(x20), s(x21))) ⇒ COND1(true, s(x3), s(s(x20)), s(s(x21)))≥COND2(gr(s(x20), s(x21)), s(x3), s(s(x20)), s(s(x21))))

We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
(5)    (COND1(true, s(x3), s(0), s(x18))≥COND2(gr(0, x18), s(x3), s(0), s(x18)))

We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (∀x22:gr(x20, x21)=false ⇒ COND1(true, s(x22), s(x20), s(x21))≥COND2(gr(x20, x21), s(x22), s(x20), s(x21))) with σ = [x22 / x3] which results in the following new constraint:
(6)    (COND1(true, s(x3), s(x20), s(x21))≥COND2(gr(x20, x21), s(x3), s(x20), s(x21)) ⇒ COND1(true, s(x3), s(s(x20)), s(s(x21)))≥COND2(gr(s(x20), s(x21)), s(x3), s(s(x20)), s(s(x21))))

For Pair COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2)) the following chains were created:

We consider the chain COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2)), COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2)) which results in the following constraint:
(7)    (COND1(and(eq(x9, x10), gr(x9, x11)), s(x9), s(x10), s(x11))=COND1(true, s(x12), s(x13), s(x14)) ⇒ COND2(false, s(x9), s(x10), s(x11))≥COND1(and(eq(x9, x10), gr(x9, x11)), s(x9), s(x10), s(x11)))

We simplified constraint (7) using rules (I), (II), (IV), (VII) which results in the following new constraint:
(8)    (eq(x9, x10)=x23∧gr(x9, x11)=x24∧and(x23, x24)=true ⇒ COND2(false, s(x9), s(x10), s(x11))≥COND1(and(eq(x9, x10), gr(x9, x11)), s(x9), s(x10), s(x11)))

We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on and(x23, x24)=true which results in the following new constraint:
(9)    (true=true∧eq(x9, x10)=true∧gr(x9, x11)=true ⇒ COND2(false, s(x9), s(x10), s(x11))≥COND1(and(eq(x9, x10), gr(x9, x11)), s(x9), s(x10), s(x11)))

We simplified constraint (9) using rules (I), (II) which results in the following new constraint:
(10)    (eq(x9, x10)=true∧gr(x9, x11)=true ⇒ COND2(false, s(x9), s(x10), s(x11))≥COND1(and(eq(x9, x10), gr(x9, x11)), s(x9), s(x10), s(x11)))

We simplified constraint (10) using rule (V) (with possible (I) afterwards) using induction on eq(x9, x10)=true which results in the following new constraints:
(11)    (true=true∧gr(0, x11)=true ⇒ COND2(false, s(0), s(0), s(x11))≥COND1(and(eq(0, 0), gr(0, x11)), s(0), s(0), s(x11)))

(12)    (eq(x29, x30)=true∧gr(s(x29), x11)=true∧(∀x31:eq(x29, x30)=true∧gr(x29, x31)=true ⇒ COND2(false, s(x29), s(x30), s(x31))≥COND1(and(eq(x29, x30), gr(x29, x31)), s(x29), s(x30), s(x31))) ⇒ COND2(false, s(s(x29)), s(s(x30)), s(x11))≥COND1(and(eq(s(x29), s(x30)), gr(s(x29), x11)), s(s(x29)), s(s(x30)), s(x11)))

We simplified constraint (11) using rules (I), (II), (VII) which results in the following new constraint:
(13)    (0=x32∧gr(x32, x11)=true ⇒ COND2(false, s(0), s(0), s(x11))≥COND1(and(eq(0, 0), gr(0, x11)), s(0), s(0), s(x11)))

We simplified constraint (12) using rule (VII) which results in the following new constraint:
(14)    (eq(x29, x30)=true∧s(x29)=x37∧gr(x37, x11)=true∧(∀x31:eq(x29, x30)=true∧gr(x29, x31)=true ⇒ COND2(false, s(x29), s(x30), s(x31))≥COND1(and(eq(x29, x30), gr(x29, x31)), s(x29), s(x30), s(x31))) ⇒ COND2(false, s(s(x29)), s(s(x30)), s(x11))≥COND1(and(eq(s(x29), s(x30)), gr(s(x29), x11)), s(s(x29)), s(s(x30)), s(x11)))

We simplified constraint (13) using rule (V) (with possible (I) afterwards) using induction on gr(x32, x11)=true which results in the following new constraints:
(15)    (true=true∧0=s(x34) ⇒ COND2(false, s(0), s(0), s(0))≥COND1(and(eq(0, 0), gr(0, 0)), s(0), s(0), s(0)))

(16)    (gr(x35, x36)=true∧0=s(x35)∧(gr(x35, x36)=true∧0=x35 ⇒ COND2(false, s(0), s(0), s(x36))≥COND1(and(eq(0, 0), gr(0, x36)), s(0), s(0), s(x36))) ⇒ COND2(false, s(0), s(0), s(s(x36)))≥COND1(and(eq(0, 0), gr(0, s(x36))), s(0), s(0), s(s(x36))))

We solved constraint (15) using rules (I), (II).We solved constraint (16) using rules (I), (II).We simplified constraint (14) using rule (V) (with possible (I) afterwards) using induction on gr(x37, x11)=true which results in the following new constraints:
(17)    (true=true∧eq(x29, x30)=true∧s(x29)=s(x39)∧(∀x31:eq(x29, x30)=true∧gr(x29, x31)=true ⇒ COND2(false, s(x29), s(x30), s(x31))≥COND1(and(eq(x29, x30), gr(x29, x31)), s(x29), s(x30), s(x31))) ⇒ COND2(false, s(s(x29)), s(s(x30)), s(0))≥COND1(and(eq(s(x29), s(x30)), gr(s(x29), 0)), s(s(x29)), s(s(x30)), s(0)))

(18)    (gr(x40, x41)=true∧eq(x29, x30)=true∧s(x29)=s(x40)∧(∀x31:eq(x29, x30)=true∧gr(x29, x31)=true ⇒ COND2(false, s(x29), s(x30), s(x31))≥COND1(and(eq(x29, x30), gr(x29, x31)), s(x29), s(x30), s(x31)))∧(∀x42,x43,x44:gr(x40, x41)=true∧eq(x42, x43)=true∧s(x42)=x40∧(∀x44:eq(x42, x43)=true∧gr(x42, x44)=true ⇒ COND2(false, s(x42), s(x43), s(x44))≥COND1(and(eq(x42, x43), gr(x42, x44)), s(x42), s(x43), s(x44))) ⇒ COND2(false, s(s(x42)), s(s(x43)), s(x41))≥COND1(and(eq(s(x42), s(x43)), gr(s(x42), x41)), s(s(x42)), s(s(x43)), s(x41))) ⇒ COND2(false, s(s(x29)), s(s(x30)), s(s(x41)))≥COND1(and(eq(s(x29), s(x30)), gr(s(x29), s(x41))), s(s(x29)), s(s(x30)), s(s(x41))))

We simplified constraint (17) using rules (I), (II), (IV) which results in the following new constraint:
(19)    (eq(x29, x30)=true ⇒ COND2(false, s(s(x29)), s(s(x30)), s(0))≥COND1(and(eq(s(x29), s(x30)), gr(s(x29), 0)), s(s(x29)), s(s(x30)), s(0)))

We simplified constraint (18) using rules (I), (II), (III) which results in the following new constraint:
(20)    (gr(x40, x41)=true∧eq(x40, x30)=true∧(∀x31:eq(x40, x30)=true∧gr(x40, x31)=true ⇒ COND2(false, s(x40), s(x30), s(x31))≥COND1(and(eq(x40, x30), gr(x40, x31)), s(x40), s(x30), s(x31)))∧(∀x42,x43,x44:gr(x40, x41)=true∧eq(x42, x43)=true∧s(x42)=x40∧(∀x44:eq(x42, x43)=true∧gr(x42, x44)=true ⇒ COND2(false, s(x42), s(x43), s(x44))≥COND1(and(eq(x42, x43), gr(x42, x44)), s(x42), s(x43), s(x44))) ⇒ COND2(false, s(s(x42)), s(s(x43)), s(x41))≥COND1(and(eq(s(x42), s(x43)), gr(s(x42), x41)), s(s(x42)), s(s(x43)), s(x41))) ⇒ COND2(false, s(s(x40)), s(s(x30)), s(s(x41)))≥COND1(and(eq(s(x40), s(x30)), gr(s(x40), s(x41))), s(s(x40)), s(s(x30)), s(s(x41))))

We simplified constraint (19) using rule (V) (with possible (I) afterwards) using induction on eq(x29, x30)=true which results in the following new constraints:
(21)    (true=true ⇒ COND2(false, s(s(0)), s(s(0)), s(0))≥COND1(and(eq(s(0), s(0)), gr(s(0), 0)), s(s(0)), s(s(0)), s(0)))

(22)    (eq(x47, x48)=true∧(eq(x47, x48)=true ⇒ COND2(false, s(s(x47)), s(s(x48)), s(0))≥COND1(and(eq(s(x47), s(x48)), gr(s(x47), 0)), s(s(x47)), s(s(x48)), s(0))) ⇒ COND2(false, s(s(s(x47))), s(s(s(x48))), s(0))≥COND1(and(eq(s(s(x47)), s(s(x48))), gr(s(s(x47)), 0)), s(s(s(x47))), s(s(s(x48))), s(0)))

We simplified constraint (21) using rules (I), (II) which results in the following new constraint:
(23)    (COND2(false, s(s(0)), s(s(0)), s(0))≥COND1(and(eq(s(0), s(0)), gr(s(0), 0)), s(s(0)), s(s(0)), s(0)))

We simplified constraint (22) using rule (VI) where we applied the induction hypothesis (eq(x47, x48)=true ⇒ COND2(false, s(s(x47)), s(s(x48)), s(0))≥COND1(and(eq(s(x47), s(x48)), gr(s(x47), 0)), s(s(x47)), s(s(x48)), s(0))) with σ = [ ] which results in the following new constraint:
(24)    (COND2(false, s(s(x47)), s(s(x48)), s(0))≥COND1(and(eq(s(x47), s(x48)), gr(s(x47), 0)), s(s(x47)), s(s(x48)), s(0)) ⇒ COND2(false, s(s(s(x47))), s(s(s(x48))), s(0))≥COND1(and(eq(s(s(x47)), s(s(x48))), gr(s(s(x47)), 0)), s(s(s(x47))), s(s(s(x48))), s(0)))

We simplified constraint (20) using rule (VI) where we applied the induction hypothesis (∀x31:eq(x40, x30)=true∧gr(x40, x31)=true ⇒ COND2(false, s(x40), s(x30), s(x31))≥COND1(and(eq(x40, x30), gr(x40, x31)), s(x40), s(x30), s(x31))) with σ = [x31 / x41] which results in the following new constraint:
(25)    (COND2(false, s(x40), s(x30), s(x41))≥COND1(and(eq(x40, x30), gr(x40, x41)), s(x40), s(x30), s(x41))∧(∀x42,x43,x44:gr(x40, x41)=true∧eq(x42, x43)=true∧s(x42)=x40∧(∀x44:eq(x42, x43)=true∧gr(x42, x44)=true ⇒ COND2(false, s(x42), s(x43), s(x44))≥COND1(and(eq(x42, x43), gr(x42, x44)), s(x42), s(x43), s(x44))) ⇒ COND2(false, s(s(x42)), s(s(x43)), s(x41))≥COND1(and(eq(s(x42), s(x43)), gr(s(x42), x41)), s(s(x42)), s(s(x43)), s(x41))) ⇒ COND2(false, s(s(x40)), s(s(x30)), s(s(x41)))≥COND1(and(eq(s(x40), s(x30)), gr(s(x40), s(x41))), s(s(x40)), s(s(x30)), s(s(x41))))

We simplified constraint (25) using rule (IV) which results in the following new constraint:
(26)    (COND2(false, s(x40), s(x30), s(x41))≥COND1(and(eq(x40, x30), gr(x40, x41)), s(x40), s(x30), s(x41)) ⇒ COND2(false, s(s(x40)), s(s(x30)), s(s(x41)))≥COND1(and(eq(s(x40), s(x30)), gr(s(x40), s(x41))), s(s(x40)), s(s(x30)), s(s(x41))))

To summarize, we get the following constraints P_≥ for the following pairs.

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
- (COND1(true, s(x3), s(0), s(x18))≥COND2(gr(0, x18), s(x3), s(0), s(x18)))
- (COND1(true, s(x3), s(x20), s(x21))≥COND2(gr(x20, x21), s(x3), s(x20), s(x21)) ⇒ COND1(true, s(x3), s(s(x20)), s(s(x21)))≥COND2(gr(s(x20), s(x21)), s(x3), s(s(x20)), s(s(x21))))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))
- (COND2(false, s(s(0)), s(s(0)), s(0))≥COND1(and(eq(s(0), s(0)), gr(s(0), 0)), s(s(0)), s(s(0)), s(0)))
- (COND2(false, s(s(x47)), s(s(x48)), s(0))≥COND1(and(eq(s(x47), s(x48)), gr(s(x47), 0)), s(s(x47)), s(s(x48)), s(0)) ⇒ COND2(false, s(s(s(x47))), s(s(s(x48))), s(0))≥COND1(and(eq(s(s(x47)), s(s(x48))), gr(s(s(x47)), 0)), s(s(s(x47))), s(s(s(x48))), s(0)))
- (COND2(false, s(x40), s(x30), s(x41))≥COND1(and(eq(x40, x30), gr(x40, x41)), s(x40), s(x30), s(x41)) ⇒ COND2(false, s(s(x40)), s(s(x30)), s(s(x41)))≥COND1(and(eq(s(x40), s(x30)), gr(s(x40), s(x41))), s(s(x40)), s(s(x30)), s(s(x41))))

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [18]:

POL(0) = 0
POL(COND1(x₁, x₂, x₃, x₄)) = -1 + x₁ + x₂
POL(COND2(x₁, x₂, x₃, x₄)) = -1 - x₁ + x₂ + x₃ - x₄
POL(and(x₁, x₂)) = 1
POL(c) = -1
POL(eq(x₁, x₂)) = 0
POL(false) = 0
POL(gr(x₁, x₂)) = 0
POL(s(x₁)) = 2 + x₁
POL(true) = 0

The following pairs are in P_>:

COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))

The following pairs are in P_bound:

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))
COND2(false, s(x0), s(z1), s(z2)) → COND1(and(eq(x0, z1), gr(x0, z2)), s(x0), s(z1), s(z2))

The following rules are usable:

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

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ Instantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ Instantiation

                                                                                                                ↳ QDP

                                                                                                                  ↳ Instantiation

                                                                                                                    ↳ QDP

                                                                                                                      ↳ ForwardInstantiation

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ AND

                                                                                                                              ↳ QDP

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QReductionProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ NonInfProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                    ↳ QDP

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

COND1(true, s(z0), s(x1), s(z2)) → COND2(gr(x1, z2), s(z0), s(x1), s(z2))

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)
eq(0, 0) → true
eq(s(x), 0) → false
eq(0, s(x)) → false
eq(s(x), s(y)) → eq(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

                                                                                                      ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

                                                                                                      ↳ UsableRulesProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QReductionProof

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

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

The TRS R consists of the following rules:

p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
p(0)
p(s(x0))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

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

gr(0, x0)
gr(s(x0), 0)
gr(s(x0), s(x1))
eq(0, 0)
eq(s(x0), 0)
eq(0, s(x0))
eq(s(x0), s(x1))
and(true, true)
and(false, x0)
and(x0, false)

↳ QTRS

  ↳ AAECC Innermost

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

                                                                                                      ↳ UsableRulesProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QReductionProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QDPSizeChangeProof

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

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

The TRS R consists of the following rules:

p(0) → 0
p(s(x)) → x

The set Q consists of the following terms:

p(0)
p(s(x0))

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

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

COND2(true, y0, s(x0), 0) → COND2(true, p(y0), x0, 0)
The graph contains the following edges 1 >= 1, 3 > 3, 4 >= 4