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:



↳ 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)

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
              ↳ 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)

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:



↳ 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)

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
              ↳ 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)

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
              ↳ 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)

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
              ↳ 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)

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
              ↳ 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:

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

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:

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

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:




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:




To summarize, we get the following constraints P for the following pairs.



The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [18]:

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

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:

falseand(false, x)
trueand(true, true)
falseand(x, 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:

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

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)

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
              ↳ 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: