Termination of the given ITRSProblem could not be shown:



↳ ITRS
  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

Cond_f(TRUE, x) → f(x)
f(x) → Cond_f(>@z(x, x), x)

The set Q consists of the following terms:

Cond_f(TRUE, x0)
f(x0)


Represented integers and predefined function symbols by Terms

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof

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

Cond_f(true, x) → f(x)
f(x) → Cond_f(greater_int(x, x), x)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))


Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

COND_F(true, x) → F(x)
F(x) → COND_F(greater_int(x, x), x)
F(x) → GREATER_INT(x, x)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(greater_int(x, x), x)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof

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

COND_F(true, x) → F(x)
F(x) → COND_F(greater_int(x, x), x)
F(x) → GREATER_INT(x, x)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(greater_int(x, x), x)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 1 less node.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP

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

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(greater_int(x, x), x)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
              ↳ QDP
              ↳ QDP

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

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

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

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ UsableRulesReductionPairsProof
              ↳ QDP
              ↳ QDP

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

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATER_INT(x1, x2)) = 2·x1 + x2   
POL(neg(x1)) = x1   
POL(s(x1)) = 2·x1   



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ UsableRulesReductionPairsProof
                          ↳ QDP
                            ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
              ↳ QDP

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

GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(greater_int(x, x), x)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
              ↳ QDP

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

GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))

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

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ UsableRulesReductionPairsProof
              ↳ QDP

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

GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATER_INT(x1, x2)) = 2·x1 + x2   
POL(pos(x1)) = x1   
POL(s(x1)) = 2·x1   



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ UsableRulesReductionPairsProof
                          ↳ QDP
                            ↳ PisEmptyProof
              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof

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

F(x) → COND_F(greater_int(x, x), x)
COND_F(true, x) → F(x)

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(greater_int(x, x), x)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof

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

F(x) → COND_F(greater_int(x, x), x)
COND_F(true, x) → F(x)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_f(true, x0)
f(x0)



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing

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

F(x) → COND_F(greater_int(x, x), x)
COND_F(true, x) → F(x)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule F(x) → COND_F(greater_int(x, x), x) at position [0] we obtained the following new rules [LPAR04]:

F(neg(0)) → COND_F(false, neg(0))
F(pos(0)) → COND_F(false, pos(0))
F(neg(s(x0))) → COND_F(greater_int(neg(x0), neg(x0)), neg(s(x0)))
F(pos(s(x0))) → COND_F(greater_int(pos(x0), pos(x0)), pos(s(x0)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ 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:

COND_F(true, x) → F(x)
F(neg(0)) → COND_F(false, neg(0))
F(pos(0)) → COND_F(false, pos(0))
F(neg(s(x0))) → COND_F(greater_int(neg(x0), neg(x0)), neg(s(x0)))
F(pos(s(x0))) → COND_F(greater_int(pos(x0), pos(x0)), pos(s(x0)))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation

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

F(neg(s(x0))) → COND_F(greater_int(neg(x0), neg(x0)), neg(s(x0)))
COND_F(true, x) → F(x)
F(pos(s(x0))) → COND_F(greater_int(pos(x0), pos(x0)), pos(s(x0)))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, x) → F(x) we obtained the following new rules [LPAR04]:

COND_F(true, pos(s(z0))) → F(pos(s(z0)))
COND_F(true, neg(s(z0))) → F(neg(s(z0)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof

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

F(neg(s(x0))) → COND_F(greater_int(neg(x0), neg(x0)), neg(s(x0)))
F(pos(s(x0))) → COND_F(greater_int(pos(x0), pos(x0)), pos(s(x0)))
COND_F(true, pos(s(z0))) → F(pos(s(z0)))
COND_F(true, neg(s(z0))) → F(neg(s(z0)))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                        ↳ QDP

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

COND_F(true, pos(s(z0))) → F(pos(s(z0)))
F(pos(s(x0))) → COND_F(greater_int(pos(x0), pos(x0)), pos(s(x0)))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                        ↳ QDP

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

COND_F(true, pos(s(z0))) → F(pos(s(z0)))
F(pos(s(x0))) → COND_F(greater_int(pos(x0), pos(x0)), pos(s(x0)))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].

greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                                ↳ QDP
                                                  ↳ MNOCProof
                                              ↳ RootLabelingFC2Proof
                                        ↳ QDP

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

COND_F(true, pos(s(z0))) → F(pos(s(z0)))
F(pos(s(x0))) → COND_F(greater_int(pos(x0), pos(x0)), pos(s(x0)))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), pos(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(pos(s(x0)), pos(s(x1)))

We have to consider all (P,Q,R)-chains.
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                                ↳ QDP
                                                  ↳ MNOCProof
                                                    ↳ QDP
                                              ↳ RootLabelingFC2Proof
                                        ↳ QDP

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

COND_F(true, pos(s(z0))) → F(pos(s(z0)))
F(pos(s(x0))) → COND_F(greater_int(pos(x0), pos(x0)), pos(s(x0)))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

Q is empty.
We have to consider all (P,Q,R)-chains.
We used root labeling (second transformation) [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
The resulting QDP is:
Q DP problem:
The TRS P consists of the following rules:

COND_F_{true,pos}(true, pos_{s}(s_{greater_int}(z0))) → F_{pos}(pos_{s}(s_{greater_int}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{pos}(z0))) → F_{pos}(pos_{s}(s_{pos}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{0}(z0))) → F_{pos}(pos_{s}(s_{0}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{false}(z0))) → F_{pos}(pos_{s}(s_{false}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{true}(z0))) → F_{pos}(pos_{s}(s_{true}(z0)))
F_{pos}(pos_{s}(s_{greater_int}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{greater_int}(x0), pos_{greater_int}(x0)), pos_{s}(s_{greater_int}(x0)))
F_{pos}(pos_{s}(s_{pos}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{pos}(x0), pos_{pos}(x0)), pos_{s}(s_{pos}(x0)))
F_{pos}(pos_{s}(s_{0}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(x0), pos_{0}(x0)), pos_{s}(s_{0}(x0)))
F_{pos}(pos_{s}(s_{false}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{false}(x0), pos_{false}(x0)), pos_{s}(s_{false}(x0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
F_{pos}(pos_{s}(s_{true}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{true}(x0), pos_{true}(x0)), pos_{s}(s_{true}(x0)))
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{pos,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{0,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{false,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{s,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{true,false}(flat0, false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → F_{false}(false)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{true,false}(flat0, false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → F_{false}(false)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → F_{true}(true)

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

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

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                        ↳ QDP

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

COND_F_{true,pos}(true, pos_{s}(s_{greater_int}(z0))) → F_{pos}(pos_{s}(s_{greater_int}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{pos}(z0))) → F_{pos}(pos_{s}(s_{pos}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{0}(z0))) → F_{pos}(pos_{s}(s_{0}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{false}(z0))) → F_{pos}(pos_{s}(s_{false}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{true}(z0))) → F_{pos}(pos_{s}(s_{true}(z0)))
F_{pos}(pos_{s}(s_{greater_int}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{greater_int}(x0), pos_{greater_int}(x0)), pos_{s}(s_{greater_int}(x0)))
F_{pos}(pos_{s}(s_{pos}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{pos}(x0), pos_{pos}(x0)), pos_{s}(s_{pos}(x0)))
F_{pos}(pos_{s}(s_{0}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(x0), pos_{0}(x0)), pos_{s}(s_{0}(x0)))
F_{pos}(pos_{s}(s_{false}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{false}(x0), pos_{false}(x0)), pos_{s}(s_{false}(x0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
F_{pos}(pos_{s}(s_{true}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{true}(x0), pos_{true}(x0)), pos_{s}(s_{true}(x0)))
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{pos,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{0,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{false,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{s,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → COND_F_{true,false}(flat0, false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → F_{false}(false)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,greater_int}(false, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,pos}(false, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,0}(false, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,false}(false, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,s}(false, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → COND_F_{false,true}(false, flat1)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{greater_int,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{pos,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{0,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{false,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{s,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → COND_F_{true,false}(flat0, false)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → COND_F_{true,false}(flat0, false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → F_{false}(false)
F_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → F_{false}(false)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,greater_int}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,0}(true, flat1)
COND_F_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,false}(true, flat1)
COND_F_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,s}(true, flat1)
COND_F_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,true}(true, flat1)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{greater_int,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{pos,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{0,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{false,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{s,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
COND_F_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → COND_F_{true,true}(flat0, true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → F_{true}(true)
F_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → F_{true}(true)

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 171 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                        ↳ QDP

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

F_{pos}(pos_{s}(s_{greater_int}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{greater_int}(x0), pos_{greater_int}(x0)), pos_{s}(s_{greater_int}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{true,pos}(true, pos_{s}(s_{greater_int}(z0))) → F_{pos}(pos_{s}(s_{greater_int}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F_{pos}(pos_{s}(s_{greater_int}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{greater_int}(x0), pos_{greater_int}(x0)), pos_{s}(s_{greater_int}(x0)))
The remaining pairs can at least be oriented weakly.

COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{true,pos}(true, pos_{s}(s_{greater_int}(z0))) → F_{pos}(pos_{s}(s_{greater_int}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_F_{greater_int,pos}(x1, x2)) = x1   
POL(COND_F_{true,pos}(x1, x2)) = 1 + x1   
POL(F_{pos}(x1)) = 1 + x1   
POL(false) = 0   
POL(greater_int_{pos,pos}(x1, x2)) = 1 + x2   
POL(pos_{0}(x1)) = 1   
POL(pos_{false}(x1)) = 0   
POL(pos_{greater_int}(x1)) = 0   
POL(pos_{pos}(x1)) = 0   
POL(pos_{s}(x1)) = 1   
POL(pos_{true}(x1)) = 0   
POL(s_{0}(x1)) = 0   
POL(s_{false}(x1)) = 1   
POL(s_{greater_int}(x1)) = x1   
POL(s_{pos}(x1)) = 0   
POL(s_{s}(x1)) = 0   
POL(s_{true}(x1)) = x1   
POL(true) = 1   

The following usable rules [FROCOS05] were oriented:

pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                        ↳ QDP

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

COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{true,pos}(true, pos_{s}(s_{greater_int}(z0))) → F_{pos}(pos_{s}(s_{greater_int}(z0)))
COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Instantiation
                                        ↳ QDP

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

COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1) we obtained the following new rules [LPAR04]:

COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Instantiation
                                                                ↳ QDP
                                                                  ↳ Instantiation
                                        ↳ QDP

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

COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1) we obtained the following new rules [LPAR04]:

COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Instantiation
                                                                ↳ QDP
                                                                  ↳ Instantiation
                                                                    ↳ QDP
                                                                      ↳ Instantiation
                                        ↳ QDP

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

COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1) we obtained the following new rules [LPAR04]:

COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Instantiation
                                                                ↳ QDP
                                                                  ↳ Instantiation
                                                                    ↳ QDP
                                                                      ↳ Instantiation
                                                                        ↳ QDP
                                                                          ↳ Instantiation
                                        ↳ QDP

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

COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1) we obtained the following new rules [LPAR04]:

COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Instantiation
                                                                ↳ QDP
                                                                  ↳ Instantiation
                                                                    ↳ QDP
                                                                      ↳ Instantiation
                                                                        ↳ QDP
                                                                          ↳ Instantiation
                                                                            ↳ QDP
                                                                              ↳ Instantiation
                                        ↳ QDP

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

COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1) we obtained the following new rules [LPAR04]:

COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Instantiation
                                                                ↳ QDP
                                                                  ↳ Instantiation
                                                                    ↳ QDP
                                                                      ↳ Instantiation
                                                                        ↳ QDP
                                                                          ↳ Instantiation
                                                                            ↳ QDP
                                                                              ↳ Instantiation
                                                                                ↳ QDP
                                                                                  ↳ Instantiation
                                        ↳ QDP

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

COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1)
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → COND_F_{true,pos}(true, flat1) we obtained the following new rules [LPAR04]:

COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP
                                              ↳ QReductionProof
                                              ↳ RootLabelingFC2Proof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Instantiation
                                                                ↳ QDP
                                                                  ↳ Instantiation
                                                                    ↳ QDP
                                                                      ↳ Instantiation
                                                                        ↳ QDP
                                                                          ↳ Instantiation
                                                                            ↳ QDP
                                                                              ↳ Instantiation
                                                                                ↳ QDP
                                                                                  ↳ Instantiation
                                                                                    ↳ QDP
                                        ↳ QDP

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

COND_F_{true,pos}(true, pos_{s}(s_{s}(z0))) → F_{pos}(pos_{s}(s_{s}(z0)))
F_{pos}(pos_{s}(s_{s}(x0))) → COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(x0), pos_{s}(x0)), pos_{s}(s_{s}(x0)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))
COND_F_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(x0)), pos_{0}(0)), pos_{s}(s_{s}(y_3))) → COND_F_{true,pos}(true, pos_{s}(s_{s}(y_3)))

The TRS R consists of the following rules:

greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{greater_int}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{greater_int}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{pos}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{pos}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{0}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{0}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{false}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{false}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{s}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{s}(x), pos_{true}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{greater_int}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{greater_int}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{pos}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{pos}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{0}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{0}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{false}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{false}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{s}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{s}(y))
greater_int_{pos,pos}(pos_{s}(s_{true}(x)), pos_{s}(s_{true}(y))) → greater_int_{pos,pos}(pos_{true}(x), pos_{true}(y))
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0)), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{0}(0))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,greater_int}(false, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,pos}(false, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,0}(false, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,false}(false, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,s}(false, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0))), flat1) → greater_int_{false,true}(false, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{greater_int,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{pos,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{0,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{false,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{s,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → greater_int_{true,false}(flat0, false)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → greater_int_{true,false}(flat0, false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → pos_{false}(false)
pos_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → pos_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{greater_int}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{pos}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{0}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{false}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{s}(_x0)))) → s_{false}(false)
s_{greater_int}(greater_int_{pos,pos}(pos_{0}(0), pos_{s}(s_{true}(_x0)))) → s_{false}(false)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,greater_int}(true, flat1)
greater_int_{greater_int,pos}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,pos}(true, flat1)
greater_int_{greater_int,0}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,0}(true, flat1)
greater_int_{greater_int,false}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,false}(true, flat1)
greater_int_{greater_int,s}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,s}(true, flat1)
greater_int_{greater_int,true}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0)), flat1) → greater_int_{true,true}(true, flat1)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{greater_int,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{greater_int,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{pos,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{pos,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{0,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{0,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{false,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{false,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{s,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{s,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
greater_int_{true,greater_int}(flat0, greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → greater_int_{true,true}(flat0, true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → pos_{true}(true)
pos_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → pos_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{greater_int}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{pos}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{0}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{false}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{s}(_x0)), pos_{0}(0))) → s_{true}(true)
s_{greater_int}(greater_int_{pos,pos}(pos_{s}(s_{true}(_x0)), pos_{0}(0))) → s_{true}(true)

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

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                        ↳ QDP
                                          ↳ UsableRulesProof

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

COND_F(true, neg(s(z0))) → F(neg(s(z0)))
F(neg(s(x0))) → COND_F(greater_int(neg(x0), neg(x0)), neg(s(x0)))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Instantiation
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                        ↳ QDP
                                          ↳ UsableRulesProof
                                            ↳ QDP

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

COND_F(true, neg(s(z0))) → F(neg(s(z0)))
F(neg(s(x0))) → COND_F(greater_int(neg(x0), neg(x0)), neg(s(x0)))

The TRS R consists of the following rules:

greater_int(neg(0), neg(0)) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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