Termination proof of /tmp/tmpVm5Z2G/unsatCond2.itrs

Termination of the given ITRSProblem could successfully be proven:

↳ 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(*@z(x, x), 0@z), 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(less_int(mult_int(x, x), pos(0)), x)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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(less_int(mult_int(x, x), pos(0)), x)
F(x) → LESS_INT(mult_int(x, x), pos(0))
F(x) → MULT_INT(x, x)
LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))
LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))
MULT_INT(pos(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(pos(x), neg(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), neg(y)) → MULT_NAT(x, y)
MULT_NAT(s(x), s(y)) → PLUS_NAT(mult_nat(x, s(y)), s(y))
MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(less_int(mult_int(x, x), pos(0)), x)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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(less_int(mult_int(x, x), pos(0)), x)
F(x) → LESS_INT(mult_int(x, x), pos(0))
F(x) → MULT_INT(x, x)
LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))
LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))
MULT_INT(pos(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(pos(x), neg(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), neg(y)) → MULT_NAT(x, y)
MULT_NAT(s(x), s(y)) → PLUS_NAT(mult_nat(x, s(y)), s(y))
MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(less_int(mult_int(x, x), pos(0)), x)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 7 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(less_int(mult_int(x, x), pos(0)), x)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(less_int(mult_int(x, x), pos(0)), x)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

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

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(less_int(mult_int(x, x), pos(0)), x)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

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

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

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

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

LESS_INT(neg(s(x)), neg(s(y))) → LESS_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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

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

The TRS R consists of the following rules:

Cond_f(true, x) → f(x)
f(x) → Cond_f(less_int(mult_int(x, x), pos(0)), x)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

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

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

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

F(x) → COND_F(less_int(mult_int(x, x), pos(0)), 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(less_int(mult_int(x, x), pos(0)), x)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

F(x) → COND_F(less_int(mult_int(x, x), pos(0)), x)
COND_F(true, x) → F(x)

The TRS R consists of the following rules:

mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

Cond_f(true, x0)
f(x0)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

F(x) → COND_F(less_int(mult_int(x, x), pos(0)), x)
COND_F(true, x) → F(x)

The TRS R consists of the following rules:

mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

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.

COND_F(true, x) → F(x)

The remaining pairs can at least be oriented weakly.

F(x) → COND_F(less_int(mult_int(x, x), pos(0)), x)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND_F(x₁, x₂)) = x₁
POL(F(x₁)) = 0
POL(false) = 0
POL(less_int(x₁, x₂)) = x₁
POL(mult_int(x₁, x₂)) = 0
POL(mult_nat(x₁, x₂)) = 0
POL(neg(x₁)) = 1 + x₁
POL(plus_nat(x₁, x₂)) = 0
POL(pos(x₁)) = 0
POL(s(x₁)) = 1
POL(true) = 1

The following usable rules [FROCOS05] were oriented:

less_int(neg(0), pos(0)) → false
less_int(pos(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), pos(0)) → false
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

F(x) → COND_F(less_int(mult_int(x, x), pos(0)), x)

The TRS R consists of the following rules:

mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

The set Q consists of the following terms:

less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.