(0) Obligation:

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

*(x, *(minus(y), y)) → *(minus(*(y, y)), x)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

*1(x, *(minus(y), y)) → *1(minus(*(y, y)), x)
*1(x, *(minus(y), y)) → *1(y, y)

The TRS R consists of the following rules:

*(x, *(minus(y), y)) → *(minus(*(y, y)), x)

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


*1(x, *(minus(y), y)) → *1(y, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
*1(x1, x2)  =  *1(x1, x2)
*(x1, x2)  =  *(x1, x2)
minus(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
*2 > *^12

The following usable rules [FROCOS05] were oriented:

*(x, *(minus(y), y)) → *(minus(*(y, y)), x)

(4) Obligation:

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

*1(x, *(minus(y), y)) → *1(minus(*(y, y)), x)

The TRS R consists of the following rules:

*(x, *(minus(y), y)) → *(minus(*(y, y)), x)

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


*1(x, *(minus(y), y)) → *1(minus(*(y, y)), x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
*1(x1, x2)  =  *1(x1, x2)
*(x1, x2)  =  *(x2)
minus(x1)  =  minus

Recursive Path Order [RPO].
Precedence:
*^12 > minus
*1 > minus

The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

*(x, *(minus(y), y)) → *(minus(*(y, y)), x)

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

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(8) TRUE