(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
*1(x0, x1, x2)  =  *1(x0, x1, x2)

Tags:
*1 has argument tags [0,0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
*1(x1, x2)  =  x2
*(x1, x2)  =  *(x2)
minus(x1)  =  minus

Lexicographic path order with status [LPO].
Quasi-Precedence:
[*1, minus]

Status:
*1: [1]
minus: []


The following usable rules [FROCOS05] were oriented: none

(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
*1(x0, x1, x2)  =  *1(x1, x2)

Tags:
*1 has argument tags [3,3,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
*1(x1, x2)  =  *1(x2)
*(x1, x2)  =  *(x1)
minus(x1)  =  minus

Lexicographic path order with status [LPO].
Quasi-Precedence:
*1 > [*^11, minus]

Status:
*^11: [1]
*1: [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