(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,
x2)
Tags:
*1 has argument tags [2,0,2] 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) =
x1
*(
x1,
x2) =
*(
x1,
x2)
minus(
x1) =
minus
Recursive path order with status [RPO].
Quasi-Precedence:
minus > *2
Status:
*2: [2,1]
minus: multiset
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(
x0)
Tags:
*1 has argument tags [2,0,3] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
*2 > [*^12, minus1]
Status:
*^12: multiset
*2: multiset
minus1: multiset
The following usable rules [FROCOS05] were oriented:
*(x, *(minus(y), y)) → *(minus(*(y, y)), x)
(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