(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
*(x, +(y, z)) → +(*(x, y), *(x, z))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
*(x, +(y, z)) → +(*(x, y), *(x, z))
The set Q consists of the following terms:
*(x0, +(x1, x2))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
*1(x, +(y, z)) → *1(x, y)
*1(x, +(y, z)) → *1(x, z)
The TRS R consists of the following rules:
*(x, +(y, z)) → +(*(x, y), *(x, z))
The set Q consists of the following terms:
*(x0, +(x1, x2))
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, +(y, z)) → *1(x, y)
*1(x, +(y, z)) → *1(x, z)
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)
*(
x1,
x2) =
x2
Recursive path order with status [RPO].
Precedence:
trivial
Status:
*^12: [1,2]
+2: multiset
The following usable rules [FROCOS05] were oriented:
*(x, +(y, z)) → +(*(x, y), *(x, z))
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
*(x, +(y, z)) → +(*(x, y), *(x, z))
The set Q consists of the following terms:
*(x0, +(x1, x2))
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