(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
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(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(-(x, y), -(x, y))
-1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(x, y)
The TRS R consists of the following rules:
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Polynomial interpretation [POLO]:
POL(-(x1, x2)) = x2
POL(neg(x1)) = 1 + x1
From the DPs we obtained the following set of size-change graphs:
- -1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(-(x, y), -(x, y)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 > 1, 2 > 2
- -1(-(neg(x), neg(x)), -(neg(y), neg(y))) → -1(x, y) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 > 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
-(-(neg(x), neg(x)), -(neg(y), neg(y))) → -(-(x, y), -(x, y))
(4) TRUE