(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
minus(minus(x)) → x
minus(h(x)) → h(minus(x))
minus(f(x, y)) → f(minus(y), minus(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:
MINUS(h(x)) → MINUS(x)
MINUS(f(x, y)) → MINUS(y)
MINUS(f(x, y)) → MINUS(x)
The TRS R consists of the following rules:
minus(minus(x)) → x
minus(h(x)) → h(minus(x))
minus(f(x, y)) → f(minus(y), minus(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
h(x1) = h(x1)
f(x1, x2) = f(x1, x2)
From the DPs we obtained the following set of size-change graphs:
- MINUS(h(x)) → MINUS(x) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
- MINUS(f(x, y)) → MINUS(y) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
- MINUS(f(x, y)) → MINUS(x) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(4) TRUE