(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
quot(0, s(y), s(z)) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(quot(x, s(z), s(z)))
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:
QUOT(s(x), s(y), z) → QUOT(x, y, z)
QUOT(x, 0, s(z)) → QUOT(x, s(z), s(z))
The TRS R consists of the following rules:
quot(0, s(y), s(z)) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(quot(x, s(z), s(z)))
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:
0 = 0
s(x1) = s(x1)
From the DPs we obtained the following set of size-change graphs:
- QUOT(s(x), s(y), z) → QUOT(x, y, z) (allowed arguments on rhs = {1, 2, 3})
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
- QUOT(x, 0, s(z)) → QUOT(x, s(z), s(z)) (allowed arguments on rhs = {1, 2, 3})
The graph contains the following edges 1 >= 1, 3 >= 2, 3 >= 3
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(4) TRUE