(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
p(m, n, s(r)) → p(m, r, n)
p(m, s(n), 0) → p(0, n, m)
p(m, 0, 0) → m
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:
P(m, n, s(r)) → P(m, r, n)
P(m, s(n), 0) → P(0, n, m)
The TRS R consists of the following rules:
p(m, n, s(r)) → p(m, r, n)
p(m, s(n), 0) → p(0, n, m)
p(m, 0, 0) → m
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:
- P(m, n, s(r)) → P(m, r, n) (allowed arguments on rhs = {1, 2, 3})
The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3
- P(m, s(n), 0) → P(0, n, m) (allowed arguments on rhs = {1, 2, 3})
The graph contains the following edges 3 >= 1, 2 > 2, 1 >= 3
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(4) TRUE