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