(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(s(x), y) → +(x, s(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(s(x), y) → +1(x, y)
+1(s(x), y) → +1(x, s(y))
The TRS R consists of the following rules:
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(s(x), y) → +(x, s(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:
s(x1) = s(x1)
From the DPs we obtained the following set of size-change graphs:
- +1(s(x), y) → +1(x, y) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
- +1(s(x), y) → +1(x, s(y)) (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