(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))
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:
F(cons(x, k), l) → G(k, l, cons(x, k))
G(a, b, c) → F(a, cons(b, c))
The TRS R consists of the following rules:
f(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))
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:
cons(x1, x2) = cons(x2)
From the DPs we obtained the following set of size-change graphs:
- G(a, b, c) → F(a, cons(b, c)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1
- F(cons(x, k), l) → G(k, l, cons(x, k)) (allowed arguments on rhs = {1, 2, 3})
The graph contains the following edges 1 > 1, 2 >= 2, 1 >= 3
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(4) TRUE