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