(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(x, y), x, z) → f(z, z, z)
g(x, y) → x
g(x, y) → 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:
F(g(x, y), x, z) → F(z, z, z)
The TRS R consists of the following rules:
f(g(x, y), x, z) → f(z, z, z)
g(x, y) → x
g(x, y) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
s =
F(
g(
x',
y),
g(
x',
y'),
z) evaluates to t =
F(
z,
z,
z)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [z / g(x', y), y' / y]
- Matcher: [ ]
Rewriting sequenceF(g(x', y), g(x', y), g(x', y)) →
F(
g(
x',
y),
x',
g(
x',
y))
with rule
g(
x'',
y') →
x'' at position [1] and matcher [
x'' /
x',
y' /
y]
F(g(x', y), x', g(x', y)) →
F(
g(
x',
y),
g(
x',
y),
g(
x',
y))
with rule
F(
g(
x,
y),
x,
z) →
F(
z,
z,
z)
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
(4) FALSE