(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, b, X) → f(X, X, X)
c → a
c → b
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(a, b, X) → F(X, X, X)
The TRS R consists of the following rules:
f(a, b, X) → f(X, X, X)
c → a
c → b
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(
c,
c,
X) evaluates to t =
F(
X,
X,
X)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [X / c]
Rewriting sequenceF(c, c, c) →
F(
c,
b,
c)
with rule
c →
b at position [1] and matcher [ ]
F(c, b, c) →
F(
a,
b,
c)
with rule
c →
a at position [0] and matcher [ ]
F(a, b, c) →
F(
c,
c,
c)
with rule
F(
a,
b,
X) →
F(
X,
X,
X)
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) NO