(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0, 1, X) → f(X, X, X)
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(0, 1, X) → F(X, X, X)
The TRS R consists of the following rules:
f(0, 1, X) → f(X, X, X)
g(X, Y) → X
g(X, Y) → Y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) NonTerminationLoopProof (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(
0,
Y),
g(
X',
1),
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: [Y / 1, X' / 0, X / g(0, 1)]
Rewriting sequenceF(g(0, 1), g(0, 1), g(0, 1)) →
F(
g(
0,
1),
1,
g(
0,
1))
with rule
g(
X',
Y') →
Y' at position [1] and matcher [
X' /
0,
Y' /
1]
F(g(0, 1), 1, g(0, 1)) →
F(
0,
1,
g(
0,
1))
with rule
g(
X',
Y) →
X' at position [0] and matcher [
X' /
0,
Y /
1]
F(0, 1, g(0, 1)) →
F(
g(
0,
1),
g(
0,
1),
g(
0,
1))
with rule
F(
0,
1,
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