(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(X, Y), Z) → f(X, f(Y, Z))
f(X, f(Y, Z)) → f(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(f(X, Y), Z) → F(X, f(Y, Z))
F(f(X, Y), Z) → F(Y, Z)
F(X, f(Y, Z)) → F(Y, Y)
The TRS R consists of the following rules:
f(f(X, Y), Z) → f(X, f(Y, Z))
f(X, f(Y, Z)) → f(Y, Y)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
F(f(X, Y), Z) → F(X, f(Y, Z))
F(f(X, Y), Z) → F(Y, Z)
F(X, f(Y, Z)) → F(Y, Y)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(F(x1, x2)) = x1 + 2·x1·x2 + 2·x12 + 2·x2
POL(f(x1, x2)) = 2 + 2·x1 + x1·x2 + 2·x12 + 2·x2
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
f(X, f(Y, Z)) → f(Y, Y)
f(f(X, Y), Z) → f(X, f(Y, Z))
(4) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(f(X, Y), Z) → f(X, f(Y, Z))
f(X, f(Y, Z)) → f(Y, Y)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(6) YES