(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(f(b, a(x)))) → f(a(a(a(x))), b)
a(a(x)) → f(b, a(f(a(x), b)))
f(a(x), b) → f(b, a(x))
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:
A(a(f(b, a(x)))) → F(a(a(a(x))), b)
A(a(f(b, a(x)))) → A(a(a(x)))
A(a(f(b, a(x)))) → A(a(x))
A(a(x)) → F(b, a(f(a(x), b)))
A(a(x)) → A(f(a(x), b))
A(a(x)) → F(a(x), b)
F(a(x), b) → F(b, a(x))
The TRS R consists of the following rules:
a(a(f(b, a(x)))) → f(a(a(a(x))), b)
a(a(x)) → f(b, a(f(a(x), b)))
f(a(x), b) → f(b, a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(f(b, a(x)))) → A(a(x))
A(a(f(b, a(x)))) → A(a(a(x)))
The TRS R consists of the following rules:
a(a(f(b, a(x)))) → f(a(a(a(x))), b)
a(a(x)) → f(b, a(f(a(x), b)))
f(a(x), b) → f(b, a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
A(a(f(b, a(x)))) → A(a(x))
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = x1
POL(a(x1)) = 1 + x1
POL(b) = 0
POL(f(x1, x2)) = x1 + x2
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(f(b, a(x)))) → A(a(a(x)))
The TRS R consists of the following rules:
a(a(f(b, a(x)))) → f(a(a(a(x))), b)
a(a(x)) → f(b, a(f(a(x), b)))
f(a(x), b) → f(b, a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
A(a(f(b, a(x)))) → A(a(a(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
| POL(f(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
a(a(x)) → f(b, a(f(a(x), b)))
a(a(f(b, a(x)))) → f(a(a(a(x))), b)
f(a(x), b) → f(b, a(x))
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(a(f(b, a(x)))) → f(a(a(a(x))), b)
a(a(x)) → f(b, a(f(a(x), b)))
f(a(x), b) → f(b, a(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE