(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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:
F(a, f(b, x)) → F(a, f(a, f(a, x)))
F(a, f(b, x)) → F(a, f(a, x))
F(a, f(b, x)) → F(a, x)
F(b, f(a, x)) → F(b, f(b, f(b, x)))
F(b, f(a, x)) → F(b, f(b, x))
F(b, f(a, x)) → F(b, x)
The TRS R consists of the following rules:
f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, 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 2 SCCs.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(b, f(a, x)) → F(b, f(b, x))
F(b, f(a, x)) → F(b, f(b, f(b, x)))
F(b, f(a, x)) → F(b, x)
The TRS R consists of the following rules:
f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04]. Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is
b1(a(x)) → b1(b(x))
b1(a(x)) → b1(b(b(x)))
b1(a(x)) → b1(x)
The a-transformed usable rules are
b(a(x)) → b(b(b(x)))
The following pairs can be oriented strictly and are deleted.
F(b, f(a, x)) → F(b, f(b, x))
F(b, f(a, x)) → F(b, f(b, f(b, x)))
F(b, f(a, x)) → F(b, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Precedence:
a1 > b11
a1 > b1
Status:
a1: multiset
b11: multiset
b1: multiset
The following usable rules [FROCOS05] were oriented:
f(b, f(a, x)) → f(b, f(b, f(b, x)))
(7) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, f(b, x)) → F(a, f(a, x))
F(a, f(b, x)) → F(a, f(a, f(a, x)))
F(a, f(b, x)) → F(a, x)
The TRS R consists of the following rules:
f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04]. Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is
a1(b(x)) → a1(a(x))
a1(b(x)) → a1(a(a(x)))
a1(b(x)) → a1(x)
The a-transformed usable rules are
a(b(x)) → a(a(a(x)))
The following pairs can be oriented strictly and are deleted.
F(a, f(b, x)) → F(a, f(a, x))
F(a, f(b, x)) → F(a, f(a, f(a, x)))
F(a, f(b, x)) → F(a, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Precedence:
b1 > a11
b1 > a1
Status:
a1: multiset
a11: multiset
b1: multiset
The following usable rules [FROCOS05] were oriented:
f(a, f(b, x)) → f(a, f(a, f(a, x)))
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(a, f(b, x)) → f(a, f(a, f(a, x)))
f(b, f(a, x)) → f(b, f(b, f(b, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) TRUE