(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].
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: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(
x1,
x2) =
F(
x2)
Tags:
F has tags [1,0]
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
a > b > f2
Status:
b: multiset
f2: [1,2]
a: 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].
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: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(
x1,
x2) =
F(
x2)
Tags:
F has tags [1,0]
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
b > a > f2
Status:
a: multiset
f2: [1,2]
b: 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