(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, 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(b, f(a, x))
F(a, f(b, x)) → F(a, x)
F(b, f(c, x)) → F(c, f(b, x))
F(b, f(c, x)) → F(b, x)
F(c, f(a, x)) → F(a, f(c, x))
F(c, f(a, x)) → F(c, x)

The TRS R consists of the following rules:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) 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)) → b1(a(x))
a1(b(x)) → a1(x)
b1(c(x)) → c1(b(x))
b1(c(x)) → b1(x)
c1(a(x)) → a1(c(x))
c1(a(x)) → c1(x)

The a-transformed usable rules are

a(b(x)) → b(a(x))
b(c(x)) → c(b(x))
c(a(x)) → a(c(x))


The following pairs can be oriented strictly and are deleted.


F(a, f(b, x)) → F(b, f(a, x))
F(a, f(b, x)) → F(a, x)
F(b, f(c, x)) → F(c, f(b, x))
F(c, f(a, x)) → F(a, f(c, x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
a1(x1)  =  x1
b(x1)  =  b(x1)
b1(x1)  =  b1
a(x1)  =  x1
c(x1)  =  c
c1(x1)  =  c1

Lexicographic path order with status [LPO].
Precedence:
b1 > b1 > c1 > c

Status:
b1: [1]
b1: []
c: []
c1: []

The following usable rules [FROCOS05] were oriented:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(b, f(c, x)) → F(b, x)
F(c, f(a, x)) → F(c, x)

The TRS R consists of the following rules:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(6) Complex Obligation (AND)

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(c, f(a, x)) → F(c, x)

The TRS R consists of the following rules:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) 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

c1(a(x)) → c1(x)

The a-transformed usable rules are

a(b(x)) → b(a(x))
b(c(x)) → c(b(x))
c(a(x)) → a(c(x))


The following pairs can be oriented strictly and are deleted.


F(c, f(a, x)) → F(c, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
c1(x1)  =  x1
a(x1)  =  a(x1)
b(x1)  =  b
c(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:
a1 > b

Status:
a1: [1]
b: []

The following usable rules [FROCOS05] were oriented:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(b, f(c, x)) → F(b, x)

The TRS R consists of the following rules:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) 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(c(x)) → b1(x)

The a-transformed usable rules are

a(b(x)) → b(a(x))
b(c(x)) → c(b(x))
c(a(x)) → a(c(x))


The following pairs can be oriented strictly and are deleted.


F(b, f(c, x)) → F(b, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
b1(x1)  =  x1
c(x1)  =  c(x1)
a(x1)  =  x1
b(x1)  =  b(x1)

Lexicographic path order with status [LPO].
Precedence:
b1 > c1

Status:
c1: [1]
b1: [1]

The following usable rules [FROCOS05] were oriented:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

(14) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f(a, f(b, x)) → f(b, f(a, x))
f(b, f(c, x)) → f(c, f(b, x))
f(c, f(a, x)) → f(a, f(c, x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE