Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

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

a(a(x)) → b(b(x))
b(b(a(x))) → a(b(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:

A(a(x)) → B(b(x))
A(a(x)) → B(x)
B(b(a(x))) → A(b(b(x)))
B(b(a(x))) → B(b(x))
B(b(a(x))) → B(x)

The TRS R consists of the following rules:

a(a(x)) → b(b(x))
b(b(a(x))) → a(b(b(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].

The following pairs can be oriented strictly and are deleted.

A(a(x)) → B(x)
B(b(a(x))) → B(b(x))
B(b(a(x))) → B(x)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[A₁, B₁] > [a₁, b₁]

Status:

A₁: [1]
a₁: [1]
B₁: [1]
b₁: [1]

The following usable rules [FROCOS05] were oriented:

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

(4) Obligation:

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

A(a(x)) → B(b(x))
B(b(a(x))) → A(b(b(x)))

The TRS R consists of the following rules:

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

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A(a(x)) → B(b(x))
B(b(a(x))) → A(b(b(x)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A(x1)  =  A(x1)
a(x1)  =  a(x1)
B(x1)  =  B(x1)
b(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[A₁, B₁] > a₁

Status:

A₁: [1]
a₁: [1]
B₁: [1]

The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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

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

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

(7) PisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE