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) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Combined order from the following AFS and order.
b(x1)  =  x1
a(x1)  =  a(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

a₁: multiset

AFS:
b(x1)  =  x1
a(x1)  =  a(x1)

From the DPs we obtained the following set of size-change graphs:

• B(b(a(x))) → A(b(b(x))) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• B(b(a(x))) → B(b(x)) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• B(b(a(x))) → B(x) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• A(a(x)) → B(b(x)) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• A(a(x)) → B(x) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

We oriented the following set of usable rules [AAECC05,FROCOS05].

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

(4) TRUE