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(b(x)) → b(a(x))
a(c(x)) → x

Q is empty.

(1) DirectTerminationProof (EQUIVALENT transformation)

We use [DIRECT_TERMINATION] with the following order to prove termination.

Knuth-Bendix order [KBO] with precedence:

a₁ > b₁ > c₁

and weight map:

a_1=1
b_1=2
c_1=1

The variable weight is 1

(2) TRUE