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:

1(0(x)) → 0(0(0(1(x))))
0(1(x)) → 1(x)
1(1(x)) → 0(0(0(0(x))))
0(0(x)) → 0(x)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Precedence:

1₁ > 0₁

Status:

1₁: [1]
0₁: [1]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

1(0(x)) → 0(0(0(1(x))))
0(1(x)) → 1(x)
1(1(x)) → 0(0(0(0(x))))
0(0(x)) → 0(x)

(2) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(3) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(4) TRUE