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:

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

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

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

a > [f₂, g₂, b]

Status:

a: []
g₂: [2,1]
f₂: [2,1]
b: []

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

g(a, x) → f(b, x)
f(a, x) → f(b, x)

(2) Obligation:

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

f(a, x) → g(a, x)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

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

f₂ > g₂
a > g₂

Status:

a: []
g₂: [2,1]
f₂: [2,1]

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

f(a, x) → g(a, x)

(4) Obligation:

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

(5) RisEmptyProof (EQUIVALENT transformation)

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

(6) TRUE