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:

g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive Path Order [RPO].
Precedence:

[g₁, f₁] > s₁
[g₁, f₁] > 0

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

g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0

(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