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(g(f(a), h(a, f(a)))) → f(h(g(f(a), a), g(f(a), f(a))))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

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

g₂ > a > f₁
g₂ > h₂ > f₁

Status:

f₁: [1]
g₂: [1,2]
a: []
h₂: [1,2]

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

f(g(f(a), h(a, f(a)))) → f(h(g(f(a), a), g(f(a), f(a))))

(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