Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
  ↳3 RisEmptyProof (⇔)
    ↳4 TRUE
  ↳5 RisEmptyProof (⇔)
    ↳6 TRUE

(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:
Polynomial interpretation [POLO]:

POL(a) = 0   
POL(f(x1)) = 2·x1   
POL(g(x1, x2)) = 2·x1 + 2·x2   
POL(h(x1, x2)) = 2 + x1 + x2   
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

(5) RisEmptyProof (EQUIVALENT transformation)

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

(6) TRUE