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(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

[f₂, s₁, a₁, c₁]

Status:

c₁: multiset
a₁: multiset
f₂: multiset
s₁: multiset

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

f(X, X) → c(X)

(2) Obligation:

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

f(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

[s₁, c₁] > [f₂, a₁]

Status:

c₁: multiset
a₁: multiset
f₂: [2,1]
s₁: multiset

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

f(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), 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