(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