(0) Obligation:

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

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

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
0 > [f1, s1]

Status:
f1: [1]
0: []
s1: [1]

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

f(s(0)) → s(0)


(2) Obligation:

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

f(0) → s(0)
f(s(s(x))) → f(f(s(x)))

Q is empty.