(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
[g1, f1, 0] > s1
Status:
f1: multiset
g1: multiset
s1: [1]
0: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0
(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