(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(x))) → +(fib(s(x)), fib(x))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive Path Order [RPO].
Precedence:
fib1 > +2 > s1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(x))) → +(fib(s(x)), fib(x))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
(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