(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Precedence:
sum1 > 0
sum1 > s1 > +2
sum11 > 0
sum11 > s1 > +2
Status:
sum1: multiset
s1: multiset
sum11: [1]
0: multiset
+2: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
(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