(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(c(x))
b(b(x)) → c(d(x))
c(c(x)) → d(d(d(x)))
d(d(d(x))) → a(c(x))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = 30 + x1
POL(b(x1)) = 27 + x1
POL(c(x1)) = 32 + x1
POL(d(x1)) = 21 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a(a(x)) → b(c(x))
b(b(x)) → c(d(x))
c(c(x)) → d(d(d(x)))
d(d(d(x))) → a(c(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) YES