(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(b(b(x)))
b(b(x)) → c(c(c(x)))
c(c(x)) → d(d(d(x)))
b(x) → d(d(x))
c(d(d(x))) → a(x)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = 80 + x1
POL(b(x1)) = 53 + x1
POL(c(x1)) = 35 + x1
POL(d(x1)) = 23 + 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(b(b(x)))
b(b(x)) → c(c(c(x)))
c(c(x)) → d(d(d(x)))
b(x) → d(d(x))
c(d(d(x))) → a(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