(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x)))) → b(b(b(b(b(b(x))))))
b(b(b(b(x)))) → c(c(c(c(c(c(x))))))
c(c(c(c(x)))) → d(d(d(d(d(d(x))))))
b(b(x)) → d(d(d(d(x))))
c(c(d(d(d(d(x)))))) → a(a(x))
Q is empty.
(1) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x)))) → b(b(b(b(b(b(x))))))
b(b(b(b(x)))) → c(c(c(c(c(c(x))))))
c(c(c(c(x)))) → d(d(d(d(d(d(x))))))
b(b(x)) → d(d(d(d(x))))
d(d(d(d(c(c(x)))))) → a(a(x))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = 134 + x1
POL(b(x1)) = 89 + x1
POL(c(x1)) = 59 + x1
POL(d(x1)) = 39 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a(a(a(a(x)))) → b(b(b(b(b(b(x))))))
b(b(b(b(x)))) → c(c(c(c(c(c(x))))))
c(c(c(c(x)))) → d(d(d(d(d(d(x))))))
b(b(x)) → d(d(d(d(x))))
d(d(d(d(c(c(x)))))) → a(a(x))
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) YES