(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))
b(x) → a(x)
c(c(x)) → d(f(x))
d(d(x)) → f(f(f(x)))
d(x) → b(x)
f(f(x)) → g(a(x))
g(g(x)) → 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(x)) → c(b(x))
b(b(x)) → d(c(x))
b(x) → a(x)
c(c(x)) → f(d(x))
d(d(x)) → f(f(f(x)))
d(x) → b(x)
f(f(x)) → a(g(x))
g(g(x)) → a(x)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = 155 + x1
POL(b(x1)) = 162 + x1
POL(c(x1)) = 147 + x1
POL(d(x1)) = 176 + x1
POL(f(x1)) = 117 + x1
POL(g(x1)) = 78 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a(a(x)) → c(b(x))
b(b(x)) → d(c(x))
b(x) → a(x)
c(c(x)) → f(d(x))
d(d(x)) → f(f(f(x)))
d(x) → b(x)
f(f(x)) → a(g(x))
g(g(x)) → 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