(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(c(a(x)))) → d(d(x))
d(b(x)) → c(c(x))
c(x) → a(a(a(a(x))))
d(x) → b(b(b(b(x))))
b(d(x)) → c(c(x))
a(c(c(c(x)))) → d(d(x))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = 3 + x1
POL(b(x1)) = 5 + x1
POL(c(x1)) = 13 + 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:
c(x) → a(a(a(a(x))))
d(x) → b(b(b(b(x))))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(c(a(x)))) → d(d(x))
d(b(x)) → c(c(x))
b(d(x)) → c(c(x))
a(c(c(c(x)))) → d(d(x))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = x1
POL(b(x1)) = 1 + x1
POL(c(x1)) = x1
POL(d(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
d(b(x)) → c(c(x))
b(d(x)) → c(c(x))
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(c(a(x)))) → d(d(x))
a(c(c(c(x)))) → d(d(x))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = 1 + x1
POL(c(x1)) = x1
POL(d(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
c(c(c(a(x)))) → d(d(x))
a(c(c(c(x)))) → d(d(x))
(6) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) YES