(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x))) → a(c(a(a(x))))
c(c(c(x))) → a(x)
a(x) → x
Q is empty.
(1) FlatCCProof (EQUIVALENT transformation)
We used flat context closure [ROOTLAB]
As Q is empty the flat context closure was sound AND complete.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x))) → a(c(a(a(x))))
a(c(c(c(x)))) → a(a(x))
c(c(c(c(x)))) → c(a(x))
a(a(x)) → a(x)
c(a(x)) → c(x)
Q is empty.
(3) RootLabelingProof (EQUIVALENT transformation)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))) → a_{a_1}(a_{a_1}(x))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))) → a_{a_1}(a_{c_1}(x))
c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))) → c_{a_1}(a_{a_1}(x))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))) → c_{a_1}(a_{c_1}(x))
a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a_{a_1}(x1)) = 3 + x1
POL(a_{c_1}(x1)) = 3 + x1
POL(c_{a_1}(x1)) = x1
POL(c_{c_1}(x1)) = 2 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))) → a_{a_1}(a_{a_1}(x))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))) → a_{a_1}(a_{c_1}(x))
c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))) → c_{a_1}(a_{a_1}(x))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))) → c_{a_1}(a_{c_1}(x))
a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a_{a_1}(x1)) = 1 + x1
POL(a_{c_1}(x1)) = x1
POL(c_{a_1}(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))
(8) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) YES