(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(x)) → c(c(c(c(x))))
c(x) → x
b(c(b(x))) → b(b(b(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:
b(c(b(x))) → b(b(b(x)))
b(b(b(x))) → b(c(c(c(c(x)))))
c(b(b(x))) → c(c(c(c(c(x)))))
b(c(x)) → b(x)
c(c(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:
b_{c_1}(c_{b_1}(b_{b_1}(x))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{c_1}(c_{b_1}(b_{c_1}(x))) → b_{b_1}(b_{b_1}(b_{c_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))
b_{b_1}(b_{b_1}(b_{c_1}(x))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
c_{b_1}(b_{b_1}(b_{b_1}(x))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))
c_{b_1}(b_{b_1}(b_{c_1}(x))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
b_{c_1}(c_{b_1}(x)) → b_{b_1}(x)
b_{c_1}(c_{c_1}(x)) → b_{c_1}(x)
c_{c_1}(c_{b_1}(x)) → c_{b_1}(x)
c_{c_1}(c_{c_1}(x)) → c_{c_1}(x)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(b_{b_1}(x1)) = 1 + x1
POL(b_{c_1}(x1)) = 2 + x1
POL(c_{b_1}(x1)) = x1
POL(c_{c_1}(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))
b_{b_1}(b_{b_1}(b_{c_1}(x))) → b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
c_{b_1}(b_{b_1}(b_{b_1}(x))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))
c_{b_1}(b_{b_1}(b_{c_1}(x))) → c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
b_{c_1}(c_{b_1}(x)) → b_{b_1}(x)
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b_{c_1}(c_{b_1}(b_{b_1}(x))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{c_1}(c_{b_1}(b_{c_1}(x))) → b_{b_1}(b_{b_1}(b_{c_1}(x)))
b_{c_1}(c_{c_1}(x)) → b_{c_1}(x)
c_{c_1}(c_{b_1}(x)) → c_{b_1}(x)
c_{c_1}(c_{c_1}(x)) → c_{c_1}(x)
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(b_{b_1}(x1)) = x1
POL(b_{c_1}(x1)) = 1 + x1
POL(c_{b_1}(x1)) = x1
POL(c_{c_1}(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
b_{c_1}(c_{b_1}(b_{b_1}(x))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{c_1}(c_{b_1}(b_{c_1}(x))) → b_{b_1}(b_{b_1}(b_{c_1}(x)))
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b_{c_1}(c_{c_1}(x)) → b_{c_1}(x)
c_{c_1}(c_{b_1}(x)) → c_{b_1}(x)
c_{c_1}(c_{c_1}(x)) → c_{c_1}(x)
Q is empty.
(9) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(b_{c_1}(x1)) = x1
POL(c_{b_1}(x1)) = x1
POL(c_{c_1}(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
b_{c_1}(c_{c_1}(x)) → b_{c_1}(x)
c_{c_1}(c_{b_1}(x)) → c_{b_1}(x)
c_{c_1}(c_{c_1}(x)) → c_{c_1}(x)
(10) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(11) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(12) YES