(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x))) → b(x)
b(b(x)) → b(a(b(x)))
b(b(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(x))) → b(x)
b(b(x)) → b(a(b(x)))
b(b(x)) → a(a(x))
Q is empty.
(3) FlatCCProof (EQUIVALENT transformation)
We used flat context closure [ROOTLAB]
As Q is empty the flat context closure was sound AND complete.
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(x)) → b(a(b(x)))
a(a(a(a(x)))) → a(b(x))
b(a(a(a(x)))) → b(b(x))
a(b(b(x))) → a(a(a(x)))
b(b(b(x))) → b(a(a(x)))
Q is empty.
(5) 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.
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b_{b_1}(b_{b_1}(x)) → b_{a_1}(a_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(x)) → b_{a_1}(a_{b_1}(b_{a_1}(x)))
a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → a_{b_1}(b_{b_1}(x))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → a_{b_1}(b_{a_1}(x))
b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → b_{b_1}(b_{b_1}(x))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → b_{b_1}(b_{a_1}(x))
a_{b_1}(b_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{a_1}(a_{b_1}(x)))
a_{b_1}(b_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{a_1}(a_{a_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(x)))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(a_{a_1}(x)))
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a_{a_1}(x1)) = 1 + x1
POL(a_{b_1}(x1)) = x1
POL(b_{a_1}(x1)) = 2 + x1
POL(b_{b_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_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → a_{b_1}(b_{b_1}(x))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → a_{b_1}(b_{a_1}(x))
b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → b_{b_1}(b_{a_1}(x))
a_{b_1}(b_{b_1}(b_{b_1}(x))) → a_{a_1}(a_{a_1}(a_{b_1}(x)))
a_{b_1}(b_{b_1}(b_{a_1}(x))) → a_{a_1}(a_{a_1}(a_{a_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(x)))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(a_{a_1}(x)))
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b_{b_1}(b_{b_1}(x)) → b_{a_1}(a_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(x)) → b_{a_1}(a_{b_1}(b_{a_1}(x)))
b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → b_{b_1}(b_{b_1}(x))
Q is empty.
(9) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a_{a_1}(x1)) = 1 + x1
POL(a_{b_1}(x1)) = x1
POL(b_{a_1}(x1)) = 1 + x1
POL(b_{b_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_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → b_{b_1}(b_{b_1}(x))
(10) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b_{b_1}(b_{b_1}(x)) → b_{a_1}(a_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(x)) → b_{a_1}(a_{b_1}(b_{a_1}(x)))
Q is empty.
(11) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a_{b_1}(x1)) = x1
POL(b_{a_1}(x1)) = x1
POL(b_{b_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_{b_1}(b_{b_1}(x)) → b_{a_1}(a_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(x)) → b_{a_1}(a_{b_1}(b_{a_1}(x)))
(12) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(13) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(14) YES