(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a) → f(b)
g(b) → g(a)
f(x) → g(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:
f(a) → f(b)
g(b) → g(a)
f(f(x)) → f(g(x))
g(f(x)) → g(g(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:
f_{a}(a) → f_{b}(b)
g_{b}(b) → g_{a}(a)
f_{f_1}(f_{f_1}(x)) → f_{g_1}(g_{f_1}(x))
f_{f_1}(f_{a}(x)) → f_{g_1}(g_{a}(x))
f_{f_1}(f_{b}(x)) → f_{g_1}(g_{b}(x))
f_{f_1}(f_{g_1}(x)) → f_{g_1}(g_{g_1}(x))
g_{f_1}(f_{f_1}(x)) → g_{g_1}(g_{f_1}(x))
g_{f_1}(f_{a}(x)) → g_{g_1}(g_{a}(x))
g_{f_1}(f_{b}(x)) → g_{g_1}(g_{b}(x))
g_{f_1}(f_{g_1}(x)) → g_{g_1}(g_{g_1}(x))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(b) = 1
POL(f_{a}(x1)) = 2 + x1
POL(f_{b}(x1)) = x1
POL(f_{f_1}(x1)) = 1 + x1
POL(f_{g_1}(x1)) = x1
POL(g_{a}(x1)) = x1
POL(g_{b}(x1)) = x1
POL(g_{f_1}(x1)) = 1 + x1
POL(g_{g_1}(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f_{a}(a) → f_{b}(b)
g_{b}(b) → g_{a}(a)
f_{f_1}(f_{f_1}(x)) → f_{g_1}(g_{f_1}(x))
f_{f_1}(f_{a}(x)) → f_{g_1}(g_{a}(x))
f_{f_1}(f_{b}(x)) → f_{g_1}(g_{b}(x))
f_{f_1}(f_{g_1}(x)) → f_{g_1}(g_{g_1}(x))
g_{f_1}(f_{f_1}(x)) → g_{g_1}(g_{f_1}(x))
g_{f_1}(f_{a}(x)) → g_{g_1}(g_{a}(x))
g_{f_1}(f_{b}(x)) → g_{g_1}(g_{b}(x))
g_{f_1}(f_{g_1}(x)) → g_{g_1}(g_{g_1}(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