YES
0 QTRS
↳1 FlatCCProof (⇔, 0 ms)
↳2 QTRS
↳3 RootLabelingProof (⇔, 0 ms)
↳4 QTRS
↳5 QTRSRRRProof (⇔, 9 ms)
↳6 QTRS
↳7 RisEmptyProof (⇔, 0 ms)
↳8 YES
g(b) → f(b)
f(a) → g(a)
b → a
g(g(b)) → g(f(b))
f(g(b)) → f(f(b))
g(f(a)) → g(g(a))
f(f(a)) → f(g(a))
g(b) → g(a)
f(b) → f(a)
g_{g_1}(g_{b}(b)) → g_{f_1}(f_{b}(b))
f_{g_1}(g_{b}(b)) → f_{f_1}(f_{b}(b))
g_{f_1}(f_{a}(a)) → g_{g_1}(g_{a}(a))
f_{f_1}(f_{a}(a)) → f_{g_1}(g_{a}(a))
g_{b}(b) → g_{a}(a)
f_{b}(b) → f_{a}(a)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(a) = 0
POL(b) = 1
POL(f_{a}(x1)) = 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)) = 2 + x1
POL(g_{f_1}(x1)) = 1 + x1
POL(g_{g_1}(x1)) = x1
g_{g_1}(g_{b}(b)) → g_{f_1}(f_{b}(b))
f_{g_1}(g_{b}(b)) → f_{f_1}(f_{b}(b))
g_{f_1}(f_{a}(a)) → g_{g_1}(g_{a}(a))
f_{f_1}(f_{a}(a)) → f_{g_1}(g_{a}(a))
g_{b}(b) → g_{a}(a)
f_{b}(b) → f_{a}(a)