NO
(ignored inputs)COMMENT from the collection of \cite{Aoto2013}
Rewrite Rules:
   [ a(?x) -> g(b(?x)),
     a(?x) -> c(?x),
     b(?x) -> g(b(?x)) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ g(b(?x)) = c(?x) ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ c(?x) = g(b(?x)),
     g(b(?x)) = c(?x) ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <c(?x_1), g(b(?x_1))> by Rules <1, 0> preceded by []
unknown Diagram Decreasing
check Non-Confluence...
obtain 13 rules by 3 steps unfolding
strenghten g(b(?x_1)) and c(?x_1)
strenghten c(a(?x_8)) and c(c(?x_8))
strenghten c(c(?x_1)) and a(c(?x_1))
strenghten g(g(b(?x_6))) and c(?x_6)
strenghten a(g(b(?x))) and c(c(?x))
strenghten c(g(b(?x_1))) and a(c(?x_1))
strenghten c(g(b(?x_11))) and c(a(?x_11))
strenghten c(g(b(?x_11))) and c(c(?x_11))
strenghten g(b(a(?x_8))) and c(c(?x_8))
strenghten g(b(c(?x_1))) and a(c(?x_1))
strenghten g(b(c(?x_12))) and c(a(?x_12))
strenghten g(b(c(?x_12))) and c(c(?x_12))
strenghten g(g(b(?x_6))) and g(b(?x_6))
strenghten g(g(g(b(?x_9)))) and c(?x_9)
strenghten c(c(a(?x))) and a(c(c(?x)))
strenghten c(g(b(?x))) and a(g(b(?x)))
strenghten c(g(b(?x_12))) and g(b(c(?x_12)))
strenghten g(b(a(?x_11))) and c(g(b(?x_11)))
strenghten g(b(a(?x_12))) and g(b(c(?x_12)))
strenghten g(b(c(?x))) and a(g(b(?x)))
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (success)
Witness for Non-Confluence: <g(b(c_1)), c(c_1)>
Direct Methods: not CR

Final result: not CR
237.trs: Success(not CR)
(14 msec.)