NO
(ignored inputs)COMMENT from the collection of \cite{Aoto2013}
Rewrite Rules:
   [ a -> c,
     a -> f(c),
     f(?x) -> h(g(?x)),
     h(?x) -> f(g(?x)) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ c = f(c) ]
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:
   [ f(c) = c,
     c = f(c) ]
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 <f(c), c> by Rules <1, 0> preceded by []
unknown Diagram Decreasing
check Non-Confluence...
obtain 14 rules by 3 steps unfolding
strenghten f(c) and c
strenghten h(g(c)) and c
strenghten f(g(c)) and h(c)
strenghten h(g(c)) and f(c)
strenghten f(g(g(c))) and c
strenghten f(g(a)) and f(g(c))
strenghten f(g(c)) and h(f(c))
strenghten h(g(a)) and h(g(c))
strenghten h(g(c)) and f(f(c))
strenghten f(g(g(c))) and f(c)
strenghten h(g(g(g(c)))) and c
strenghten f(g(g(?x_4))) and h(g(?x_4))
strenghten f(g(g(a))) and h(g(c))
strenghten f(g(g(c))) and h(g(c))
strenghten f(h(g(c))) and h(g(c))
strenghten h(h(g(c))) and f(g(c))
strenghten h(g(g(g(c)))) and f(c)
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Root-Approximation (success)
Witness for Non-Confluence: <f(c), c>
Direct Methods: not CR

Final result: not CR
244.trs: Success(not CR)
(19 msec.)