YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
f(a(),x) -> f(a(),g(x))
a() -> b()
g(x) -> x

CP(R):
f(b(),___y78) -> f(a(),g(___y78))

critical-pair-closing system C:
a() -> b()
g(x) -> x

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
a() -> b()
g(x) -> x

CP(R):


critical-pair-closing system C:


CR(C): see below.

---- CR(R)
---- R is empty.