YES

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

CP(R):


critical-pair-closing system C:
a() -> f(c())

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
a() -> f(c())
CP(R):