YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
F(x,y) -> c(A())
G(x) -> x
h(x) -> c(x)

CP(R):


critical-pair-closing system C:
F(x,y) -> c(A())
h(x) -> c(x)

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
F(x,y) -> c(A())
h(x) -> c(x)
CP(R):