YES

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

CP(R):
f(c(),b()) -> d()
f(a(),c()) -> d()
f(c(),c()) -> f(c(),c())
f(c(),c()) -> f(c(),c())
f(a(),c()) -> f(c(),b())
f(c(),b()) -> f(a(),c())

critical-pair-closing system C:
a() -> c()
f(a(),b()) -> d()
f(c(),x) -> f(c(),c())
d() -> f(c(),b())

CR(C): see below.

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