YES

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

CP(R):
f(a'(),a()) -> b()
f(a(),a'()) -> b()
f(a'(),a'()) -> f(a'(),a'())
f(a'(),a'()) -> b()
f(a'(),a'()) -> f(a'(),a'())
f(a'(),a'()) -> b()
b() -> f(a'(),a'())
b() -> f(a'(),a'())

critical-pair-closing system C:
f(a(),a()) -> b()
a() -> a'()
f(a'(),a'()) -> b()
b() -> f(a'(),a'())

CR(C): see below.

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

CP(R):
f(a'(),a()) -> b()
f(a(),a'()) -> b()

critical-pair-closing system C:
a() -> a'()
f(a'(),a'()) -> b()

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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