YES

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

CP(R):
b(b(___y420)) -> b(a(c(a(___y420))))
b(a(c(a(___y425)))) -> b(b(___y425))
a(c(___y435)) -> b(a(c(a(___y435))))
c(___y447) -> c(___y447)

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

CR(C): see below.

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

CP(R):
c(___y465) -> c(___y465)

critical-pair-closing system C:


CR(C): see below.

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