YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
d(0()) -> 0()
d(s(x)) -> s(s(d(x)))
f(0()) -> 0()
f(s(x)) -> +(+(s(x),s(x)),s(x))
f(g(0())) -> +(+(g(0()),g(0())),g(0()))
g(x) -> s(d(x))

CP(R):
f(s(d(0()))) -> +(+(g(0()),g(0())),g(0()))

critical-pair-closing system C:
f(s(x)) -> +(+(s(x),s(x)),s(x))
g(x) -> s(d(x))

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
f(s(x)) -> +(+(s(x),s(x)),s(x))
g(x) -> s(d(x))

CP(R):


critical-pair-closing system C:


CR(C): see below.

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