YES

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

CP(R):


critical-pair-closing system C:
+(0(),y) -> y
+(s(s(x)),y) -> s(s(+(y,x)))

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
+(0(),y) -> y
+(s(s(x)),y) -> s(s(+(y,x)))
CP(R):