YES

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

CP(R):
0() -> 0()
s(___y10) -> s(+(0(),___y10))
s(+(___y18,0())) -> s(___y18)
s(+(___y21,s(___y24))) -> s(+(s(___y21),___y24))
0() -> 0()
s(___y28) -> s(+(___y28,0()))
s(+(0(),___y36)) -> s(___y36)
s(+(s(___y40),___y39)) -> s(+(___y40,s(___y39)))

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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
+(0(),y) -> y
+(s(x),y) -> s(+(x,y))
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
CP(R):
0() -> 0()
s(___y58) -> s(+(0(),___y58))
s(+(___y66,0())) -> s(___y66)
s(+(___y69,s(___y72))) -> s(+(s(___y69),___y72))
0() -> 0()
s(___y76) -> s(+(___y76,0()))
s(+(0(),___y84)) -> s(___y84)
s(+(s(___y88),___y87)) -> s(+(___y88,s(___y87)))