YES

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

CP(R):
0() -> 0()
s(___y10) -> s(+(0(),___y10))
___y11 -> +(___y11,0())
0() -> 0()
s(___y19) -> s(+(___y19,0()))
___y24 -> +(0(),___y24)
s(+(___y30,0())) -> s(___y30)
s(+(___y37,s(___y40))) -> s(+(s(___y37),___y40))
s(+(___y41,___y42)) -> +(___y42,s(___y41))
s(+(0(),___y46)) -> s(___y46)
s(+(s(___y53),___y52)) -> s(+(___y53,s(___y52)))
s(+(___y59,___y60)) -> +(s(___y60),___y59)
+(___y64,0()) -> ___y64
+(0(),___y66) -> ___y66
+(___y70,s(___y71)) -> s(+(___y71,___y70))
+(s(___y76),___y73) -> s(+(___y73,___y76))

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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
+(0(),y) -> y
+(x,0()) -> x
+(s(x),y) -> s(+(x,y))
+(x,s(y)) -> s(+(x,y))
+(x,y) -> +(y,x)
CP(R):
0() -> 0()
s(___y254) -> s(+(0(),___y254))
___y255 -> +(___y255,0())
0() -> 0()
s(___y263) -> s(+(___y263,0()))
___y268 -> +(0(),___y268)
s(+(___y274,0())) -> s(___y274)
s(+(___y281,s(___y284))) -> s(+(s(___y281),___y284))
s(+(___y285,___y286)) -> +(___y286,s(___y285))
s(+(0(),___y290)) -> s(___y290)
s(+(s(___y297),___y296)) -> s(+(___y297,s(___y296)))
s(+(___y303,___y304)) -> +(s(___y304),___y303)
+(___y308,0()) -> ___y308
+(0(),___y310) -> ___y310
+(___y314,s(___y315)) -> s(+(___y315,___y314))
+(s(___y320),___y317) -> s(+(___y317,___y320))