YES

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

CP(R):
-(0()) -> 0()

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

CR(C): see below.

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