YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
nats() -> :(0(),inc(nats()))
inc(:(x,y)) -> :(s(x),inc(y))
inc(tl(nats())) -> tl(inc(nats()))
hd(:(x,y)) -> x
tl(:(x,y)) -> y
d(:(x,y)) -> :(x,:(x,d(y)))

CP(R):
inc(tl(:(0(),inc(nats())))) -> tl(inc(nats()))

critical-pair-closing system C:
nats() -> :(0(),inc(nats()))
inc(:(x,y)) -> :(s(x),inc(y))
tl(:(x,y)) -> y

CR(C): see below.

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

CP(R):


critical-pair-closing system C:


CR(C): see below.

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