YES

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

CP(R):
p(___y79) -> p(___y79)

critical-pair-closing system C:
foo(0(x)) -> 0(s(p(p(p(s(s(s(p(s(x))))))))))
bar(0(x)) -> 0(p(s(s(s(x)))))
p(p(s(x))) -> p(x)
p(0(x)) -> 0(s(s(s(s(x)))))

CR(C): see below.

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