Try to understand 3.1-3.4 in this paper, and you'll find that the correctness proof doesn't prove anything.

In particular, when they define <_c, they do this in terms of rule1, rule2, and rule3, but these are defined in terms of <_c, so this is just a circular definition, and therefore actually not a definition at all, but just wishful thinking. They then prove that <_c is a total order, but that proof doesn't matter, because <_c does not exist with the given properties in the first place.