I'm pretty far into "Concrete Semantics With Isabelle HOL" and I've come to a section on complete lattices. Their definition of "complete lattice" goes like this:
A type $'a$ with a partial order $\leq$ is a complete lattice if every set $S ::\; 'a\; set$ has a greatest lower bound $\sqcap S ::\; 'a$
By "a type 'a with a partial order $\leq$", they mean that the values of 'a (not the type hierarchy) form a partial order. The definition of partial order is conventional (ie reflexive, anti-symmetric, transitive).
The greatest lower bound $\sqcap S$ is defined as:
$\forall s \in S. \sqcap S \leq s$
If $\forall s \in S. l \leq s$ then $l \leq \sqcap S$
I am far from understanding this but, to me, this looks like a partial lattice, not complete. In particular, I don't understand how this is supposed to provide a glb (I can see how this produces an lub). The following exercise from the book sums up my confusion:
Where is the confusion in the following argument? The natural numbers form a complete lattice because any set of natural numbers has an infimum, its least element.
I know that the naturals do not form a complete lattice - e.g. ${x > 3}$ has no glb. And I assume that the answer to this question is that the empty set doesn't work but I might be wrong since it seems fine by their definition to define that {0} is the glb of {}.
Extra points for an answer that tells me how to prove the existence of the lub given just that glb (ie why is the lattice complete, which I always defined as having an lub AND a glb?). I know from other books that this is the key takeaway from this theorem and I figure its about time I understood how.
