Lemmas, Theorems, Corollaries, and Propositions

I thought I’d attempt to come up with a consistent convention of classifying mathematical statements.  To do so we will simply think of the collection of statements as a partially ordered set where \phi\leq\psi is interpreted as \phi\Rightarrow\psi.  And for each statement/node/point there will be two \mathbb{N}-valued functions that represent the number of incoming and outgoing edges (i.e. immediate antecedents and consequences).  A point will be called a corollary if it has one incoming edge.  A point will be called a theorem if its set of successors is a “large” subset of the poset (even if it is a corollary).  A point will be called a lemma if at least one of its immediate successors is a theorem.  A point will be called an axiom if it has zero incoming edges.  And a point will be called a proposition if it is none of the above.

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