Some Facts about FOb1 (and hence Ob1) Note, in the following E is an environment, J is a judgement, a is a term, and A is a type. Lemma (environments must be well-formed) If E |- J, then E |- <>. Lemma (types are independent of environments in FOb1) E |- A if and only if {} |- A and E |- <>. Corollary. In FOb1: If E |- A and E' |- <>, then E' |- A. If E |- A and E,E' |- <>, then E,E' |- A. Lemma (domains of well-formed environments are disjoint) E,E' |- <> if and only if E |- <>, E' |- <>, and dom(E) and dom(E') are disjoint. Lemma (free variables of well-typed terms are bound in the environment) If E |- a:A, then FV(a) is a subset of dom(E). Lemma (environment extension or weakening) If E |- a:A and E,E' |- <>, then E,E' |- a:A.