This homework is due on Monday, 7 March.

1. (Trivial) In lecture notes 7, we claimed that the Type Safety
   Theorem followed from the Preservation and Progress Lemmas.  Prove
   that this is the case.

2. (Boring, but important) In lecture notes 7, we proved the
   Preservation Lemma by appealing to the Substitution Lemma.
   Complete the proof by proving the Substitution Lemma.

3. (Trivial) Give a large-step, call-by-name operational semantics for
   the lambda calculus.

4. Prove that the small-step and large-step call-by-value operational
   semantics coincide.  That is, prove:
   a) e => v'   implies  e ->* v'
   b) e ->* v'  implies  e => v'

5. Prove the Normalization Theorem for the call-by-name operational
   semantics.

6. (Trivial) Pierce proves normalization using the following logical
   predicates and lemma:

         E'[int] = { e | |- e : int and e ->* v }
    E'[t1 -> t2] = { e | |- e : t1 -> t2 and e ->* v and
                         All e' in E'[t1]. e e' in E'[t2] }

    Lemma:
    If G |- e : t and g in G[G],
    then g(e) in E'[t].

   Show that Pierce's definitions are equivalent to the ones used in
   lecture notes 8; i.e., prove that forall t, E[t] = E'[t].
   Hint: There is an easy proof and a hard proof.

7. Consider extending the simply-typed lambda calculus with a fix
   combinator:

    e := ... | fix e

    small-step:

    (FixBeta) fix (\x:ta. e') -> e'[fix (\x:ta. e') / x']

                  e -> e'
    (FixArg) -----------------
              fix e -> fix e'

    large-step:

               e => \x:ta. e'   e'[fix (\x:ta. e') / x'] => v
    (FixBeta) ------------------------------------------------
                                 fix e => v

    typing-rule:

           G |- e : t -> t
    (Fix) -----------------
            G |- fix e : t

   Clearly, the extended language is no longer normalizing, as the
   well-typed expression

    D = fix (\x:int. x)

   diverges.  Briefly explain where the proof of the Normalization
   breaks for this extended language.