By Frank Pfenning

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Exp, D has the form D2 D1 `E M2 * exp `E M1 * exp conapp `E app M1 M2 * exp By the induction hypothesis on D1 and D2 , xM1y and xM2y are de ned and therefore xM y = xM1y xM2y is also de ned. Furthermore, pxM yq = pxM1 y xM2 yq = app pxM1 yq pxM2 yq = app M1 M2 , where the last equality follows by the induction hypothesis on M1 and M2 . Subcase: c = letv. Then c has two arguments and, suppressing the premiss E (letv) = exp ! (exp ! exp) ! exp, D has the form D2 D1 `E M2 * exp ! exp `E M1 * exp conapp `E letv M1 M2 * exp There is only one inference rule which could have been used as the last inference in D2 , namely carrow.

Instead, we have to inject the values into a disjoint sum type: pred 0 = lam x: case x of z ) inl h i | s x0 ) inr x0 so that . pred 0 : nat ! (1 | nat) Optional values of type can be modelled in general by using the type (1 | ). 1. Give appropriate rules for evaluation and typing. 2. Extend the notion of value. 3. 1). 4. 5). 7 Consider a language extension ::= : : : j : where is the type of lists whose members have type . 6. 2. 1. De ne x free in e which should hold when the variable x occurs free in e.

Then the last inference rule of any derivation of . e : 0 for some 0 is uniquely determined. Proof: By inspection of the inference rules. 2 Note that this does not imply that types are unique; they are not in this formulation of Mini-ML as remarked above. 6 Type Preservation Before we come to the statement and proof of type preservation in Mini-ML, we need a few preparatory lemmas. The reader may wish to skip ahead and reexamine these lemmas wherever they are needed. We rst note the properties of exchange and weakening and then state and prove a substitution lemma for typing derivations.