Harvard School of Engineering and Applied Sciences — CS 152: Programming Languages
Simply-typed lambda calculus; More types (Lectures 11-12)
Section and Practice Problems
Week 7: Tue Mar 12–Fri Mar 15, 2019
1 Simply-typed lambda calculus
(a) Add appropriate type annotations to the following expressions, and state the type of the expression.
(i) λa. a + 4
(ii) λf. 3 + f()
(iii) (λx. x) (λf. f (f42))
(b) For each of the following expressions, give a derivation showing that the expression is well typed.
(i) (λf :int →int. f 38) (λa:int. a + 4)
(ii) λg :(int →int)→(int →int). g (λc:int. c + 1) 7
(iii) λf :int →int. λg :int →int. λx :int. g (f x)
2 Type soundness
(a) Recall the substitution lemma that we used in the proof of type soundness.
Lemma (Substitution).If x:τ0`e:τand `v:τ0then `e{v/x}:τ.
Using the definition of substitution given in Assignment 2, prove this lemma. You may assume that v
does not have any free variables (i.e., FV (v) = ∅).
Remember to state what set you are performing induction on and what the property is that you are
proving for every element in that set. If you are not sure what cases you need to consider, or what
you are able to assume in each case of the inductive proof, we strongly suggest that you write down
the inductive reasoning principle for the inductively defined set.
(b) Recall the context lemma that we used in the proof of type soundness.
Lemma (Context).If `E[e0]:τand `e0:τ0and `e1:τ0then `E[e1]:τ.
Prove this lemma.
Remember to state what set you are performing induction on and what the property is that you are
proving for every element in that set. If you are not sure what cases you need to consider, or what
you are able to assume in each case of the inductive proof, we strongly suggest that you write down
the inductive reasoning principle for the inductively defined set.
