+
+Booleans, Church numbers, and Church lists in OCaml
+---------------------------------------------------
+
+These questions adapted from web materials written by some smart dude named Acar.
+The idea is to get booleans, Church numbers, "Church" lists, and
+binary trees working in OCaml.
+
+ Recall from class System F, or the polymorphic λ-calculus.
+
+ τ ::= 'α | τ1 → τ2 | ∀'α. τ | c
+ e ::= x | λx:τ. e | e1 e2 | Λ'α. e | e [τ ]
+
+ Recall that bool may be encoded as follows:
+
+ bool := ∀α. α → α → α
+ true := Λα. λt:α. λf :α. t
+ false := Λα. λt:α. λf :α. f
+
+ (where τ indicates the type of e1 and e2)
+
+ Note that each of the following terms, when applied to the
+ appropriate arguments, return a result of type bool.
+
+ (a) the term not that takes an argument of type bool and computes its negation;
+ (b) the term and that takes two arguments of type bool and computes their conjunction;
+ (c) the term or that takes two arguments of type bool and computes their disjunction.
+
+ The type nat (for "natural number") may be encoded as follows:
+
+ nat := ∀α. α → (α → α) → α
+ zero := Λα. λz:α. λs:α → α. z
+ succ := λn:nat. Λα. λz:α. λs:α → α. s (n [α] z s)
+
+ A nat n is defined by what it can do, which is to compute a function iterated n times. In the polymorphic
+ encoding above, the result of that iteration can be any type α, as long as you have a base element z : α and
+ a function s : α → α.
+
+ **Excercise**: get booleans and Church numbers working in OCaml,
+ including OCaml versions of bool, true, false, zero, succ, and pred.
+ It's especially useful to do a version of pred, starting with one
+ of the (untyped) versions available in the lambda library
+ accessible from the main wiki page. The point of the excercise
+ is to do these things on your own, so avoid using the built-in
+ OCaml booleans and list predicates.
+
+ Consider the following list type:
+
+ type ’a list = Nil | Cons of ’a * ’a list
+
+ We can encode τ lists, lists of elements of type τ as follows:
+
+ τ list := ∀α. α → (τ → α → α) → α
+ nilτ := Λα. λn:α. λc:τ → α → α. n
+ makeListτ := λh:τ. λt:τ list. Λα. λn:α. λc:τ → α → α. c h (t [α] n c)
+
+ As with nats, recursion is built into the datatype.
+
+ We can write functions like head, isNil, and map:
+
+ map : (σ → τ ) → σ list → τ list
+
+ We've given you the type for map, you only need to give the term.
+
+ With regard to `head`, think about what value to give back if the
+ argument is the empty list. Ultimately, we might want to make use
+ of our `'a option` technique, but for this assignment, just pick a
+ strategy, no matter how clunky.
+
+ Please provide both the terms and the types for each item.