(* CIS341 HW1 --------------------------------------------------------------- *)
(*
   hellocaml.ml

   This is an _individual_ project -- please complete all of the work yourself!
*)

(*
  GOAL: get up to speed with OCaml; write a simple arithmetic interpreter
 
  This tutorial-style project refers to parts of the book "Introduction to 
  Objective Caml", by Jason Hickey (available on the couse web pages).
  It provides a very good reference for learning OCaml.  In the problems below
  when you see a note like "See IOC 5.2" please refer to that section of the
  book.
 
  This project covers most of the content of the first five chapters of IOC,
  and parts of the sixth and ninth chapters.  You might want to skim through 
  this material first; parts of the project below will point you to specific 
  sections of the book.
  
  Please also take some time to skim the other available resources on the course
  homepage -- there are links to the standard libraries, and on-line tutorials. 
  It is not possible to cover all of OCaml's features in just a couple days,
  so we expect you to be able to learn most of what you need on your own.
 
  Also, please feel ask a course staff member for help if you get stuck.
  Learning OCaml is essential for your successful completion of later projects, 
  but it is not supposed to be too onerous in itself!
*)

(*
  If you are using VSCode, please see the instructions on the course web
  page about how to get started.  In particular, you should start a 
  "sandboxed terminal" and then run the command `make dev` while working on 
  this file.
*)

(*
   Files in this project:

   bin/hellocaml.ml 
     -- (this file) the project file you will need to modify
   bin/main.ml
     -- the main executable for this project

   util/assert.ml   
     -- a unit test framework used throughout the course
   
   test/gradedtests.ml   
     -- some tests that we will run to assess the correctness of your homework
     -- our test suite includes more tests that those given here
                       
   test/studenttests.ml 
     -- tests that you write to help with debugging and for a grade 
        (as requested below)
   
*)

(******************************************************************************)
(*                                                                            *)
(* PART 1: OCaml Basics                                                       *)
(*                                                                            *)
(******************************************************************************)

(* OCaml comments are written using '(*' and '*)' delimiters.  They nest.     *)

(*
  OCaml is strongly typed and provides a standard suite of base types:
    int  -  31 bit integers 0, 1, (-1), etc.
    int64 - 64 bit integers 0l, 1l, (-1l), etc.
    bool -  'true' and 'false'
    unit -  ()   a 'trivial' type that has only one member, namely ()
    string - "a string" or "another"
    float, char, etc. will also crop up occasionally
  For more details, see IOC Chapter 2, pay particular attention to 2.4 about
  typechecking.
*)

(*
   OCaml is an expression-oriented, functional language.  This mean you typically
   don't do imperative updates to variables as you do in languages like C, Java, etc.

   Instead, you 'name' computations using the keyword 'let':
*)

let an_int : int =  8
let another_int : int = 3 * 14

(*
   The  ': int' part of the 'let' bindings above is an type ascription -- these
   are optional in OCaml programs (type inference will figure them out), but it
   is good style to use type ascriptions on the top-level definitions you make
   because it will improve the error messages that the OCaml typechecker
   generates when you make a mistake.

   For example, it is an error to add a float and an int:
*)
(* Uncomment to get a type error: *)
(*
   let an_error : int = 3 + 1.0
*) 

(*
  'let' expressions can be nested.  The scope of a let-bound variable is 
  delimited by the 'in' keyword.
  See also IOC Chapter 3:
*)

(* bind z to the value 39 *)
let z : int =
  let x = 3 in
  (* x is in scope here *)
  let y = x + x in
  (* x and y are both in scope here *)  
  (y * y) + x


(*
   Scoping is sometimes easier to see by writing (optional) 'begin'-'end'
   delimiters.  This is equivalent to the binding for z above.
*)
(* bind z to the value 39 *)
let z : int =
  let x = 3 in begin
    let y = x + x in begin
      (y * y) + x
    end
  end

(*
   Here and elsewhere 'begin'-'end' are treated exactly the same as parentheses:
   (but it is bad format to abuse that fact).
*)
(* bind z to the value 39 *)
let z : int =
  let x = 3 in
  let y = x + x in
  begin y * y end + x

(*
  Once bound by a 'let', binding between a variable (like 'z' above) and its
  value (like 39) never changes.  Variables bindings can be shadowed, though.
  each subsequent definition of 'z' above 'shadows' the previous one.
*)

(*
   The most important type of values in OCaml is the function type.
   Function types are written like 'int -> int', which is the type of a function
   that takes an int argument and produces an int result.
   See IOC Chapter 3.1

   Functions values are introduced using the 'fun' keyword and the '->' sytnax:
    fun (x:int) -> x + x    (* a function that takes an int and doubles it *)

   Functions are first class -- they can passed around just like integers or
   other primitive data.
*)

(* bind the variable 'double' of type 'int -> int' to a function: *)
let double : int -> int = fun (x : int) -> x + x

(*
  Functions are called or 'applied' by juxtaposition -- the space ' ' 
  between a function name and its arguments is the function application site. 
  Unlike Java or C, no parentheses are needed, exept for grouping and 
  precedence:
*)
let doubled_z : int = double z (* call double on z  *)

let quadrupled_z : int = double (double z) (* parens needed for grouping *)

let sextupled_z : int = quadrupled_z + double z

(*
  Functions with more than one argument have types like:
  'int -> int -> int', which is the type of a function that takes an int
  and returns a function that itself takes an int and returns an int.
  i.e. 'int -> int -> int' is just 'int -> (int -> int)'
*)

let mult : int -> int -> int = fun (x : int) (y : int) -> x * y
let squared_z : int = mult z z (* multiply z times z *)

(*
  Because functions like 'mult' above return functions, they can be
  partially applied: 
*)
let mult_by_3 : int -> int = mult 3 (* partially apply mult to 3 *)

let mult_by_4 : int -> int = mult 4 (* partially apply mult to 4 *)

let meaning_of_life : int = mult_by_3 14
(* call the partially applied function *)

let excellent_score : int = mult_by_4 25 (* compute 100 *)

(*
   The let-fun syntax above is a bit heavy, so OCaml provides syntactic sugar
   for abbreviating function definitions, avoiding the need for 'fun' and
   redundant-type annotations.

   For example, we can write double like this:
*)
let double (x : int) : int = x + x (* this definition shadows the earlier one *)

(* and mult like this: *)
let mult (x : int) (y : int) : int = x * y

(* We still call them in the same way as before: *)
let quadrupled_z : int = double (double z) (* parens needed for grouping *)

let mult_by_3 : int -> int = mult 3 (* partially apply mult to 3 *)

(*
   Note the use of type annotations
      let f (arg1:t1) (arg2:t2) ... (argN:tN) : retT = ...
   Defines f, a function of type t1 -> t2 -> ... -> tN -> retT
*)

(* Functions are first-class values, they can be passed to other functions: *)
let twice (f : int -> int) (x : int) : int =
  (* f is a function from ints to ints *)
  f (f x)

let quadrupled_z_again : int = twice double z (* pass double to twice *)

(* OCaml's top-level-loop --------------------------------------------------- *)
(*
 
  You can play around with OCaml programs in the 'top-level-loop'.  This is an
  interactive OCaml session that accepts OCaml expressions on the command line,
  compiles them, runs the resulting bytecode, and then prints out the result.
  The top-level-loop is hard to use once programs get big (and are distributed
  into many source files), but it is a great way to learn the OCaml basics, and 
  for trying out one-file programs like 'hellocaml.ml'.
 
  We recommend using 'utop' -- a featureful toplevel loop that is well 
  integrated with VSCode. 
 
  When you run utop, you get a welcome message and then a prompt:

  utop #
 
  You can type ocaml expressions at the prompt like this:
 
  # 3 + 4;;
  -: int = 7
 
  To quit, use the '#quit' directive:
 
  # #quit;;
 
  Note that, in the top-level-loop (unlike in a source file) top-level def-
  -initions and expressions to be evaluated must be terminated by ';;'.  OCaml 
  will compile and run the result (in this case computing the int 7).
 
  (Restart utop now.)
  You can (re)load .ml files into the top-level loop via the '#use' directive:
 
  # #use "bin/hellocaml.ml";;
  ...
 
  In this case, OCaml behaves as though you entered in all of the contents of 
  the .ml file at the command prompt (except that ';;' is not needed). 
 
  Try it now -- load this "bin/hellocaml.ml" file into the OCaml top level.
  If you succeed, you should see a long list of definitions that this file 
  makes.  (Note that utop, by default, will start in whatever directory you
  run it in; by default, that will probably be the root directory of
  your project.  You can change utop's working directory using the #cd command.)

  Once you '#use' a file, you can then interact with the functions and other 
  values it defines at the top-level.  This can be very useful when playing 
  around with your programs, debugging, or testing functions.  

  Try it now: after using Hellocaml.ml, type  `twice ((+) 2) 4;;` a the 
  prompt.  Utop should respond with:

  - : int = 8

  In many cases are lots of files that you want to load into utop, so we have
  provided a file called `.initocaml`, which sets up utop with all of the 
  code in your project.  (If you make changes you may want to stop and 
  then restart utop).  Rather than being directly imported into the toplevel
  as with  #use, the loaded code will follow the module structure of the 
  project.  For example, the modules Hellocaml, Main, Util.Assert, etc., are
  all part of this project.
 
  Try it now -- restart utop (remember #quit) and then begin typing 
  "Hellocaml.tw...".  Utop will offer code completions at the bottom of the
  terminal, which can be chosen using tab.  For example, you can find 
  the `twice` function we defined earlier.
  
  utop # Hellocaml.tw

*)

(* Compiling ---------------------------------------------------------------- *)
(*
   You can also run your project using the test harness and the 'oatc' file.
   
   If you are using VScode, you can work with the command line in its built-in
   shell.  Note that you can start more than one shell (which is useful to 
   have `make dev` going at the same time as other work).

   We recommend using the command line to build and ineract with your code, and
   will provide you with a suitable Makefile for each assignment.  The Makefile
   includes suitable targets for building, testing, cleaning, and zipping your
   homework files for submission.  Just doing `make` will build the project:

   > make
   dune build
   [[ warnings omitted ]]

   If the project successfully compiles, you should see the `oatc` executable
   in the working directory. For now, `oatc` is a simple wrapper for running
   test cases.  Later projects will turn `oatc` into a (more) fully-fledged 
   compiler.

   `make test` will execute the test harness, which is equivalent to invoking
   `oatc --test` from the command line.

   The test harness is a useful way to determine how much of the assignment you
   have completed.  We will be using (a variant of) this test harness to
   partially automate grading of your project code.  We will also do some
   manual grading of your code, and we may withhold some of the tests cases
   we run -- you should definitely test your projects thoroughly on your own.
   Later in the course we will encourage you (and may even require you) to
   submit test cases so that you can help each other test your implementations.

   Hint: examining the test cases in gradedtests.ml can help you figure out
   the specifications of the code you are supposed to implement.
*)

(* PART 1 Problems ---------------------------------------------------------- *)
(*
   Complete the following definitions as directed -- you can tell when you get
   them right because the unit tests specified in part1_tests of the
   gradedtests.ml file will succeed when you run main with the '--test' flag.

   Note that (fun _ -> failwith "unimplemented") is a function that takes any
   argument and raises a Failure exception -- you will have to replace these
   definitions with other function definitions to pass the tests.
*)

(* Problem 1-1 *)
(*
   The 'pieces' variable below is bound to the wrong value.  Bind it to one that
   makes the first case of part1_tests "Problem 1" succeed. See the
   gradedtests.ml file.
*)
let pieces : int = -1

(* Implement a function cube that takes an int value and produces its cube. *)
let cube : int -> int = fun _ -> failwith "cube unimplemented"

(* Problem 1-2 *)
(*
   Write a function "cents_of" that takes
    q - a number of quarters
    d - a number of dimes
    n - a number of nickels
    p - a number of pennies
   (all numbers non-negative)
   and computes the total value of the coins in cents:
*)
let cents_of : int -> int -> int -> int -> int =
   fun _ -> failwith "cents_of unimplemented"

(* Problem 1-3 *)
(*
   Edit the function argument of the "Student-Provided Problem 3" test in
   test/studenttests.ml so that "case1" passes, given the definition below.  You
   will need to remove the function body starting with failwith and replace it
   with something else.
*)
let prob3_ans : int = 42

(*
  Edit the function argument of the "Student-Provided Problem 3" test in 
  test/studenttests.ml so that "case2" passes, given the definition below:
*)
let prob3_case2 (x : int) : int = prob3_ans - x

(*
  Replace 0 with a literal integer argument in the "Student-Provided Problem 3"
  test in test/studenttest.ml so that "case3" passes, given the definition below:
*)
let prob3_case3 : int =
  let aux = prob3_case2 10 in
  double aux

(*
  In this and later projects, you can add your own test cases to the
  test/studenttests.ml file.  They are automatically run by the test harness
  when you do `make test` or `oatc --test` from the command line.
*)

(******************************************************************************)
(*                                                                            *)
(* PART 2: Tuples, Generics, Pattern Matching                                 *)
(*                                                                            *)
(******************************************************************************)

(*
   NOTE: See IOC 5.2
   Tuples are a built-in aggregate datatype that generalizes
   pairs, triples, etc.   Tuples have types like:
       int * bool * string
   At the value level, tuples are written using ',':
*)

let triple : int * bool * string = (3, true, "some string")

(* NOTE: technically, the parentheses are _optional_, so we could have done: *)
let triple : int * bool * string = 3, true, "some string"

(* Tuples can nest *)
let pair_of_triples : (int * bool * string) * (int * bool * string) =
  (triple, triple)

(*
   IMPORTANT!!  Be sure to learn this!

   You can destruct tuples and most other kinds of data by "pattern-matching".

   Pattern matching is a fundamental concept in OCaml: most
   non-trivial OCaml types are usually "destructed" or "inspected" by
   pattern matching using the 'match-with' notation.  See IOC Chapter 4.

   A "pattern" looks like a value, except that it can have 'holes'
   marked by _ that indicate irrelevant parts of the pattern, and
   binders, indicated by variables.

   Consider:

   begin match exp with
     | pat1 -> case1
     | pat2 -> case2
     ...
   end

   This evaluates exp until it reaches a value.  Then, that value is
   'matched' against the patterns pat1, pat2, etc. until a match is
   found.  When the first match is found, the variables appearing in
   the pattern are bound to the corresponding parts of the value and
   the case associated with the pattern is executed.

   If no match is found, OCaml will raise a Match_failure exception.
   If your patterns are not exhaustive -- i.e. they do not cover
   all of the possible cases, the compiler will issue a (usually very
   helpful) warning.
*)

(*
  Tuples are "generic" or "polymorphic" -- you can create tuples of any 
  datatypes.  See IOC 5.1
 
  The generic parts of types in OCaml are written using tick ' notation:
  as shown in the examples below.  What you might write in Java as
   List<A>  you would write in OCaml as 'a list -- type parameters
  are written in prefix.
  Similarly, Map<A,B> would be written as ('a,'b) map -- multiple
  type parameters use a 'tuple' notation.
*)

(*
   NOTE:
   'a is pronounced "tick a" or "alpha".
   'b is pronounced "tick b" or "beta".
   'c is pronounced "tick c" or "gamma".

   Using Greek letters -- this an ML tradition that dates back to its use 
   for developing formal logics and proof systems.
*)

(*
  TIP: In VSCode you can get the type of an expression by hovering the
  mouse cursor over the program text.  This only works if the program
  successfully compiled.  
 
  If you're using Emacs or Vim with the merlin plugin, you can get the 
  type of a an expression at the cursor too: ^C ^T on Emacs
*)

(* Example pattern matching against tuples: *)
let first_of_three (t : 'a * 'b * 'c) : 'a =
  (* t is a generic triple *)
  match t with x, _, _ -> x

let t1 : int = first_of_three triple (* binds t1 to 3 *)

let second_of_three (t : 'a * 'b * 'c) : 'b = match t with _, x, _ -> x
let t2 : bool = second_of_three triple (* binds t2 to true *)

(*
   This generic function takes an arbitrary input, x, and
   returns a pair both of whose components are the given input:
*)
let pair_up (x : 'a) : 'a * 'a = (x, x)

(* Part 2 Problems ---------------------------------------------------------- *)

(*
   Problem 2-1

   Complete the definition of third_of_three; be sure to give it
   the correct type signature (we will grade that part manually):
*)
let third_of_three _ = failwith "third_of_three unimplemented"

(*
  Problem 2-2
 
  Implement a function compose_pair of the given type that takes 
  a pair of functions and composes them in sequence.  Note that
  you must return a function.  See the test cases in gradedtests.ml
  for examples of its use.
*)

let compose_pair (p : ('b -> 'c) * ('a -> 'b)) : 'a -> 'c =
  failwith "compose_pair unimplemented"

(******************************************************************************)
(*                                                                            *)
(* PART 3: Lists and Recursion                                                *)
(*                                                                            *)
(******************************************************************************)

(*
  OCaml has a built-in datatype of generic lists: See IOC 5.3
 
  [] is the nil list
  if  
    h   is a head-value of type t
    tl  is a list of elements of type t
  then
   h::tl  is a list with h as the head and tl as the tail
 
  `::`  is pronounced "cons", as it constructs a list.
*)

let list1 : int list = [ 3; 2; 1 ]

(* Lists can also be written using the [v1;v2;v3] notation: *)

let list1' = [ 3; 2; 1 ] (* this is equivalent to list1 *)

(* Lists are homogeneous -- they hold values of only one type: *)
(* Uncomment to get a type error; recomment to compile:
   let bad_list = [1;"hello";true]
*)

(*
  As usual in OCaml, we use pattern matching to destruct lists.
  For example, to determine whether a list has length 0 we need to 
  do a case-analysis (via pattern matching) to see whether it is nil
  or is non-empty.  The following function takes a list l and
  determines whether l is empty: 
*)
let is_empty (l : 'a list) : bool =
  match l with [] -> true (* nil case -- return true *) | _ -> false
(* non-nil case -- return false *)

let ans1 : bool = is_empty [] (* evaluates to true *)

let ans2 : bool = is_empty list1 (* evaluates to false *)

(*
  Lists are an example of a "disjoint union" data type -- they are either
  empty [] or have some head value consed on to a tail h::tl.  
  OCaml provides programmers with mechanisms to define their own generic 
  disjoint union and potentially recursive types using the 'type' keyword.
*)

(*
  A user-defined generic type ('a mylist) can be defined within OCaml by:
*)

type 'a mylist =
  | Nil
  (* my version of [] *)
  | Cons of 'a * 'a mylist
(* Cons(h,tl) is my version of h::tl *)

(*
   We build a mylist by using its 'constructors' (specified in the branches)
   For example, compare mylist1 below to list1 defined by built-in lists
   above:
*)
let mylist1 : int mylist = Cons (3, Cons (2, Cons (1, Nil)))

(*
   Pattern matching against a user-defined datatype works the same as for
   a built-in type.  The cases we need to consider in a pattern are given by
   the cases of the type definition.  For example, to write the is_empty
   function for mylist we do the following.  Compare it with is_empty:
*)
let is_mylist_empty (l : 'a mylist) : bool =
  match l with Nil -> true | Cons _ -> false

(*
   IMPORTANT!! Be sure to learn this!

   Recursion:  The built in list type and the mylist type we defined above
   are recursive -- they are defined in terms of themselves.  To implement
   useful functions over such datatypes, we often need to use recursion.

   Recursive functions in OCaml use the 'rec' keyword -- inside the body
   of a recursive function, you can call the function being defined.  As usual
   you must be careful not to introduce infinite loops.

   Recursion plus pattern matching is a very powerful combination.  Here is
   a recursive function that sums the elements of an integer list:
*)

let rec sum (l : int list) : int =
  (* note the 'rec' keyword! *)
  match l with [] -> 0 | x :: xs -> x + sum xs
(* note the recursive call to sum *)

let sum_ans1 : int = sum [ 1; 2; 3 ] (* evaluates to 6 *)

(*
  Here is a function that takes a list and determines whether it is 
  sorted and contains no duplicates according to the built-in 
  generic inequality test <
 
  Note that it uses nested pattern matching to name the
  first two elements of the list in the third case of the match:
*)
let rec is_sorted (l : 'a list) : bool =
  match l with
  | [] -> true
  | [ _ ] -> true
  | h1 :: h2 :: tl -> h1 < h2 && is_sorted (h2 :: tl)

let is_sorted_ans1 : bool = is_sorted [ 1; 2; 3 ] (* true *)

let is_sorted_ans2 : bool = is_sorted [ 1; 3; 2 ] (* false *)

(*
   The List library
   (see http://caml.inria.fr/pub/docs/manual-ocaml/libref/List.html)
   implements many useful functions for list maniuplation.  You will
   recreate some of them in the exercises below.

   Here is map, one of the most useful:
*)
let rec map (f : 'a -> 'b) (l : 'a list) : 'b list =
  match l with [] -> [] | h :: tl -> f h :: map f tl

let map_ans1 : int list = map double [ 1; 2; 3 ] (* evaluates to [2;4;6]  *)

let map_ans2 : (int * int) list = map pair_up [ 1; 2; 3 ]
(* evaluates to [(1,1);(2,2);(3,3)] *)

(*
   The mylist type is isomorphic to the built-in lists.
   The recursive function below converts a mylist to a built-in list.
*)

let rec mylist_to_list (l : 'a mylist) : 'a list =
  match l with Nil -> [] | Cons (h, tl) -> h :: mylist_to_list tl

(* Part 3 Problems ---------------------------------------------------------- *)

(*
  Problem 3-1
 
  Implement list_to_mylist with the type signature below; this is 
  the inverse of the mylist_to_list function given above.
*)
let rec list_to_mylist (l : 'a list) : 'a mylist =
  failwith "list_to_mylist unimplemented"

(*
  Problem 3-2
  
  Implement the library function append, which takes two lists (of the same
  types) and concatenates them.  Do not use the library function or the built-in
  short-hand '@'. 
 
  (append [1;2;3] [4;5]) should evaluate to [1;2;3;4;5]
  (append [] []) should evaluate to []
 
  Note that OCaml provides the infix fuction @ as an alternate way of writing
  append.  So (List.append [1;2] [3]) is the same as  ([1;2] @ [3]).
*)
let rec append (l1 : 'a list) (l2 : 'a list) : 'a list =
  failwith "append unimplemented"

(*
  Problem 3-3
  
  Implement the library function rev, which reverses a list. In this solution,
  you might want to call append.  Do not use the library function.
*)
let rec rev (l : 'a list) : 'a list =
  failwith "rev unimplemented"

(*
  Problem 3-4
 
  Read IOC 5.4 about "tail recursion" and implement a tail recursive version of
  rev.  Note that you will need a helper function that takes an extra parameter
  -- it should be defined using a local let definition.  The rev_t function
  itself should not be recursive. Tail recursion is important to efficiency --
  OCaml will compile a tail recursive function to a simple loop.
*)
let rev_t (l : 'a list) : 'a list =
  failwith "rev_t unimplemented"

(*
  Problem 3-5
 
  Implement insert, a function that, given an element x and a sorted list l
  (i.e. one for which is_sorted returns true) returns the list obtained by 
  inserting x into the list l at the proper location.  Note that if x is 
  already in the list then insert should just return the original list.
 
  You will need to use the if-then-else expression (see IOC 2.2). Remember that
  OCaml is expression-oriented; "if t then e1 else e2" evaluates to either the
  value computed by e1 or the value computed by e2 depending on whether t 
  evaluates to true or false.
*)
let rec insert (x : 'a) (l : 'a list) : 'a list =
  failwith "insert unimplemented"

(*
  Problem 3-6
 
  Implement union, a function that takes two sorted lists and returns the 
  sorted list containing all of the elements from both of the two input lists.  
  Hint: you might want to use the insert function that you just defined.
*)
let rec union (l1 : 'a list) (l2 : 'a list) : 'a list =
   failwith "union unimplemented"

(******************************************************************************)
(*                                                                            *)
(* PART 3: Expression Trees and Interpreters                                  *)
(*                                                                            *)
(******************************************************************************)

(* TERMINOLOGY: "Object" level vs. "Meta" level

   When we implement a compiler, we use code in one programming language to
   implement the features of another language.  The language we are implementing
   is called the "object language" -- it is the "object of study".  In contrast,
   the language we use to implement the object language in is called the
   "meta language" -- it is used to "talk about" the object language.  In this
   course OCaml will usually be the "meta language" (indeed the 'm' and 'l' in
   OCaml come from ML - meta language).

   We will implement several different object languages in this course.  Within
   the metalanguage, we use ordinary datatypes: lists, tuples, trees, unions,etc.
   to represent the features of the object language.  A compiler is just a
   function that translates one representation of an object language into
   another (usually while preserving some notion of a program's 'behavior').

   The representation of a language is often best done using an "abstract syntax
   tree" -- a representation that hides concrete details about parsing,
   infix syntax, syntactic sugar, etc.
*)

(*
  In this course we will be working with many kinds of abstract syntax trees.
  Such trees are a convenient way of representing the syntax of a programming
  language.  In OCaml, we build such datatypes using the technology we have
  already seen above.  
 
  Here is a simple datatype of arithemetic expressions.  To make things slightly
  more interesting, and to familiarize you with another important library, these
  expressions trees denote 64-bit integers.  See the Int64 module of the OCaml
  standard library.  We have to use Int64.add to work with these kinds of numbers.
  (The OCaml notation Int64.add accesses the 'add' function defined in the Int64
  library -- you can use similar notation for List.map, etc.
 
  64-bit literal constants are written with the `L` suffix: 0L, 1L, 2L, etc.
*)

(* An object language: a simple datatype of 64-bit integer expressions *)
type exp =
  | Var of string (* string representing an object-language variable *)
  | Const of int64 (* a constant int64 value -- use the 'L' suffix *)
  | Add of exp * exp (* sum of two expressions *)
  | Mult of exp * exp (* product of two expressions *)
  | Neg of exp (* negation of an expression *)

(*
   An object-language arithmetic expression whose concrete (ASCII) syntax is
   "2 * 3" could be represented like this:
*)
let e1 : exp = Mult (Const 2L, Const 3L) (* "2 * 3" *)

(*
  If the object-level expression contains variables, we represent them as
  strings, like this:
*)
let e2 : exp = Add (Var "x", Const 1L) (* "x + 1" *)

(* Here is a more complex expression that involves multiple variables: *)

let e3 : exp = Mult (Var "y", Mult (e2, Neg e2)) (* "y * ((x+1) * -(x+1))" *)

(*
   Problem 4-1
 
  Implement vars_of -- a function that, given en expression e returns
  a list containing exactly the strings representing variables that appear
  in the expression.  The result should be a set -- that is, it should
  contain no duplicates and be sorted (according to is_sorted). 
  
  For example:
    vars_of e1 should produce []
    vars_of e2 should produce ["x"]
    vars_of e3 should produce ["x";"y"]
 
  Hint: you need to pattern match on the exp e.
  Hint: you probably want to use the 'union' function you wrote for Problem 3-5.
*)
let rec vars_of (e : exp) : string list =
  failwith "vars_of unimplemented"

(*
  How should we _interpret_ (i.e. give meaning to) an expression?
 
   Some examples:
     Add(Const 3L, Const 5L)    should have the interpretation 8L
     Mult(Const 2L, (Add(Const 3L, Const 5L)))   denotes 16L
 
  What about Add(Var "x", Const 1L)?
  What should we do with (Var "x")?  
 
  Each expression denotes an int64 value, but since it can contain variables,
  we need an "evaluation context" that maps variable names to int64 values.  
  
  NOTE: an _evaluation context_ is just a finite map from variables to values.

  One simple (but not particularly efficient) way to represent a context is as
  an "association list" that is just a list of (string * int64) pairs: 
*)

type ctxt = (string * int64) list

(* Here are some example evalution contexts: *)
let ctxt1 : ctxt = [ ("x", 3L) ] (* maps "x" to 3L *)

let ctxt2 : ctxt = [ ("x", 2L); ("y", 7L) ] (* maps "x" to 2L, "y" to 7L *)

(*
  When interpreting an expression, we need to look up the value
  associated with a variable in an evaluation context.  
   For example:
      lookup "x" ctxt1    should yield 3L
      lookup "x" ctxt2    should yield 2L
      lookup "y" cxtx2    should yield 7L
 
  What if we lookup a variable that doesn't appear in the context?  In that 
  case we raise an exception.  OCaml provides user-defined and some pre-
  defined exceptions.  For container-like structures (like ctxt), the 
  Not_found exception is appropriate.
  
  See IOC 9.2.1  (and 9.2 about exception handlers)
  
  To throw an exception Exception just use:
          raise Exception
  For example:
      lookup "y" ctxt1    should raise Not_found
 
  There is one other case to consider -- if the context has two bindings for a
  variable, the one closer to the head of the list should take precedence.  
   For example:
   lookup "x" [("x", 1L);("x", 2L)]    should yield 1L (not 2L)
*)

(*
  Problem 4-2
 
  Implement the lookup function with the signature given below. It should find 
  the int64 value associated with a given string in the ctxt c.  If there is no
  such value, it should raise the Not_found exception.
*)
let rec lookup (x : string) (c : ctxt) : int64 =
  failwith "unimplemented"

(*
   Problem 4-3

   At last, we can write an interpreter for exp's.  This is just a function
   'interpret' that, given an evaluation context c and an expression e, computes
   an int64 value corresponding to the expression.  If the expression mentions a
   variable that does not appear in the context, interpret should throw the
   Not_found exception (note that this amounts to trying to lookup the variable
   and *not* catching any resulting Not_found exception.)

   For example:
      interpret ctxt1 e1    should yield 6L
      interpret ctxt1 e2    should yield 4L
      interpret ctxt1 e3    should raise a Not_found exception

   Note that the interpreter should recursively call itself to obtain the int64
   values of subexpressions.  You will need to use the Int64 library to
   implement addition and multiplication.

   You should test your interpeter on more examples than just those provided in
   gradedtests.ml.
*)

let rec interpret (c : ctxt) (e : exp) : int64 =
  failwith "unimplemented"

(*
  Problem 4-4
 
  Now, write an _optimizer_ for expressions.  An optimizer is just a function 
  that tries to reduce the number of operations by doing some work "at compile
  time" rather than at "run time".    
 
  For example:
     optimize (Add(Const 3L, Const 4L))     might yield (Const 7L)
     optimize (Mult(Const 0L, Var "x"))     might yield (Const 0L)
 
  No matter what optimizations your function performs, it should not change the
  value computed by the expression (if there is one).  That is, for every
  context c and exp e, if c has bindings for all the variables in e, it should
  be the case that: 
 
      (interpret c e)  =  (interpret c (optimize e))
 
  Note that it is not always possible to reduce the size of an expression: 
  (Var "x")  is already "optimal"). You will also have to optimize sub-
  expressions before applying further optimizations -- this means that you will
  need to use nested match expressions.
 
  Here is a (slightly) more complex example:
   optimize (Add(Const 3L, Mult(Const 0L, Var "x")))    yields (Const 3L)
 
  Note that your optimizer won't be complete (unless you work *very* hard)
  for example, 
    Add(Var "x", Neg(Add(Var "x", Neg(Const 1L))))      "x - (x - 1)"
  is equivalent to Const 1l, but we do not expect your optimizer to
  discover that.  Indeed, there are many algebraic laws that would be hard
  to fully exploit.
 
  You can get full credit for this problem if your optimizer:
    (1) doesn't change the meaning of an expression in any context
        that provides bindings for all of the expression's variables
    (2) recursively reduces the size of the expression in "obvious" cases 
          -- adding 0, multiplying by 0 or 1, constants, etc.

  Hint: what simple optimizations can you do with Neg?
*)

let rec optimize (e : exp) : exp =
  failwith "optimize unimplemented"

(******************************************************************************)
(*                                                                            *)
(* PART 4: Compiling the Expression Language to a Stack-Based Language        *)
(*                                                                            *)
(******************************************************************************)

(*
  The interpreter for the expression language above is simple, but the language
  itself is fairly high-level in the sense that we are using many meta-language
  (OCaml) features to define the behavior of the object language.  For example,
  the natural way of defining the 'interpret' function as a recursive OCaml
  function means that we rely on OCaml's function calls and its order of
  evaluation. 
 
  Let's look at a closely-related but "lower-level" language for defining
  arithmetic expressions.
 
  Unlike the language 'exp' defined above, which has nested subexpressions, a
  program in the new language will simply be a list of instructions that specify
  a sequence of actions to carry out. Also unlike the 'exp' language, which uses
  OCaml's recursive functions and hence a (meta-level) call stack to keep track
  of the order in which subexpressions are processed, this new language will  
  explicitly manipulate its own stack of Int64 values.
 
  (ASIDE: historically, several companies actually built calculators that used
   this "reverse polish notation" programming language to express their
   computations.  Google for it to find out more...) 
 *)

(*
   The instructions of the new language are defined as follows. Note that the
   instructions IMul, IAdd, and INeg pop their input values from the stack and
   push their results onto the stack.
*)
type insn =
  | IPushC of int64 (* push an int64 constant onto the stack *)
  | IPushV of string (* push (lookup string ctxt)  onto the stack *)
  | IMul (* multiply the top two values on the stack *)
  | IAdd (* add the top two values on the stack *)
  | INeg
(* negate the top value on the stack *)

(* A stack program is just a list of instructions. *)
type program = insn list

(*
  The stack itself is represented as a list of int64 values, where the head of
  the list is the top of the stack. 
*)
type stack = int64 list

(*
  The operational semantics (i.e. behavior) of this new language can easily be
  specified by saying how one instruction step is performed.  Each step takes a
  stack and returns the new stack resulting from carrying out the computation. 
 
  Note that this function is not recursive, and also that it might raise an
  exception if the stack doesn't have enough entries.  As with the 'exp'
  language interpreter, we use a context to lookup the value of variables.  
*)

let step (c : ctxt) (s : stack) (i : insn) : stack =
  match (i, s) with
  | IPushC n, _ -> n :: s (* push n onto the stack *)
  | IPushV x, _ -> lookup x c :: s (* lookup x, push it *)
  | IMul, v1 :: v2 :: s -> Int64.mul v1 v2 :: s
  | IAdd, v1 :: v2 :: s -> Int64.add v1 v2 :: s
  | INeg, v1 :: s -> Int64.neg v1 :: s
  | _ -> failwith "Stack had too few values"

(*
   To define how a program executes, we simply iterate over the
   instructions, threading the stack through.
*)
let rec execute (c : ctxt) (s : stack) (p : program) : stack =
  match p with
  | [] -> s (* no more instructions to execute *)
  | i :: cont -> execute c (step c s i) cont

(*
   If you want to be slick, you can write the above equivalently using
   List.fold_left (which is tail recrusive and hence iterative):
*)
let execute' (c : ctxt) = List.fold_left (step c)

(*
  Let us define 'answer' to mean the sole value of a stack containing only one
  element.  This makes sense since: if a program left the stack empty, there
  wouldn't be any int64 value computed and if there is more than one value on
  the stack, that means that the program 'stopped' too early.
*)
let answer (s : stack) : int64 =
  match s with [ n ] -> n | _ -> failwith "no answer"

(*
  Finally, we can 'run' a program in a given context by executing it starting
  from the empty stack and returning the answer that the program computes.
 *)
let run (c : ctxt) (p : program) : int64 = answer (execute c [] p)

(*
  As an example program in this stack language, consider the following program
  that runs to produce the answer 6. 
 
  Compare this program to the 'exp' program called e1 above. They both compute
  the value 2 * 3. 
*)
let p1 = [ IPushC 2L; IPushC 3L; IMul ]
let ans1 = run [] p1

(*
  Problem 5
 
  Implement a function 'compile' that takes a program in the 'exp' language and
  translates it to an equivalent program in the 'stack' language.  
 
  For example, (compile e1) should yield the program p1 given above.
   
  Correctness means that: 
  For all expressions e, contexts c (defining the variables in e), and int64
  values v:

     (interpret c e) = v       if and only if
     (run c (compile e)) = v
 
  Hints: 
   - Think about how to define your compiler compositionally so that you build
     up the sequence of instructions for a compound expression like Add(e1, e2)
     from the programs that compute e1 and e2.
 
   - You may want to use the append function (or the built in @) function to
     glue together two programs. 

   - You should test the correctness of your compiler on several examples.
*)
let rec compile (e : exp) : program =
  failwith "compile unimplemented"

(************)
(* Epilogue *)
(************)

(*
   Whew!  That was a whirl-wind tour of OCaml.  There are still quite a few
   features we haven't seen -- we'll mention them in the next project and later
   as needed.  However, even with this little bit of OCaml and a few basic ideas
   you've probably seen in other languages, you can already go quite far.

   Take a quick read through lib/util/assert.ml -- this is a small libary for writing
   unit tests that we'll use for grading throughout this course. It makes
   extensive use of unions, lists, strings, ints, and a few List and Printf
   library functions, but otherwise it should be fairly readable to you already...
*)