7 files changed, 658 insertions, 1 deletions
diff --git a/users/wpcarro/scratch/compiler/.gitignore b/users/wpcarro/scratch/compiler/.gitignore
index 25414be2a981..96261d3fc7e9 100644
--- a/users/wpcarro/scratch/compiler/.gitignore
+++ b/users/wpcarro/scratch/compiler/.gitignore
@@ -1,2 +1,5 @@
 a.out
-*.cmi
\ No newline at end of file
+*.cmi
+*.cmo
+*.cmx
+*.o
\ No newline at end of file
diff --git a/users/wpcarro/scratch/compiler/expr_parser.ml b/users/wpcarro/scratch/compiler/expr_parser.ml
new file mode 100644
index 000000000000..7d24efc80de3
--- /dev/null
+++ b/users/wpcarro/scratch/compiler/expr_parser.ml
@@ -0,0 +1,195 @@
+(*******************************************************************************
+ * CLI REPL for an s-expression Lambda Calculus.
+ *
+ * Lambda Calculus Expression Language:
+ *
+ *   Helpers:
+ *     symbol     -> [-a-z]+
+ *     boolean    -> 'true' | 'false'
+ *     integer    -> [1-9][0-9]*
+ *
+ *   Core:
+ *     expression -> funcdef
+ *     binding    -> '(' 'let' symbol expr expr ')'
+ *     funcdef    -> '(' 'fn' symbol expr ')'
+ *     funccall   -> '(' ( symbol | funcdef) expr ')'
+ *     literal    -> boolean | integer
+ *     variable   -> symbol
+ *
+ * Example Usage:
+ *   $ ocamlopt types.ml str.cmxa inference.ml parser.ml expr_parser.ml && ./a.out
+ *   repl> true
+ *   tokens: [ "true" ]
+ *   ast: ValueLiteral (LiteralBool true)
+ *   Boolean
+ *   repl>
+ *
+ ******************************************************************************)
+
+open Parser
+open Inference
+
+let to_array (q : 'a Queue.t) : 'a array =
+  let result = Array.make (Queue.length q) "" in
+  let i = ref 0 in
+  Queue.iter
+    (fun x ->
+      result.(!i) <- x;
+      i := !i + 1)
+    q;
+  result
+
+type literal = LiteralBool of bool | LiteralInt of int
+
+let ( let* ) = Option.bind
+let map = Option.map
+
+let tokenize (x : string) : token array =
+  let q = Queue.create () in
+  let i = ref 0 in
+  while !i < String.length x do
+    match x.[!i] with
+    | ' ' -> i := !i + 1
+    | '(' ->
+        Queue.push "(" q;
+        i := !i + 1
+    | ')' ->
+        Queue.push ")" q;
+        i := !i + 1
+    | _ ->
+        let token = ref "" in
+        while !i < String.length x && not (String.contains "() " x.[!i]) do
+          token := !token ^ String.make 1 x.[!i];
+          i := !i + 1
+        done;
+        Queue.push !token q
+  done;
+  to_array q
+
+let parse_symbol (p : parser) : string option =
+  let* x = p#curr in
+  if Str.string_match (Str.regexp "[-a-z][0-9]*") x 0 then
+    begin
+      p#advance;
+      Some x
+    end
+  else
+    None
+
+let parse_variable (p : parser) : Types.value option =
+  let* x = parse_symbol p in
+  Some (Types.ValueVariable x)
+
+let parse_literal (p : parser) : Types.value option =
+  match p#curr with
+  | Some "true" ->
+     p#advance;
+     Some (ValueLiteral (LiteralBool true))
+  | Some "false" ->
+     p#advance;
+     Some (ValueLiteral (LiteralBool false))
+  | Some x ->
+     (match int_of_string_opt x with
+      | Some n ->
+         p#advance;
+         Some (ValueLiteral (LiteralInt n))
+      | _ -> parse_variable p)
+  | _ -> None
+
+let rec parse_expression (p : parser) : Types.value option =
+  parse_binding p
+
+and parse_funccall (p : parser) : Types.value option =
+  match (p#curr, p#next) with
+  | (Some "(", Some "(") ->
+     p#advance;
+     let* f = parse_funcdef p in
+     let* x = parse_expression p in
+     p#expect ")";
+     Some (Types.ValueApplication (f, x))
+  | (Some "(", _) ->
+     p#advance;
+     let* f = parse_symbol p in
+     let* x = parse_expression p in
+     p#expect ")";
+     Some (Types.ValueVarApplication (f, x))
+  | _ -> parse_literal p
+
+and parse_funcdef (p : parser) : Types.value option =
+  match (p#curr, p#next) with
+  | (Some "(", Some "fn") ->
+     p#advance;
+     p#advance;
+     let* name = parse_symbol p in
+     let* body = parse_expression p in
+     p#expect ")";
+     Some (Types.ValueFunction (name, body))
+  | _ -> parse_funccall p
+
+and parse_binding (p : parser) : Types.value option =
+  match (p#curr, p#next) with
+  | (Some "(", Some "let") ->
+     p#advance;
+     p#advance;
+     let* name = parse_symbol p in
+     let* value = parse_expression p in
+     let* body = parse_expression p in
+     Some (Types.ValueBinder (name, value, body))
+  | _ -> parse_funcdef p
+
+let print_tokens (xs : string array) =
+  xs |> Array.to_list
+  |> List.map (Printf.sprintf "\"%s\"")
+  |> String.concat ", "
+  |> Printf.sprintf "tokens: [ %s ]"
+  |> print_string |> print_newline
+
+let parse_language (x : string) : Types.value option =
+  let tokens = tokenize x in
+  print_tokens tokens;
+  parse_expression (new parser tokens)
+
+let rec debug (ast : Types.value) : string =
+  match ast with
+  | ValueLiteral (LiteralBool x) ->
+     Printf.sprintf "ValueLiteral (LiteralBool %s)" (string_of_bool x)
+  | ValueLiteral (LiteralInt x) ->
+     Printf.sprintf "ValueLiteral (LiteralInt %s)" (string_of_int x)
+  | ValueVariable x ->
+     Printf.sprintf "ValueVariable %s" x
+  | ValueFunction (x, body) ->
+     Printf.sprintf "ValueFunction (%s, %s)" x (debug body)
+  | ValueApplication (f, x) ->
+     Printf.sprintf "ValueApplication (%s, %s)" (debug f) (debug x)
+  | ValueVarApplication (f, x) ->
+     Printf.sprintf "ValueVarApplication (%s, %s)" f (debug x)
+  | ValueBinder (k, v, x) ->
+      Printf.sprintf "ValueBinder (%s, %s, %s)" k (debug v) (debug x)
+
+let debug_ast (ast : Types.value) : Types.value =
+  ast |> debug |> Printf.sprintf "ast: %s" |> print_string |> print_newline;
+  ast
+
+let main =
+  while true do
+    begin
+      print_string "repl> ";
+      let x = read_line () in
+      match parse_language x with
+      | Some ast ->
+         (match ast |> debug_ast |> do_infer with
+          | None ->
+             "Type-check failed"
+             |> print_string
+             |> print_newline
+          | Some x ->
+             x
+             |> Types.pretty
+             |> print_string
+             |> print_newline)
+      | None ->
+         "Could not parse"
+         |> print_string
+         |> print_newline
+    end
+  done
diff --git a/users/wpcarro/scratch/compiler/inference.ml b/users/wpcarro/scratch/compiler/inference.ml
new file mode 100644
index 000000000000..52838f85e649
--- /dev/null
+++ b/users/wpcarro/scratch/compiler/inference.ml
@@ -0,0 +1,177 @@
+(*******************************************************************************
+ * WIP implementation of the Hindley-Milner type system primarily for learning
+ * purposes.
+ *
+ * Wish List:
+ * - TODO Debug this inference (let f (fn x x) f)
+ ******************************************************************************)
+
+open Types
+
+(*******************************************************************************
+ * Library
+ ******************************************************************************)
+
+let ( let* ) = Option.bind
+
+let set_from_list (xs : string list) : set =
+  xs |> List.fold_left (fun acc x -> FromString.add x true acc) FromString.empty
+
+(* Map union that favors the rhs values (i.e. "last writer wins"). *)
+let lww (xs : 'a FromString.t) (ys : 'a FromString.t) : 'a FromString.t =
+  FromString.union (fun k x y -> Some y) xs ys
+
+let emptyEnv : env = FromString.empty
+
+let rec free_type_vars (t : _type) : set =
+  match t with
+  | TypeVariable k -> FromString.singleton k true
+  | TypeInt -> FromString.empty
+  | TypeBool -> FromString.empty
+  | TypeArrow (a, b) -> lww (free_type_vars a) (free_type_vars b)
+
+let i : int ref = ref 0
+
+let make_type_var () : _type =
+  let res = Printf.sprintf "a%d" !i in
+  i := !i + 1;
+  TypeVariable res
+
+exception OccursCheck
+
+let bind_var (k : string) (t : _type) : substitution =
+  if t == TypeVariable k then FromString.empty
+  else if FromString.exists (fun name _ -> name == k) (free_type_vars t) then
+    raise OccursCheck
+  else FromString.singleton k t
+
+let rec instantiate (q : quantified_type) : _type =
+  let (QuantifiedType (names, t)) = q in
+  match t with
+  | TypeInt -> TypeInt
+  | TypeBool -> TypeBool
+  | TypeVariable k ->
+      if List.exists (( == ) k) names then make_type_var () else TypeVariable k
+  | TypeArrow (a, b) ->
+      TypeArrow
+        (instantiate (QuantifiedType (names, a)), instantiate (QuantifiedType (names, b)))
+
+let quantified_type_ftvs (q : quantified_type) : set =
+  let (QuantifiedType (names, t)) = q in
+  lww (free_type_vars t) (names |> set_from_list)
+
+let generalize (env : env) (t : _type) : quantified_type =
+  let envftv =
+    env |> FromString.bindings
+    |> List.map (fun (_, v) -> quantified_type_ftvs v)
+    |> List.fold_left lww FromString.empty
+  in
+  let names =
+    lww (free_type_vars t) envftv
+    |> FromString.bindings
+    |> List.map (fun (k, _) -> k)
+  in
+  QuantifiedType (names, t)
+
+let rec substitute_type (s : substitution) (t : _type) : _type =
+  match t with
+  | TypeVariable k as tvar ->
+     (match FromString.find_opt k s with
+      | Some v -> substitute_type s v
+      | None -> tvar)
+  | TypeArrow (a, b) -> TypeArrow (substitute_type s a, substitute_type s b)
+  | TypeInt -> TypeInt
+  | TypeBool -> TypeBool
+
+let substitute_quantified_type (s : substitution) (q : quantified_type) : quantified_type =
+  let (QuantifiedType (names, t)) = q in
+  let s1 =
+    FromString.filter (fun k v -> List.exists (fun x -> k != x) names) s
+  in
+  QuantifiedType (names, substitute_type s1 t)
+
+let substitute_env (s : substitution) (env : env) : env =
+  FromString.map (fun q -> substitute_quantified_type s q) env
+
+let compose_substitutions (xs : substitution list) : substitution =
+  let do_compose_substitutions s1 s2 = lww s2 (FromString.map (substitute_type s2) s1) in
+  List.fold_left do_compose_substitutions FromString.empty xs
+
+let rec unify (a : _type) (b : _type) : substitution option =
+  match (a, b) with
+  | TypeInt, TypeInt -> Some FromString.empty
+  | TypeBool, TypeBool -> Some FromString.empty
+  | TypeVariable k, _ -> Some (bind_var k b)
+  | _, TypeVariable k -> Some (bind_var k a)
+  | TypeArrow (a, b), TypeArrow (c, d) ->
+      let* s1 = unify a c in
+      let* s2 = unify (substitute_type s1 b) (substitute_type s1 d) in
+      let s3 = compose_substitutions [s1; s2] in
+      s1 |> Types.debug_substitution |> Printf.sprintf "s1: %s\n" |> print_string;
+      s2 |> Types.debug_substitution |> Printf.sprintf "s2: %s\n" |> print_string;
+      s3 |> Types.debug_substitution |> Printf.sprintf "s3: %s\n" |> print_string;
+      Some s3
+  | _ -> None
+
+let print_env (env : env) =
+  Printf.sprintf "env: %s\n" (Types.debug_env env)
+  |> print_string
+
+let print_val (x : value) =
+  Printf.sprintf "val: %s\n" (Types.debug_value x)
+  |> print_string
+
+let print_inference (x : inference option) =
+  match x with
+  | None -> "no inference\n" |> print_string
+  | Some x ->
+     Printf.sprintf "inf: %s\n" (Types.debug_inference x)
+     |> print_string
+
+let rec infer (env : env) (x : value) : inference option =
+  print_env env;
+  print_val x;
+  let res = match x with
+  | ValueLiteral lit -> (
+      match lit with
+      | LiteralInt _ -> Some (Inference (FromString.empty, TypeInt))
+      | LiteralBool _ -> Some (Inference (FromString.empty, TypeBool)))
+  | ValueVariable k ->
+      let* v = FromString.find_opt k env in
+      Some (Inference (FromString.empty, instantiate v))
+  | ValueFunction (param, body) ->
+      let typevar = make_type_var () in
+      let env1 = FromString.remove param env in
+      let env2 = lww (FromString.singleton param (QuantifiedType ([], typevar))) env1 in
+      let* (Inference (s1, t1)) = infer env2 body in
+      Some (Inference (s1, TypeArrow (substitute_type s1 typevar, t1)))
+  | ValueApplication (f, x) ->
+      let result = make_type_var () in
+      let* (Inference (s1, t1)) = infer env f in
+      let* (Inference (s2, t2)) = infer (substitute_env s1 env) x in
+      let* s3 = unify (substitute_type s2 t1) (TypeArrow (t2, result)) in
+      Some (Inference
+              ( compose_substitutions [s3; s2; s1],
+                substitute_type s3 result ))
+  | ValueVarApplication (name, x) ->
+      let* v = FromString.find_opt name env in
+      let t1 = instantiate v in
+      let typevar = make_type_var () in
+      let* (Inference (s2, t2)) = infer env x in
+      let* s3 = unify (substitute_type s2 t1) (TypeArrow (t2, typevar)) in
+      Some (Inference
+              ( compose_substitutions [s2; s3],
+                substitute_type s3 typevar ))
+  | ValueBinder (k, v, body) ->
+      let* (Inference (s1, t1)) = infer env v in
+      let env1 = FromString.remove k env in
+      let tg = generalize (substitute_env s1 env) t1 in
+      let env2 = FromString.add k tg env1 in
+      let* (Inference (s2, t2)) = infer (substitute_env s1 env2) body in
+      Some (Inference (compose_substitutions [s1; s2], t2)) in
+  print_inference res;
+  res
+
+let do_infer (x : value) : _type option =
+  let* Inference (_, t) = infer FromString.empty x in
+  Some t
diff --git a/users/wpcarro/scratch/compiler/parser.ml b/users/wpcarro/scratch/compiler/parser.ml
new file mode 100644
index 000000000000..75cbe04a3f72
--- /dev/null
+++ b/users/wpcarro/scratch/compiler/parser.ml
@@ -0,0 +1,48 @@
+(*******************************************************************************
+ * Defines a generic parser class.
+ ******************************************************************************)
+
+exception ParseError of string
+
+type token = string
+type state = { i : int; tokens : token array }
+
+let get (i : int) (xs : 'a array) : 'a option =
+  if i >= Array.length xs then None else Some xs.(i)
+
+class parser (tokens : token array) =
+  object (self)
+    val mutable tokens : token array = tokens
+    val mutable i = ref 0
+    method print_state = Printf.sprintf "{ i = %d; }" !i
+    method advance = i := !i + 1
+    method prev : token option = get (!i - 1) tokens
+    method curr : token option = get !i tokens
+    method next : token option = get (!i + 1) tokens
+
+    method consume : token option =
+      match self#curr with
+      | None -> None
+      | Some x as res ->
+          self#advance;
+          res
+
+    method expect (x : token) =
+      match self#curr with
+      | Some y when x = y -> self#advance
+      | _ -> raise (ParseError (Printf.sprintf "Expected %s" x))
+
+    method matches (x : token) : bool =
+      match self#curr with
+      | None -> false
+      | Some y ->
+          if x = y then
+            begin
+              self#advance;
+              true
+            end
+          else false
+
+    method exhausted : bool = !i >= Array.length tokens
+    method state : state = { i = !i; tokens }
+  end
diff --git a/users/wpcarro/scratch/compiler/tests.ml b/users/wpcarro/scratch/compiler/tests.ml
new file mode 100644
index 000000000000..828cbd16f090
--- /dev/null
+++ b/users/wpcarro/scratch/compiler/tests.ml
@@ -0,0 +1,43 @@
+open Expr_parser
+open Type_parser
+open Inference
+
+type test = { input : string; expect : string; }
+(* type sub_test = { s1 : string; s2 : string; s3 : string } *)
+
+let ( let* ) = Option.bind
+
+let tests = [
+    { input = "((fn x x) 10)"; expect = "Integer"; };
+    { input = "(let f (fn x x) f)"; expect = "a -> a"; };
+]
+
+(* let sub_tests = [ *)
+(*     { *)
+(*       s1 = "{b |-> b -> Int}"; *)
+(*       s2 = "{a: Bool, b: Int, c: Bool}"; *)
+(*       s3 = "{a: Bool, b: Int -> Int, c: Bool}"; *)
+(*     } *)
+(* ] *)
+
+exception FailedAssertion
+exception TestError
+
+let main =
+  tests
+  |> List.iter (fun { input; expect } ->
+         Printf.sprintf ":t %s == %s\n" input expect |> print_string;
+         match (parse_language input, parse_input expect) with
+         | Some ast, Some expected ->
+            (match do_infer ast with
+             | Some actual ->
+                if actual != expected then
+                  begin
+                    print_type actual;
+                    raise FailedAssertion
+                  end
+                else
+                  print_string "Test passed.\n"
+             | _ -> raise TestError)
+         | _ -> raise TestError);
+  print_string "All tests pass!"
diff --git a/users/wpcarro/scratch/compiler/type_parser.ml b/users/wpcarro/scratch/compiler/type_parser.ml
new file mode 100644
index 000000000000..9b6c0a309309
--- /dev/null
+++ b/users/wpcarro/scratch/compiler/type_parser.ml
@@ -0,0 +1,115 @@
+(******************************************************************************
+ * Type Expression Language:
+ *
+ * Helpers:
+ *   symbol   -> [a-z]
+ *
+ * Core:
+ *   type     -> function
+ *   function -> ( variable | literal ) '->' type
+ *   literal  -> 'Integer' | 'Boolean'
+ *   variable -> symbol
+ ******************************************************************************)
+
+open Types
+open Parser
+open Inference
+
+type side = LHS | RHS
+
+let ( let* ) = Option.bind
+
+let printsub (s : substitution) =
+  FromString.fold (fun k v acc -> Printf.sprintf "%s\"%s\" |-> %s;" acc k (pretty v)) s ""
+  |> Printf.sprintf "Sub { %s }"
+  |> print_string
+  |> print_newline
+
+let to_array (q : 'a Queue.t) : 'a array =
+  let result = Array.make (Queue.length q) "" in
+  let i = ref 0 in
+  Queue.iter
+    (fun x ->
+      result.(!i) <- x;
+      i := !i + 1)
+    q;
+  result
+
+let tokenize (x : string) : token array =
+  let q = Queue.create () in
+  let i = ref 0 in
+  while !i < String.length x do
+    match x.[!i] with
+    | ' ' -> i := !i + 1
+    | _ ->
+       let beg = !i in
+       while (!i < String.length x) && (x.[!i] != ' ') do
+         i := !i + 1
+       done;
+       Queue.push (String.sub x beg (!i - beg)) q
+  done;
+  to_array q
+
+let rec parse_type (p : parser) : _type option =
+  parse_function p
+and parse_function (p : parser) : _type option =
+  match p#next with
+  | Some "->" ->
+     let* a = parse_literal p in
+     p#advance;
+     let* b = parse_type p in
+     Some (TypeArrow (a, b))
+  | _ -> parse_literal p
+and parse_literal (p : parser) : _type option =
+  match p#curr with
+  | Some "Integer" | Some "Int" -> p#advance; Some TypeInt
+  | Some "Boolean" | Some "Bool" -> p#advance; Some TypeBool
+  | Some _ -> parse_variable p
+  | None -> None
+and parse_variable (p : parser) : _type option =
+  match p#curr with
+  | Some x when String.length x = 1 -> p#advance; Some (TypeVariable x)
+  | _ -> None
+
+let print_tokens (xs : string array) =
+  xs
+  |> Array.to_list
+  |> List.map (Printf.sprintf "\"%s\"")
+  |> String.concat ", "
+  |> Printf.sprintf "tokens: [ %s ]"
+  |> print_string |> print_newline
+
+let print_type (t : _type) =
+  t |> pretty |> Printf.sprintf "type: %s" |> print_string |> print_newline
+
+let parse_input (x : string) : _type option =
+  let tokens = tokenize x in
+  print_tokens tokens;
+  parse_type (new parser tokens)
+
+(* Continually prompt until user provides a parseable type expression *)
+let rec read_type (arg : side) : _type =
+  let prompt = match arg with
+    | LHS -> "lhs> "
+    | RHS -> "rhs> " in
+  print_string prompt;
+  let x = read_line () in
+  match parse_input x with
+  | None ->
+     print_string "Failed to parse input.\n";
+     read_type arg
+  | Some ast ->
+     print_type ast;
+     ast
+
+let main =
+  while true do
+    begin
+      let lhs = read_type LHS in
+      let rhs = read_type RHS in
+      match unify lhs rhs with
+      | None ->
+         Printf.printf "Cannot unify \"%s\" with \"%s\"\n" (pretty lhs) (pretty rhs)
+      | Some x -> printsub x
+    end
+  done
diff --git a/users/wpcarro/scratch/compiler/types.ml b/users/wpcarro/scratch/compiler/types.ml
new file mode 100644
index 000000000000..99f35cd5a57d
--- /dev/null
+++ b/users/wpcarro/scratch/compiler/types.ml
@@ -0,0 +1,76 @@
+type literal = LiteralInt of int | LiteralBool of bool
+
+(* Lambda Calculus definition *)
+type value =
+  | ValueLiteral of literal
+  | ValueVariable of string
+  | ValueFunction of string * value
+  | ValueApplication of value * value
+  | ValueVarApplication of string * value
+  | ValueBinder of string * value * value
+
+let rec debug_value (x : value) : string =
+  match x with
+  | ValueLiteral (LiteralInt x) ->
+     Printf.sprintf "Int %d" x
+  | ValueLiteral (LiteralBool x) ->
+     Printf.sprintf "Bool %b" x
+  | ValueVariable x ->
+     Printf.sprintf "Var %s" x
+  | ValueFunction (name, x) ->
+     Printf.sprintf "Fn %s %s" name (debug_value x)
+  | ValueApplication (f, x) ->
+     Printf.sprintf "App %s %s" (debug_value f) (debug_value x)
+  | ValueVarApplication (name, x) ->
+     Printf.sprintf "App %s %s" name (debug_value x)
+  | ValueBinder (name, x, body) ->
+     Printf.sprintf "Bind %s %s %s" name (debug_value x) (debug_value body)
+
+module FromString = Map.Make (String)
+
+type _type =
+  | TypeInt
+  | TypeBool
+  | TypeVariable of string
+  | TypeArrow of _type * _type
+
+let rec debug_type (t : _type) : string =
+  match t with
+  | TypeInt -> "Integer"
+  | TypeBool -> "Boolean"
+  | TypeVariable k -> Printf.sprintf "%s" k
+  | TypeArrow (a, b) -> Printf.sprintf "%s -> %s" (debug_type a) (debug_type b)
+
+type quantified_type = QuantifiedType of string list * _type
+
+let debug_quantified_type (q : quantified_type) : string =
+  let QuantifiedType (vars, t) = q in
+  if List.length vars == 0 then
+    Printf.sprintf "%s" (debug_type t)
+  else
+    Printf.sprintf "forall %s. %s" (String.concat "," vars) (debug_type t)
+
+type set = bool FromString.t
+type substitution = _type FromString.t
+
+let debug_substitution (s : substitution) : string =
+  FromString.fold (fun k v acc -> Printf.sprintf "%s\"%s\" |-> %s;" acc k (debug_type v)) s ""
+  |> Printf.sprintf "{ %s }"
+
+type env = quantified_type FromString.t
+
+let debug_env (s : env) : string =
+  FromString.fold (fun k v acc -> Printf.sprintf "%s\"%s\" |-> %s;" acc k (debug_quantified_type v)) s ""
+  |> Printf.sprintf "{ %s }"
+
+type inference = Inference of substitution * _type
+
+let debug_inference (Inference (s, t)) =
+  Printf.sprintf "type: %s; sub: %s" (debug_type t) (debug_substitution s)
+
+let rec pretty (t : _type) : string =
+  match t with
+  | TypeInt -> "Integer"
+  | TypeBool -> "Boolean"
+  | TypeVariable k -> Printf.sprintf "%s" k
+  | TypeArrow (a, b) -> Printf.sprintf "%s -> %s" (pretty a) (pretty b)