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+(*******************************************************************************
+ * 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