type t =
  | Var of Id.t
  | Lit of int
  | Add of t * t
  | Let of t * bound1
  | Hole of t Id.subst

and bound1 = B of Id.t * t

let rec pp ff = function
  | Var x ->
     Id.pp ff x
  | Lit n ->
     Format.fprintf ff "%d" n
  | Add (e1, e2) ->
     Format.fprintf ff "add(@[@[%a@],@ @[%a@]@])"
       pp e1
       pp e2
  | Let (e1, e2) ->
     Format.fprintf ff "let(@[@[%a@],@ @[%a@]@])"
       pp e1
       pp_bound1 e2
  | Hole s ->
     Format.fprintf ff "[]{@[%a@]}"
       pp_substitution s

and pp_bound1 ff (B (x, e))  =
  Format.fprintf ff "@[%a.@,@[%a@]@]"
    Id.pp x
    pp e

and pp_substitution ff xs =
  Fmt.list ~sep:Fmt.comma
    (fun ff (x, e) ->
      Format.fprintf ff "%a\\@[%a@]" Id.pp x pp e)
    ff xs

module Builder = struct
  let v x = Var x
  let l i = Lit i
  let ( + ) e1 e2 = Add (e1, e2)
  let ( let* ) e1 mk_e2 = let x = Id.fresh () in Let (e1, B (x, mk_e2 (Var x)))
  let h = Hole []
end

let rec subst s = function
  | Var x ->
     begin try Id.assoc x s with Not_found -> Var x end
  | Lit k ->
     Lit k
  | Add (e1, e2) ->
     Add (subst s e1, subst s e2)
  | Let (e1, e2) ->
     Let (subst s e1, subst_bound1 s e2)
  | Hole s' ->
     Hole (compose s' s)

and subst_bound1 s (B (x, e)) =
  let x' = Id.fresh () in
  B (x', subst (Id.extend x (Var x') s) e)

and compose s1 s2 =
  match s1 with
  | [] -> s2
  | (x, t) :: s1 -> (x, subst s2 t) :: compose s1 s2

let rec equal e e' =
  match e, e' with
  | Var x, Var x' ->
     x = x'
  | Lit i, Lit i' ->
     i = i'
  | Add (e1, e2), Add (e1', e2') ->
     equal e1 e1' && equal e2 e2'
  | Let (e1, e2), Let (e1', e2') ->
     equal e1 e1' && equal_bound1 e2 e2'
  | Hole sub1, Hole sub2 ->
     equal_subst sub1 sub2
  | _ ->
     false

and equal_bound1 (B (x, e)) (B (x', e')) =
  let y = Id.fresh () in
  equal
    (subst Id.(singleton x (Var y)) e)
    (subst Id.(singleton x' (Var y)) e')

and equal_subst sub1 sub2 = Id.equal_subst equal sub1 sub2

let%expect_test "e1" =
  Id.reset ();
  let print e1 e2 =
    Format.printf "@[@[%a@]@ = @[%a@]: %b@]@."
      pp e1
      pp e2
      (equal e1 e2)
  in
  let open Builder in
  print (let* x = l 1 in x) (let* y = l 1 in y);
  [%expect {| let(1, _x2._x2) = let(1, _x1._x1): true |}];
  print (let* x = l 1 in x) (let* y = l 2 in y);
  [%expect {| let(1, _x5._x5) = let(2, _x4._x4): false |}];
  print
    (let* x = l 1 in let* k = l 2 in x + k)
    (let* y = l 1 in let* k = l 2 in y + k);
  [%expect {|
    let(1, _x8.let(2, _x9.add(_x8, _x9)))
    = let(1, _x6.let(2, _x7.add(_x6, _x7))): true |}];
  print
    (let* x = l 1 in let* k = l 2 in x + k)
    (let* y = l 1 in let* k = l 2 in k + y);
  [%expect {|
    let(1, _x16.let(2, _x17.add(_x16, _x17)))
    = let(1, _x14.let(2, _x15.add(_x15, _x14))): false |}];;

let close_with (B (x, e)) e' = subst Id.(singleton x e') e

let rec plug k e =
  match k with
  | (Var _ | Lit _) as k -> k
  | Add (k1, k2) -> Add (plug k1 e, plug k2 e)
  | Let (k1, k2) -> Let (plug k1 e, plug_bound1 k2 e)
  | Hole s -> subst s e

and plug_bound1 (B (x, k)) e = B (x, plug k e)

let beta0 : t -> t option
  = function
  | Add (Lit k1, Lit k2) ->
     Some (Lit (k1 + k2))
  | Let (e1, e2) ->
     Some (close_with e2 e1)
  | _ ->
     None

let wrap f (k, e) = (f k, e)

let rec decompose t =
  Seq.cons (Hole [], t) (decompose_subterms t)

and decompose_subterms = function
  | Var _ | Lit _ | Hole _ ->
     Seq.empty
  | Add (e1, e2) ->
     Seq.interleave
       (decompose e1 |> Seq.map (wrap (fun k -> Add (k, e2))))
       (decompose e2 |> Seq.map (wrap (fun k -> Add (e1, k))))
  | Let (e1, B (x, e2)) ->
     Seq.interleave
       (decompose e1 |> Seq.map (wrap (fun k -> Let (k, B (x, e2)))))
       (decompose e2 |> Seq.map (wrap (fun k -> Let (e1, B (x, k)))))

let rewrite e =
  decompose e
  |> Seq.filter_map (fun (k, e) -> beta0 e |> Option.map (fun e -> (k, e)))
  |> Seq.map (fun (k, e) -> plug k e)

(* Test fixtures *)

let e1, e2, e3, e4 =
  Id.reset ();
  let open Builder in
  (l 1 + l 1),
  (let* x = l 1 + l 1 in x),
  (let* x = l 1 + l 1 in l 40 + x),
  (let* x = l 1 + l 1 in let* y = x + l 1 in let* x = l 3 in x + y)

let%expect_test "rew0" =
  Id.reset ();
  let e = Builder.(let* x = l 1 + l 1 in l 28 + l 12 + x) in
  Format.printf "%a@." pp e;
  [%expect {| let(add(1, 1), _x1.add(add(28, 12), _x1)) |}];
  Format.printf "@[@[%a@]@ -> @[<hv 2>%a@]@]@."
    pp e
    Fmt.(list ~sep:comma pp) (rewrite e |> List.of_seq);
  [%expect {|
    let(add(1, 1), _x1.add(add(28, 12), _x1))
    -> add(add(28, 12), add(1, 1)),
         let(2, _x1.add(add(28, 12), _x1)),
         let(add(1, 1), _x1.add(40, _x1)) |}]