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(********************************************************************)
(*                                                                  *)
(*  The Why3 Verification Platform   /   The Why3 Development Team  *)
(*  Copyright 2010-2012   --   INRIA - CNRS - Paris-Sud University  *)
(*                                                                  *)
(*  This software is distributed under the terms of the GNU Lesser  *)
(*  General Public License version 2.1, with the special exception  *)
(*  on linking described in file LICENSE.                           *)
(*                                                                  *)
(********************************************************************)
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open Stdlib
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open Ident
open Ty
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open Term
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(** value types (w/o effects) *)
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module rec T : sig
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  type varset = {
    vars_tv  : Stv.t;
    vars_reg : Reg.S.t;
  }
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  type varmap = varset Mid.t

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  type itysymbol = {
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    its_ts   : tysymbol;
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    its_regs : region list;
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    its_def  : ity option;
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    its_inv  : bool;
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    its_abst : bool;
    its_priv : bool;
  }
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  and ity = {
    ity_node : ity_node;
    ity_vars : varset;
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    ity_tag  : Weakhtbl.tag;
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  }
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  and ity_node =
    | Ityvar of tvsymbol
    | Itypur of tysymbol * ity list
    | Ityapp of itysymbol * ity list * region list
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  and region = {
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    reg_name : ident;
    reg_ity  : ity;
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  }

end = struct

  type varset = {
    vars_tv  : Stv.t;
    vars_reg : Reg.S.t;
  }

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  type varmap = varset Mid.t

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  type itysymbol = {
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    its_ts   : tysymbol;
    its_regs : region list;
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    its_def  : ity option;
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    its_inv  : bool;
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    its_abst : bool;
    its_priv : bool;
  }

  and ity = {
    ity_node : ity_node;
    ity_vars : varset;
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    ity_tag  : Weakhtbl.tag;
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  }

  and ity_node =
    | Ityvar of tvsymbol
    | Itypur of tysymbol * ity list
    | Ityapp of itysymbol * ity list * region list

  and region = {
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    reg_name : ident;
    reg_ity  : ity;
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  }

end

and Reg : sig
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  module M : Extmap.S with type key = T.region
  module S : Extset.S with module M = M
  module H : Exthtbl.S with type key = T.region
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  module W : Weakhtbl.S with type key = T.region
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end = MakeMSHW (struct
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  type t = T.region
  let tag r = r.T.reg_name.id_tag
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end)

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open T

(** regions *)

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module Sreg = Reg.S
module Mreg = Reg.M
module Hreg = Reg.H
module Wreg = Reg.W

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let reg_equal : region -> region -> bool = (==)
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let reg_hash r = id_hash r.reg_name
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let create_region id ty = { reg_name = id_register id; reg_ity = ty }
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(* variable sets *)

let vars_empty = { vars_tv = Stv.empty ; vars_reg = Sreg.empty }

let vars_union s1 s2 = {
  vars_tv  = Stv.union s1.vars_tv s2.vars_tv;
  vars_reg = Sreg.union s1.vars_reg s2.vars_reg;
}

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let vars_merge = Mid.fold (fun _ -> vars_union)

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let create_varset tvs regs = {
  vars_tv = Sreg.fold (fun r -> Stv.union r.reg_ity.ity_vars.vars_tv) regs tvs;
  vars_reg = regs;
}

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(* value type symbols *)
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module Itsym = MakeMSHW (struct
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  type t = itysymbol
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  let tag its = its.its_ts.ts_name.id_tag
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end)

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module Sits = Itsym.S
module Mits = Itsym.M
module Hits = Itsym.H
module Wits = Itsym.W
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let its_equal : itysymbol -> itysymbol -> bool = (==)
let ity_equal : ity       -> ity       -> bool = (==)
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let its_hash its = id_hash its.its_ts.ts_name
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let ity_hash ity = Weakhtbl.tag_hash ity.ity_tag
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module Hsity = Hashcons.Make (struct
  type t = ity
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  let equal ity1 ity2 = match ity1.ity_node, ity2.ity_node with
    | Ityvar n1, Ityvar n2 ->
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        tv_equal n1 n2
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    | Itypur (s1,l1), Itypur (s2,l2) ->
        ts_equal s1 s2 && List.for_all2 ity_equal l1 l2
    | Ityapp (s1,l1,r1), Ityapp (s2,l2,r2) ->
        its_equal s1 s2 && List.for_all2 ity_equal l1 l2
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          && List.for_all2 reg_equal r1 r2
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    | _ ->
        false
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  let hash ity = match ity.ity_node with
    | Ityvar v -> tv_hash v
    | Itypur (s,tl) -> Hashcons.combine_list ity_hash (ts_hash s) tl
    | Ityapp (s,tl,rl) ->
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        Hashcons.combine_list reg_hash
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          (Hashcons.combine_list ity_hash (its_hash s) tl) rl
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  let ity_vars s ity = vars_union s ity.ity_vars
  let reg_vars s r = { s with vars_reg = Sreg.add r s.vars_reg }

  let vars s ity = match ity.ity_node with
    | Ityvar v ->
        { s with vars_tv = Stv.add v s.vars_tv }
    | Itypur (_,tl) ->
        List.fold_left ity_vars s tl
    | Ityapp (_,tl,rl) ->
        List.fold_left reg_vars (List.fold_left ity_vars s tl) rl

  let tag n ity = { ity with
    ity_vars = vars vars_empty ity;
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    ity_tag  = Weakhtbl.create_tag n }
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end)

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module Ity = MakeMSHW (struct
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  type t = ity
  let tag ity = ity.ity_tag
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end)

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module Sity = Ity.S
module Mity = Ity.M
module Hity = Ity.H
module Wity = Ity.W
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let mk_ity n = {
  ity_node = n;
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  ity_vars = vars_empty;
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  ity_tag  = Weakhtbl.dummy_tag;
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}

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let ity_var n = Hsity.hashcons (mk_ity (Ityvar n))
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let ity_pur_unsafe s tl = Hsity.hashcons (mk_ity (Itypur (s,tl)))
let ity_app_unsafe s tl rl = Hsity.hashcons (mk_ity (Ityapp (s,tl,rl)))
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(* generic traversal functions *)

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let ity_map fn ity = match ity.ity_node with
  | Ityvar _ -> ity
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  | Itypur (f,tl) -> ity_pur_unsafe f (List.map fn tl)
  | Ityapp (f,tl,rl) -> ity_app_unsafe f (List.map fn tl) rl
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let ity_fold fn acc ity = match ity.ity_node with
  | Ityvar _ -> acc
  | Itypur (_,tl)
  | Ityapp (_,tl,_) -> List.fold_left fn acc tl
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let ity_all pr ity =
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  try ity_fold (Util.all_fn pr) true ity with Util.FoldSkip -> false
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let ity_any pr ity =
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  try ity_fold (Util.any_fn pr) false ity with Util.FoldSkip -> true
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(* symbol-wise map/fold *)

let rec ity_s_fold fn fts acc ity = match ity.ity_node with
  | Ityvar _ -> acc
  | Itypur (ts, tl) -> List.fold_left (ity_s_fold fn fts) (fts acc ts) tl
  | Ityapp (f, tl, rl) ->
      let acc = List.fold_left (ity_s_fold fn fts) (fn acc f) tl in
      List.fold_left (fun acc r -> ity_s_fold fn fts acc r.reg_ity) acc rl

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let ity_s_all pr pts ity =
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  try ity_s_fold (Util.all_fn pr) (Util.all_fn pts) true ity
  with Util.FoldSkip -> false
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let ity_s_any pr pts ity =
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  try ity_s_fold (Util.any_fn pr) (Util.any_fn pts) false ity
  with Util.FoldSkip -> true
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(* traversal functions on type variables and regions *)

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let rec ity_v_map fnv fnr ity = match ity.ity_node with
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  | Ityvar v ->
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      fnv v
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  | Itypur (f,tl) ->
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      ity_pur_unsafe f (List.map (ity_v_map fnv fnr) tl)
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  | Ityapp (f,tl,rl) ->
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      ity_app_unsafe f (List.map (ity_v_map fnv fnr) tl) (List.map fnr rl)
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let ity_subst_unsafe mv mr ity =
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  ity_v_map (fun v -> Mtv.find v mv) (fun r -> Mreg.find r mr) ity
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let ity_closed ity = Stv.is_empty ity.ity_vars.vars_tv
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let ity_immutable ity = Sreg.is_empty ity.ity_vars.vars_reg
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let rec ity_has_inv ity = match ity.ity_node with
  | Ityapp (its,_,_) -> its.its_inv || ity_any ity_has_inv ity
  | _ -> ity_any ity_has_inv ity
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let rec reg_fold fn vars acc =
  let on_reg r acc = reg_fold fn r.reg_ity.ity_vars (fn r acc) in
  Sreg.fold on_reg vars.vars_reg acc

let rec reg_all fn vars =
  let on_reg r = fn r && reg_all fn r.reg_ity.ity_vars in
  Sreg.for_all on_reg vars.vars_reg

let rec reg_any fn vars =
  let on_reg r = fn r || reg_any fn r.reg_ity.ity_vars in
  Sreg.exists on_reg vars.vars_reg

let rec reg_iter fn vars =
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  let on_reg r = fn r; reg_iter fn r.reg_ity.ity_vars in
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  Sreg.iter on_reg vars.vars_reg

let reg_occurs r vars = reg_any (reg_equal r) vars

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(* smart constructors *)

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exception BadItyArity of itysymbol * int * int
exception BadRegArity of itysymbol * int * int
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exception DuplicateRegion of region
exception UnboundRegion of region
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type ity_subst = {
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  ity_subst_tv  : ity Mtv.t;
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  ity_subst_reg : region Mreg.t; (* must preserve ghost-ness *)
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}

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let ity_subst_empty = {
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  ity_subst_tv  = Mtv.empty;
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  ity_subst_reg = Mreg.empty;
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}

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exception RegionMismatch of region * region * ity_subst
exception TypeMismatch of ity * ity * ity_subst
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let ity_equal_check ty1 ty2 =
  if not (ity_equal ty1 ty2) then raise (TypeMismatch (ty1,ty2,ity_subst_empty))

let reg_equal_check r1 r2 =
  if not (reg_equal r1 r2) then raise (RegionMismatch (r1,r2,ity_subst_empty))
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let reg_full_inst s r = Mreg.find r s.ity_subst_reg
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let ity_full_inst s ity =
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  if ity_closed ity && ity_immutable ity then ity
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  else ity_subst_unsafe s.ity_subst_tv s.ity_subst_reg ity
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let rec ity_match s ity1 ity2 =
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  let set = function
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    | None -> Some ity2
    | Some ity3 as r when ity_equal ity3 ity2 -> r
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    | _ -> raise Exit
  in
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  match ity1.ity_node, ity2.ity_node with
  | Ityapp (s1, l1, r1), Ityapp (s2, l2, r2) when its_equal s1 s2 ->
      let s = List.fold_left2 ity_match s l1 l2 in
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      List.fold_left2 reg_match s r1 r2
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  | Itypur (s1, l1), Itypur (s2, l2) when ts_equal s1 s2 ->
      List.fold_left2 ity_match s l1 l2
  | Ityvar tv1, _ ->
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      { s with ity_subst_tv = Mtv.change set tv1 s.ity_subst_tv }
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  | _ ->
      raise Exit

and reg_match s r1 r2 =
  let is_new = ref false in
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  let set = function
    | None -> is_new := true; Some r2
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    | Some r3 as r when reg_equal r3 r2 -> r
    | _ -> raise Exit
  in
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  let reg_map = Mreg.change set r1 s.ity_subst_reg in
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  let s = { s with ity_subst_reg = reg_map } in
  if !is_new then ity_match s r1.reg_ity r2.reg_ity else s
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let ity_match s ity1 ity2 =
  try ity_match s ity1 ity2
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  with Exit -> raise (TypeMismatch (ity1,ity2,s))

let reg_match s r1 r2 =
  try reg_match s r1 r2
  with Exit -> raise (RegionMismatch (r1,r2,s))
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let rec ty_of_ity ity = match ity.ity_node with
  | Ityvar v -> ty_var v
  | Itypur (s,tl) -> ty_app s (List.map ty_of_ity tl)
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  | Ityapp (s,tl,_) -> ty_app s.its_ts (List.map ty_of_ity tl)
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let rec ity_of_ty ty = match ty.ty_node with
  | Tyvar v -> ity_var v
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  | Tyapp (s,tl) -> ity_pur_unsafe s (List.map ity_of_ty tl)
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let rec ity_inst_fresh mv mr ity = match ity.ity_node with
  | Ityvar v ->
      mr, Mtv.find v mv
  | Itypur (s,tl) ->
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      let mr,tl = Lists.map_fold_left (ity_inst_fresh mv) mr tl in
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      mr, ity_pur_unsafe s tl
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  | Ityapp (s,tl,rl) ->
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      let mr,tl = Lists.map_fold_left (ity_inst_fresh mv) mr tl in
      let mr,rl = Lists.map_fold_left (reg_refresh mv) mr rl in
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      mr, ity_app_unsafe s tl rl
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and reg_refresh mv mr r = match Mreg.find_opt r mr with
  | Some r ->
      mr, r
  | None ->
      let mr,ity = ity_inst_fresh mv mr r.reg_ity in
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      let id = id_clone r.reg_name in
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      let reg = create_region id ity in
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      Mreg.add r reg mr, reg

let ity_app_fresh s tl =
  (* type variable map *)
  let add m v t = Mtv.add v t m in
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  let mv = try List.fold_left2 add Mtv.empty s.its_ts.ts_args tl
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    with Invalid_argument _ ->
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      raise (BadItyArity (s, List.length s.its_ts.ts_args, List.length tl)) in
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  (* refresh regions *)
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  let mr,rl = Lists.map_fold_left (reg_refresh mv) Mreg.empty s.its_regs in
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  let sub = { ity_subst_tv = mv; ity_subst_reg = mr } in
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  (* every top region in def is guaranteed to be in mr *)
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  match s.its_def with
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  | Some ity -> ity_full_inst sub ity
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  | None -> ity_app_unsafe s tl rl
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let ity_app s tl rl =
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  (* type variable map *)
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  let add m v t = Mtv.add v t m in
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  let mv = try List.fold_left2 add Mtv.empty s.its_ts.ts_args tl
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    with Invalid_argument _ ->
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      raise (BadItyArity (s, List.length s.its_ts.ts_args, List.length tl)) in
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  (* region map *)
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  let sub = { ity_subst_tv = mv; ity_subst_reg = Mreg.empty } in
  let sub = try List.fold_left2 reg_match sub s.its_regs rl
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    with Invalid_argument _ ->
      raise (BadRegArity (s, List.length s.its_regs, List.length rl)) in
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  (* every type var and top region in def are in its_ts.ts_args and its_regs *)
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  match s.its_def with
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  | Some ity -> ity_full_inst sub ity
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  | None -> ity_app_unsafe s tl rl
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let ity_pur s tl =
  (* type variable map *)
  let add m v t = Mtv.add v t m in
  let mv = try List.fold_left2 add Mtv.empty s.ts_args tl
    with Invalid_argument _ ->
      raise (Ty.BadTypeArity (s, List.length s.ts_args, List.length tl)) in
  let sub = { ity_subst_tv = mv; ity_subst_reg = Mreg.empty } in
  (* every top region in def is guaranteed to be in mr *)
  match s.ts_def with
  | Some ty -> ity_full_inst sub (ity_of_ty ty)
  | None -> ity_pur_unsafe s tl

(* itysymbol creation *)
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let create_itysymbol_unsafe, restore_its =
  let ts_to_its = Wts.create 17 in
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  (fun ts ~abst ~priv ~inv regs def ->
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    let its = {
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      its_ts    = ts;
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      its_regs  = regs;
      its_def   = def;
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      its_inv   = inv;
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      its_abst  = abst;
      its_priv  = priv;
    } in
    Wts.set ts_to_its ts its;
    its),
  Wts.find ts_to_its
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let create_itysymbol
      name ?(abst=false) ?(priv=false) ?(inv=false) args regs def =
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  let puredef = Opt.map ty_of_ity def in
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  let purets = create_tysymbol name args puredef in
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  (* all regions *)
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  let add s v = Sreg.add_new (DuplicateRegion v) v s in
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  let sregs = List.fold_left add Sreg.empty regs in
  (* all type variables *)
  let sargs = List.fold_right Stv.add args Stv.empty in
  (* all type variables in [regs] must be in [args] *)
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  let check dtvs = if not (Stv.subset dtvs sargs) then
    raise (UnboundTypeVar (Stv.choose (Stv.diff dtvs sargs))) in
  List.iter (fun r -> check r.reg_ity.ity_vars.vars_tv) regs;
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  (* all regions in [def] must be in [regs] *)
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  let check dregs = if not (Sreg.subset dregs sregs) then
    raise (UnboundRegion (Sreg.choose (Sreg.diff dregs sregs))) in
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  Opt.iter (fun d -> check d.ity_vars.vars_reg) def;
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  (* if a type is an alias then it cannot be abstract or private *)
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  if def <> None then begin
    if abst then Loc.errorm "Type aliases cannot be abstract";
    if priv then Loc.errorm "Type aliases cannot be private";
    if inv  then Loc.errorm "Type aliases cannot have invariants"
  end;
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  create_itysymbol_unsafe purets ~abst ~priv ~inv regs def
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let ts_unit = ts_tuple 0
let ty_unit = ty_tuple []

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let ity_int  = ity_of_ty Ty.ty_int
let ity_bool = ity_of_ty Ty.ty_bool
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let ity_unit = ity_of_ty ty_unit
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let vars_freeze s =
  let sbs = Stv.fold (fun v -> Mtv.add v (ity_var v)) s.vars_tv Mtv.empty in
  let sbs = { ity_subst_tv = sbs ; ity_subst_reg = Mreg.empty } in
  Sreg.fold (fun r s -> reg_match s r r) s.vars_reg sbs

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(** cloning *)

let its_clone sm =
  let itsh = Hits.create 3 in
  let regh = Hreg.create 3 in
  let rec add_ts oits nts =
    let nits = try restore_its nts with Not_found ->
      let abst = oits.its_abst in
      let priv = oits.its_priv in
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      let inv  = oits.its_inv in
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      let regs = List.map conv_reg oits.its_regs in
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      let def  = Opt.map conv_ity oits.its_def in
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      create_itysymbol_unsafe nts ~abst ~priv ~inv regs def
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    in
    Hits.replace itsh oits nits;
    nits
  and conv_reg r =
    try Hreg.find regh r with Not_found ->
      let ity = conv_ity r.reg_ity in
      let nr = create_region (id_clone r.reg_name) ity in
      Hreg.replace regh r nr;
      nr
  and conv_ity ity = match ity.ity_node with
    | Ityapp (its,tl,rl) ->
        let tl = List.map conv_ity tl in
        let rl = List.map conv_reg rl in
        ity_app_unsafe (conv_its its) tl rl
    | Itypur (ts,tl) ->
        let tl = List.map conv_ity tl in
        ity_pur_unsafe (conv_ts ts) tl
    | Ityvar _ -> ity
  and conv_its its =
    try Hits.find itsh its with Not_found ->
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      try add_ts its (Mts.find its.its_ts sm.Theory.sm_ts)
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      with Not_found -> its
  and conv_ts ts =
    Mts.find_def ts ts sm.Theory.sm_ts
  in
  let add_ts ots nts =
    try ignore (add_ts (restore_its ots) nts)
    with Not_found -> ()
  in
  Mts.iter add_ts sm.Theory.sm_ts;
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  Hits.fold Mits.add itsh Mits.empty,
  Hreg.fold Mreg.add regh Mreg.empty
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(** computation types (with effects) *)

(* exception symbols *)
type xsymbol = {
  xs_name : ident;
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  xs_ity  : ity; (* closed and pure *)
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}
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exception PolymorphicException of ident * ity
exception MutableException of ident * ity
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let xs_equal : xsymbol -> xsymbol -> bool = (==)

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let create_xsymbol id ity =
  let id = id_register id in
  if not (ity_closed ity) then raise (PolymorphicException (id, ity));
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  if not (ity_immutable ity) then raise (MutableException (id, ity));
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  { xs_name = id; xs_ity = ity; }
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module Exn = MakeMSH (struct
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  type t = xsymbol
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  let tag xs = Weakhtbl.tag_hash xs.xs_name.id_tag
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end)
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module Sexn = Exn.S
module Mexn = Exn.M

(* effects *)
type effect = {
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  eff_reads  : Sreg.t;
  eff_writes : Sreg.t;
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  eff_raises : Sexn.t;
  eff_ghostr : Sreg.t; (* ghost reads *)
  eff_ghostw : Sreg.t; (* ghost writes *)
  eff_ghostx : Sexn.t; (* ghost raises *)
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  (* if r1 -> Some r2 then r1 appears in ty(r2) *)
  eff_resets : region option Mreg.t;
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}
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let eff_empty = {
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  eff_reads  = Sreg.empty;
  eff_writes = Sreg.empty;
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  eff_raises = Sexn.empty;
  eff_ghostr = Sreg.empty;
  eff_ghostw = Sreg.empty;
  eff_ghostx = Sexn.empty;
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  eff_resets = Mreg.empty;
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}
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let eff_is_empty e =
  Sreg.is_empty e.eff_reads &&
  Sreg.is_empty e.eff_writes &&
  Sexn.is_empty e.eff_raises &&
  Sreg.is_empty e.eff_ghostr &&
  Sreg.is_empty e.eff_ghostw &&
  Sexn.is_empty e.eff_ghostx &&
  Mreg.is_empty e.eff_resets

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let join_reset _key v1 v2 = match v1, v2 with
  | Some r1, Some r2 ->
      if reg_equal r1 r2 then Some v1 else
      if reg_occurs r1 r2.reg_ity.ity_vars then Some v2 else
      if reg_occurs r2 r1.reg_ity.ity_vars then Some v1 else
      Some None
  | _ -> Some None
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let eff_union x y = {
  eff_reads  = Sreg.union x.eff_reads  y.eff_reads;
  eff_writes = Sreg.union x.eff_writes y.eff_writes;
  eff_raises = Sexn.union x.eff_raises y.eff_raises;
  eff_ghostr = Sreg.union x.eff_ghostr y.eff_ghostr;
  eff_ghostw = Sreg.union x.eff_ghostw y.eff_ghostw;
  eff_ghostx = Sexn.union x.eff_ghostx y.eff_ghostx;
  eff_resets = Mreg.union join_reset x.eff_resets y.eff_resets;
}

let eff_ghostify e = {
  eff_reads  = Sreg.empty;
  eff_writes = Sreg.empty;
  eff_raises = Sexn.empty;
  eff_ghostr = Sreg.union e.eff_reads  e.eff_ghostr;
  eff_ghostw = Sreg.union e.eff_writes e.eff_ghostw;
  eff_ghostx = Sexn.union e.eff_raises e.eff_ghostx;
  eff_resets = e.eff_resets;
}

let eff_read e ?(ghost=false) r = if ghost
  then { e with eff_ghostr = Sreg.add r e.eff_ghostr }
  else { e with eff_reads  = Sreg.add r e.eff_reads  }

let eff_write e ?(ghost=false) r = if ghost
  then { e with eff_ghostw = Sreg.add r e.eff_ghostw }
  else { e with eff_writes = Sreg.add r e.eff_writes }

let eff_raise e ?(ghost=false) x = if ghost
  then { e with eff_ghostx = Sexn.add x e.eff_ghostx }
  else { e with eff_raises = Sexn.add x e.eff_raises }
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let eff_reset e r = { e with eff_resets = Mreg.add r None e.eff_resets }

exception IllegalAlias of region

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let eff_refresh e r u =
  if not (reg_occurs r u.reg_ity.ity_vars) then
    invalid_arg "Mlw_ty.eff_refresh";
  let reset = Mreg.singleton r (Some u) in
  { e with eff_resets = Mreg.union join_reset e.eff_resets reset }

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let eff_assign e ?(ghost=false) r ty =
  let e = eff_write e ~ghost r in
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  let sub = ity_match ity_subst_empty r.reg_ity ty in
  (* assignment cannot instantiate type variables *)
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  let check tv ity = match ity.ity_node with
    | Ityvar tv' -> tv_equal tv tv'
    | _ -> false in
  if not (Mtv.for_all check sub.ity_subst_tv) then
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    raise (TypeMismatch (r.reg_ity,ty,ity_subst_empty));
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  (* r:t[r1,r2] <- t[r1,r1] introduces an alias *)
  let add_right _ v s = Sreg.add_new (IllegalAlias v) v s in
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  ignore (Mreg.fold add_right sub.ity_subst_reg Sreg.empty);
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  (* every region on the rhs must be erased *)
  let add_right k v m = if reg_equal k v then m else Mreg.add v None m in
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  let reset = Mreg.fold add_right sub.ity_subst_reg Mreg.empty in
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  (* ...except those which occur on the lhs : they are preserved under r *)
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  let add_left k v m = if reg_equal k v then m else Mreg.add k (Some r) m in
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  let reset = Mreg.fold add_left sub.ity_subst_reg reset in
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  { e with eff_resets = Mreg.union join_reset e.eff_resets reset }

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let eff_remove_raise e x = { e with
  eff_raises = Sexn.remove x e.eff_raises;
  eff_ghostx = Sexn.remove x e.eff_ghostx;
}
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let eff_full_inst s e =
  let s = s.ity_subst_reg in
  (* modified or reset regions in e *)
  let wr = Mreg.map (Util.const ()) e.eff_resets in
  let wr = Sreg.union e.eff_writes wr in
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  let wr = Sreg.union e.eff_ghostw wr in
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  (* read-only regions in e *)
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  let ro = Sreg.diff (Mreg.map (Util.const ()) s) wr in
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  (* all modified or reset regions are instantiated into distinct regions *)
  let add_affected r acc =
    let r = Mreg.find r s in Sreg.add_new (IllegalAlias r) r acc in
  let wr = Sreg.fold add_affected wr Sreg.empty in
  (* all read-only regions are instantiated outside wr *)
  let add_readonly r =
    let r = Mreg.find r s in if Sreg.mem r wr then raise (IllegalAlias r) in
  Sreg.iter add_readonly ro;
  (* calculate instantiated effect *)
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  let add_sreg r acc = Sreg.add (Mreg.find r s) acc in
  let add_mreg r v acc =
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    Mreg.add (Mreg.find r s) (Opt.map (fun v -> Mreg.find v s) v) acc in
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  { e with
    eff_reads  = Sreg.fold add_sreg e.eff_reads  Sreg.empty;
    eff_writes = Sreg.fold add_sreg e.eff_writes Sreg.empty;
    eff_ghostr = Sreg.fold add_sreg e.eff_ghostr Sreg.empty;
    eff_ghostw = Sreg.fold add_sreg e.eff_ghostw Sreg.empty;
    eff_resets = Mreg.fold add_mreg e.eff_resets Mreg.empty;
  }
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let eff_filter vars e =
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  let check r = reg_occurs r vars in
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  let reset r = function
    | _ when not (check r) -> None
    | Some u as v when check u -> Some v
    | _ -> Some None
  in
  { e with
    eff_reads  = Sreg.filter check e.eff_reads;
    eff_writes = Sreg.filter check e.eff_writes;
    eff_ghostr = Sreg.filter check e.eff_ghostr;
    eff_ghostw = Sreg.filter check e.eff_ghostw;
    eff_resets = Mreg.mapi_filter reset e.eff_resets;
  }

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let eff_stale_region eff vars =
  let check_reset r u =
    let rec check_reg reg =
      reg_equal r reg || match u with
        | Some u when reg_equal u reg -> false
        | _ -> Sreg.exists check_reg reg.reg_ity.ity_vars.vars_reg in
    Sreg.exists check_reg vars.vars_reg in
  Mreg.exists check_reset eff.eff_resets

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(** specification *)

type pre = term          (* precondition: pre_fmla *)
type post = term         (* postcondition: eps result . post_fmla *)
type xpost = post Mexn.t (* exceptional postconditions *)

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type variant = term * lsymbol option (* tau * (tau -> tau -> prop) *)

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let create_post vs f = t_eps_close vs f

let open_post f = match f.t_node with
  | Teps bf -> t_open_bound bf
  | _ -> Loc.errorm "invalid post-condition"

let check_post ty f = match f.t_node with
  | Teps _ -> Ty.ty_equal_check ty (t_type f)
  | _ -> Loc.errorm "invalid post-condition"

type spec = {
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  c_pre     : pre;
  c_post    : post;
  c_xpost   : xpost;
  c_effect  : effect;
  c_variant : variant list;
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  c_letrec  : int;
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}

let spec_empty ty = {
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  c_pre     = t_true;
  c_post    = create_post (create_vsymbol (id_fresh "dummy") ty) t_true;
  c_xpost   = Mexn.empty;
  c_effect  = eff_empty;
  c_variant = [];
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  c_letrec  = 0;
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}

let spec_full_inst sbs tvm vsm c =
  let subst = t_ty_subst tvm vsm in {
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    c_pre     = subst c.c_pre;
    c_post    = subst c.c_post;
    c_xpost   = Mexn.map subst c.c_xpost;
    c_effect  = eff_full_inst sbs c.c_effect;
    c_variant = List.map (fun (t,rel) -> subst t, rel) c.c_variant;
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    c_letrec  = c.c_letrec;
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  }

let spec_subst sbs c =
  let subst = t_subst sbs in {
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    c_pre     = subst c.c_pre;
    c_post    = subst c.c_post;
    c_xpost   = Mexn.map subst c.c_xpost;
    c_effect  = c.c_effect;
    c_variant = List.map (fun (t,rel) -> subst t, rel) c.c_variant;
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    c_letrec  = c.c_letrec;
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  }

let spec_filter varm vars c =
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  let add f s = Mvs.set_union f.t_vars s in
  let vss = add c.c_pre c.c_post.t_vars in
  let vss = Mexn.fold (fun _ -> add) c.c_xpost vss in
  let vss = List.fold_right (fun (t,_) -> add t) c.c_variant vss in
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  let check { vs_name = id } _ = if not (Mid.mem id varm) then
    Loc.errorm "Local variable %s escapes from its scope" id.id_string in
  Mvs.iter check vss;
  { c with c_effect = eff_filter vars c.c_effect }

exception UnboundException of xsymbol

let spec_check c ty =
  if c.c_pre.t_ty <> None then
    Loc.error ?loc:c.c_pre.t_loc (Term.FmlaExpected c.c_pre);
  check_post ty c.c_post;
  Mexn.iter (fun xs q -> check_post (ty_of_ity xs.xs_ity) q) c.c_xpost;
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  let check_variant (t,rel) = match rel with
    | Some ps -> ignore (ps_app ps [t;t])
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    | None -> ignore (t_type t) in
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  List.iter check_variant c.c_variant;
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  let sexn = Sexn.union c.c_effect.eff_raises c.c_effect.eff_ghostx in
  let sexn = Mexn.set_diff sexn c.c_xpost in
  if not (Sexn.is_empty sexn) then
    raise (UnboundException (Sexn.choose sexn))

(** program variables *)
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type vty_value = {
  vtv_ity   : ity;
  vtv_ghost : bool;
  vtv_mut   : region option;
}

let vty_value ?(ghost=false) ?mut ity =
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  Opt.iter (fun r -> ity_equal_check ity r.reg_ity) mut;
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  { vtv_ity   = ity;
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    vtv_ghost = ghost;
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    vtv_mut   = mut }

let vtv_vars {vtv_ity = {ity_vars = vars}; vtv_mut = mut} = match mut with
  | Some r -> { vars with vars_reg = Sreg.add r vars.vars_reg }
  | None   -> vars
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let vtv_unmut vtv =
  if vtv.vtv_mut = None then vtv else
    vty_value ~ghost:vtv.vtv_ghost vtv.vtv_ity

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type pvsymbol = {
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  pv_vs   : vsymbol;
  pv_vtv  : vty_value;
  pv_vars : varset;
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}

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module PVsym = MakeMSHW (struct
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  type t = pvsymbol
  let tag pv = pv.pv_vs.vs_name.id_tag
end)

module Spv = PVsym.S
module Mpv = PVsym.M
module Hpv = PVsym.H
module Wpv = PVsym.W

let pv_equal : pvsymbol -> pvsymbol -> bool = (==)

let create_pvsymbol id vtv = {
  pv_vs   = create_vsymbol id (ty_of_ity vtv.vtv_ity);
  pv_vtv  = vtv;
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  pv_vars = vtv_vars vtv;
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}

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let create_pvsymbol, restore_pv, restore_pv_by_id =
  let id_to_pv = Wid.create 17 in
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  (fun id vtv ->
    let pv = create_pvsymbol id vtv in
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    Wid.set id_to_pv pv.pv_vs.vs_name pv;
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    pv),
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  (fun vs -> Wid.find id_to_pv vs.vs_name),
  (fun id -> Wid.find id_to_pv id)
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(** program types *)

type vty =
  | VTvalue of vty_value
  | VTarrow of vty_arrow

and vty_arrow = {
  vta_args   : pvsymbol list;
  vta_result : vty;
  vta_spec   : spec;
  vta_ghost  : bool;
}

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let rec vta_vars vta =
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  let add_arg vars pv = vars_union vars pv.pv_vars in
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  List.fold_left add_arg (vty_vars vta.vta_result) vta.vta_args

and vty_vars = function
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  | VTvalue vtv -> vtv_vars vtv
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  | VTarrow vta -> vta_vars vta
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let vty_ghost = function
  | VTvalue vtv -> vtv.vtv_ghost
  | VTarrow vta -> vta.vta_ghost

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let ity_of_vty = function
  | VTvalue vtv -> vtv.vtv_ity
  | VTarrow _   -> ity_unit

let ty_of_vty = function
  | VTvalue vtv -> ty_of_ity vtv.vtv_ity
  | VTarrow _   -> ty_unit

let spec_check spec vty = spec_check spec (ty_of_vty vty)

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let vty_arrow_unsafe argl spec ghost vty = {
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  vta_args   = argl;
  vta_result = vty;
  vta_spec   = spec;
  vta_ghost  = ghost || vty_ghost vty;
}
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let vty_arrow argl ?spec ?(ghost=false) vty =
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  let exn = Invalid_argument "Mlw.vty_arrow" in
  (* the arguments must be all distinct *)
  if argl = [] then raise exn;
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  let add_arg pvs pv =
    (* mutable arguments are rejected outright *)
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    if pv.pv_vtv.vtv_mut <> None then raise exn;
    Spv.add_new exn pv pvs in
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  ignore (List.fold_left add_arg Spv.empty argl);
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  (* only projections may return mutable values *)
  begin match vty with
    | VTvalue { vtv_mut = Some _ } -> raise exn
    | _ -> ()
  end;
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  let spec = match spec with
    | Some spec -> spec_check spec vty; spec
    | None -> spec_empty (ty_of_vty vty) in
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  (* we admit non-empty variant list even for null letrec,
     so that we can store there external variables from
     user-written effects to save them from spec_filter *)
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  vty_arrow_unsafe argl spec ghost vty
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(* this only compares the types of arguments and results, and ignores
   the spec. In other words, only the type variables and regions
   in .vta_vars are matched. The caller should supply a "freezing"
   substitution that covers all external type variables and regions. *)
let rec vta_vars_match s a1 a2 =
  let vtv_match s v1 v2 = ity_match s v1.vtv_ity v2.vtv_ity in
  let rec match_args s l1 l2 = match l1, l2 with
    | [],[] -> s, a1.vta_result, a2.vta_result
    | [], _ -> s, a1.vta_result, VTarrow { a2 with vta_args = l2 }
    | _, [] -> s, VTarrow { a1 with vta_args = l1 }, a2.vta_result
    | {pv_vtv = v1}::l1, {pv_vtv = v2}::l2 ->
        match_args (vtv_match s v1 v2) l1 l2
  in
  let s, vty1, vty2 = match_args s a1.vta_args a2.vta_args in
  match vty1, vty2 with
  | VTarrow a1, VTarrow a2 -> vta_vars_match s a1 a2
  | VTvalue v1, VTvalue v2 -> vtv_match s v1 v2
  | _ -> invalid_arg "Mlw_ty.vta_vars_match"

(* the substitution must cover not only vta.vta_tvs and vta.vta_regs
   but also every type variable and every region in vta_spec *)
let vta_full_inst sbs vta =
  let tvm = Mtv.map ty_of_ity sbs.ity_subst_tv in
  let vtv_inst { vtv_ity = ity; vtv_ghost = ghost } =
    vty_value ~ghost (ity_full_inst sbs ity) in
  let pv_inst { pv_vs = vs; pv_vtv = vtv } =
    create_pvsymbol (id_clone vs.vs_name) (vtv_inst vtv) in
  let add_arg vsm pv =
    let nv = pv_inst pv in
    Mvs.add pv.pv_vs (t_var nv.pv_vs) vsm, nv in
  let rec vta_inst vsm vta =
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    let spec = spec_full_inst sbs tvm vsm vta.vta_spec in
    let vty = match vta.vta_result with
      | VTarrow vta -> VTarrow (vta_inst vsm vta)
      | VTvalue vtv -> VTvalue (vtv_inst vtv) in
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    vty_arrow_unsafe args spec vta.vta_ghost vty
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  in
  vta_inst Mvs.empty vta

(* remove from the given arrow every effect that is covered
   neither by the arrow's vta_vars nor by the given varmap *)
let rec vta_filter varm vars vta =
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  let add_m pv m = Mid.add pv.pv_vs.vs_name pv.pv_vars m in
  let add_s pv s = vars_union pv.pv_vars s in
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  let varm = List.fold_right add_m vta.vta_args varm in
  let vars = List.fold_right add_s vta.vta_args vars in
  let vty = match vta.vta_result with
    | VTarrow a -> VTarrow (vta_filter varm vars a)
    | VTvalue _ -> vta.vta_result in
  (* reads and writes must come from the context,
     resets may affect the regions in the result *)
  let spec = spec_filter varm vars vta.vta_spec in
  let rst = vta.vta_spec.c_effect.eff_resets in
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  let spec = if Mreg.is_empty rst then spec else
    let vars = vars_union vars (vty_vars vty) in
    let rst = { eff_empty with eff_resets = rst } in
    let rst = (eff_filter vars rst).eff_resets in
    let eff = { spec.c_effect with eff_resets = rst } in
    { spec with c_effect = eff } in
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  (* we must also reset every fresh region in the value *)
  let spec = match vta.vta_result with
    | VTvalue v ->
        let on_reg r e = if reg_occurs r vars then e else eff_reset e r in
        let eff = reg_fold on_reg v.vtv_ity.ity_vars spec.c_effect in
        { spec with c_effect = eff }
    | VTarrow _ -> spec in
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  vty_arrow_unsafe vta.vta_args spec vta.vta_ghost vty
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let vta_filter varm vta =
  vta_filter varm (vars_merge varm vars_empty) vta

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let vtv_ghostify vtv = { vtv with vtv_ghost = true }
let vta_ghostify vta = { vta with vta_ghost = true }

let vty_ghostify = function
  | VTvalue vtv -> VTvalue (vtv_ghostify vtv)
  | VTarrow vta -> VTarrow (vta_ghostify vta)

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let vta_app vta pv =
  let vtv = pv.pv_vtv in
  let arg, rest = match vta.vta_args with
    | arg::rest -> arg,rest | _ -> assert false in
  ity_equal_check arg.pv_vtv.vtv_ity vtv.vtv_ity;
  let sbs = Mvs.singleton arg.pv_vs (t_var pv.pv_vs) in
  let rec vty_subst = function
    | VTarrow a when not (List.exists (pv_equal arg) a.vta_args) ->
        let result = vty_subst a.vta_result in
        let spec = spec_subst sbs a.vta_spec in
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        VTarrow (vty_arrow_unsafe a.vta_args spec a.vta_ghost result)
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    | vty -> vty in
  let result = vty_subst vta.vta_result in
  let spec = spec_subst sbs vta.vta_spec in
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  if not vtv.vtv_ghost && arg.pv_vtv.vtv_ghost then
    Loc.errorm "non-ghost value passed as a ghost argument";
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  let ghost =
    vta.vta_ghost || (vtv.vtv_ghost && not arg.pv_vtv.vtv_ghost) in
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  if rest = [] then spec, (if ghost then vty_ghostify result else result)
  else spec_empty ty_unit, VTarrow (vty_arrow_unsafe rest spec ghost result)