mlw_wp.ml 64.2 KB
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(********************************************************************)
(*                                                                  *)
(*  The Why3 Verification Platform   /   The Why3 Development Team  *)
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(*  Copyright 2010-2013   --   INRIA - CNRS - Paris-Sud University  *)
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(*                                                                  *)
(*  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
open Term
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open Decl
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open Theory
open Mlw_ty
open Mlw_ty.T
open Mlw_expr

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let debug = Debug.register_info_flag "whyml_wp"
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  ~desc:"Print@ details@ of@ verification@ conditions@ generation."
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let no_track = Debug.register_flag "wp_no_track"
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  ~desc:"Do@ not@ remove@ redundant@ type@ invariant@ conditions@ from@ VCs."

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let no_eval = Debug.register_flag "wp_no_eval"
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  ~desc:"Do@ not@ simplify@ pattern@ matching@ on@ record@ datatypes@ in@ VCs."
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let lemma_label = Ident.create_label "why3:lemma"

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(** Marks *)
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let ts_mark = create_tysymbol (id_fresh "'mark") [] None
let ty_mark = ty_app ts_mark []

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let ity_mark = ity_pur ts_mark []
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let fresh_mark () = create_vsymbol (id_fresh "'mark") ty_mark
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let fs_at =
  let ty = ty_var (create_tvsymbol (id_fresh "a")) in
  create_lsymbol (id_fresh "at") [ty; ty_mark] (Some ty)

let fs_old =
  let ty = ty_var (create_tvsymbol (id_fresh "a")) in
  create_lsymbol (id_fresh "old") [ty] (Some ty)

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let th_mark_at =
  let uc = create_theory (id_fresh "WP builtins: at") in
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  let uc = add_ty_decl uc ts_mark in
  let uc = add_param_decl uc fs_at in
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  close_theory uc

let th_mark_old =
  let uc = create_theory (id_fresh "WP builtins: old") in
  let uc = use_export uc th_mark_at in
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  let uc = add_param_decl uc fs_old in
  close_theory uc

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let fs_now = create_lsymbol (id_fresh "%now") [] (Some ty_mark)
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let t_now = fs_app fs_now [] ty_mark
let e_now = e_lapp fs_now [] (ity_pur ts_mark [])
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(* [vs_old] appears in the postconditions given to the core API,
   which expects every vsymbol to be a pure part of a pvsymbol *)
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let pv_old = create_pvsymbol (id_fresh "%old") ity_mark
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let vs_old = pv_old.pv_vs
let t_old  = t_var vs_old
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let t_at_old t = t_app fs_at [t; t_old] t.t_ty

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let ls_absurd = create_lsymbol (id_fresh "absurd") [] None
let t_absurd  = ps_app ls_absurd []
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let mk_t_if f = t_if f t_bool_true t_bool_false
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let to_term t = if t.t_ty = None then mk_t_if t else t
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(* any vs in post/xpost is either a pvsymbol or a fresh mark *)
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let ity_of_vs vs =
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  if Ty.ty_equal vs.vs_ty ty_mark then ity_mark else (restore_pv vs).pv_ity
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(* replace every occurrence of [old(t)] with [at(t,'old)] *)
let rec remove_old f = match f.t_node with
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  | Tapp (ls,[t]) when ls_equal ls fs_old -> t_at_old (remove_old t)
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  | _ -> t_map remove_old f

(* replace every occurrence of [at(t,'now)] with [t] *)
let rec remove_at f = match f.t_node with
  | Tapp (ls, [t; { t_node = Tapp (fs,[]) }])
    when ls_equal ls fs_at && ls_equal fs fs_now -> remove_at t
  | _ -> t_map remove_at f

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(* replace [at(t,'old)] with [at(t,lab)] everywhere in formula [f] *)
let old_mark lab t = t_subst_single vs_old (t_var lab) t

(* replace [at(t,lab)] with [at(t,'now)] everywhere in formula [f] *)
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let erase_mark lab t = t_subst_single lab t_now t

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(* retreat to the point of the current postcondition's ['old] *)
let backstep fn q xq =
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  let lab = fresh_mark () in
  let f = fn (old_mark lab q) (Mexn.map (old_mark lab) xq) in
  erase_mark lab f
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(** WP utilities *)
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let default_exn_post xs _ =
  let vs = create_vsymbol (id_fresh "result") (ty_of_ity xs.xs_ity) in
  create_post vs t_true

let default_post vty ef =
  let vs = create_vsymbol (id_fresh "result") (ty_of_vty vty) in
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  create_post vs t_true, Mexn.mapi default_exn_post ef.eff_raises
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let wp_label e f =
  let loc = if f.t_loc = None then e.e_loc else f.t_loc in
  let lab = Ident.Slab.union e.e_label f.t_label in
  t_label ?loc lab f

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let expl_pre       = Ident.create_label "expl:precondition"
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let expl_post      = Ident.create_label "expl:postcondition"
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let expl_xpost     = Ident.create_label "expl:exceptional postcondition"
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let expl_assume    = Ident.create_label "expl:assumption"
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let expl_assert    = Ident.create_label "expl:assertion"
let expl_check     = Ident.create_label "expl:check"
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let expl_absurd    = Ident.create_label "expl:unreachable point"
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let expl_type_inv  = Ident.create_label "expl:type invariant"
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let expl_loop_init = Ident.create_label "expl:loop invariant init"
let expl_loop_keep = Ident.create_label "expl:loop invariant preservation"
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let expl_loopvar   = Ident.create_label "expl:loop variant decrease"
let expl_variant   = Ident.create_label "expl:variant decrease"
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let rec wp_expl l f = match f.t_node with
  | _ when Slab.mem Split_goal.stop_split f.t_label -> t_label_add l f
  | Tbinop (Tand,f1,f2) -> t_label_copy f (t_and (wp_expl l f1) (wp_expl l f2))
  | Teps _ -> t_label_add l f (* post-condition, push down later *)
  | _ -> f
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let wp_and ~sym f1 f2 =
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  if sym then t_and_simp f1 f2 else t_and_asym_simp f1 f2

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let wp_ands ~sym fl =
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  if sym then t_and_simp_l fl else t_and_asym_simp_l fl

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let wp_implies f1 f2 = t_implies_simp f1 f2
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let wp_let v t f = t_let_close_simp v t f

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let wp_forall vl f = t_forall_close_simp vl [] f

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let is_equality_for v f = match f.t_node with
  | Tapp (ps, [{ t_node = Tvar u }; t])
    when ls_equal ps ps_equ && vs_equal u v && not (Mvs.mem v t.t_vars) ->
      Some t
  | _ ->
      None

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let wp_forall_post v p f =
  (* we optimize for the case when a postcondition
     is of the form (... /\ result = t /\ ...) *)
  let rec down p = match p.t_node with
    | Tbinop (Tand,l,r) ->
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        let t, l, r =
          let t, l = down l in
          if t <> None then t, l, r else
            let t, r = down r in t, l, r
        in
        t, if t = None then p else t_label_copy p (t_and_simp l r)
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    | _ ->
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        let t = is_equality_for v p in
        t, if t = None then p else t_true
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  in
  if ty_equal v.vs_ty ty_unit then
    t_subst_single v t_void (wp_implies p f)
  else match down p with
    | Some t, p -> wp_let v t (wp_implies p f)
    | _ -> wp_forall [v] (wp_implies p f)
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let t_and_subst v t1 t2 =
  (* if [t1] defines variable [v], return [t2] with [v] replaced by its
     definition. Otherwise return [t1 /\ t2] *)
  match is_equality_for v t1 with
  | Some def -> t_subst_single v def t2
  | None -> t_and_simp t1 t2

let t_implies_subst v t1 t2 =
  (* if [t1] defines variable [v], return [t2] with [v] replaced by its
     definition. Otherwise return [t1 -> t2] *)
  match is_equality_for v t1 with
  | Some def -> t_subst_single v def t2
  | None -> t_implies_simp t1 t2

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let regs_of_writes eff = Sreg.union eff.eff_writes eff.eff_ghostw
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let exns_of_raises eff = Sexn.union eff.eff_raises eff.eff_ghostx
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let open_post q =
  let v, f = open_post q in
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  v, Slab.fold wp_expl q.t_label f
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let open_unit_post q =
  let v, q = open_post q in
  t_subst_single v t_void q

let create_unit_post =
  let v = create_vsymbol (id_fresh "void") ty_unit in
  fun q -> create_post v q

let vs_result e =
  create_vsymbol (id_fresh ?loc:e.e_loc "result") (ty_of_vty e.e_vty)

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(** WP state *)

type wp_env = {
  prog_known : Mlw_decl.known_map;
  pure_known : Decl.known_map;
  global_env : Env.env;
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  ps_int_le  : Term.lsymbol;
  ps_int_ge  : Term.lsymbol;
  ps_int_lt  : Term.lsymbol;
  ps_int_gt  : Term.lsymbol;
  fs_int_pl  : Term.lsymbol;
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  fs_int_mn  : Term.lsymbol;
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  letrec_var : term list Mint.t;
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}
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let decrease_alg ?loc env old_t t =
  let oty = t_type old_t in
  let nty = t_type t in
  let quit () =
    Loc.errorm ?loc "no default order for %a" Pretty.print_term t in
  let ts = match oty with { ty_node = Tyapp (ts,_) } -> ts | _ -> quit () in
  let csl = Decl.find_constructors env.pure_known ts in
  if csl = [] then quit ();
  let sbs = ty_match Mtv.empty (ty_app ts (List.map ty_var ts.ts_args)) oty in
  let add_arg acc fty =
    let fty = ty_inst sbs fty in
    if ty_equal fty nty then
      let vs = create_vsymbol (id_fresh "f") nty in
      t_or_simp acc (t_equ (t_var vs) t), pat_var vs
    else acc, pat_wild fty in
  let add_cs (cs,_) =
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    let f, pl = Lists.map_fold_left add_arg t_false cs.ls_args in
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    t_close_branch (pat_app cs pl oty) f in
  t_case old_t (List.map add_cs csl)

let decrease_rel ?loc env old_t t = function
  | Some ls -> ps_app ls [t; old_t]
  | None when ty_equal (t_type t) ty_int ->
      t_and
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        (ps_app env.ps_int_le [t_nat_const 0; old_t])
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        (ps_app env.ps_int_lt [t; old_t])
  | None -> decrease_alg ?loc env old_t t

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let decrease loc lab env olds varl =
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  let rec decr pr olds varl = match olds, varl with
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    | [], [] -> (* empty variant *)
        t_true
    | [old_t], [t, rel] ->
        t_and_simp pr (decrease_rel ?loc env old_t t rel)
    | old_t::_, (t,_)::_ when not (oty_equal old_t.t_ty t.t_ty) ->
        Loc.errorm ?loc "cannot use lexicographic ordering"
    | old_t::olds, (t,rel)::varl ->
        let dt = t_and_simp pr (decrease_rel ?loc env old_t t rel) in
        let pr = t_and_simp pr (t_equ old_t t) in
        t_or_simp dt (decr pr olds varl)
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    | _ -> assert false
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  in
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  t_label ?loc lab (decr t_true olds varl)

let expl_variant = Slab.add Split_goal.stop_split (Slab.singleton expl_variant)
let expl_loopvar = Slab.add Split_goal.stop_split (Slab.singleton expl_loopvar)
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(** Reconstruct pure values after writes *)

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let model1_lab = Slab.singleton (create_label "model:1")
let model2_lab = Slab.singleton (create_label "model:quantify(2)")
let model3_lab = Slab.singleton (create_label "model:cond")

let mk_var id label ty = create_vsymbol (id_clone ~label id) ty

(* replace "contemporary" variables with fresh ones *)
let rec subst_at_now now mvs t = match t.t_node with
  | Tvar vs when now ->
      begin try t_var (Mvs.find vs mvs) with Not_found -> t end
  | Tapp (ls, _) when ls_equal ls fs_old -> assert false
  | Tapp (ls, [_; mark]) when ls_equal ls fs_at ->
      let now = match mark.t_node with
        | Tvar vs when vs_equal vs vs_old -> assert false
        | Tapp (ls,[]) when ls_equal ls fs_now -> true
        | _ -> false in
      t_map (subst_at_now now mvs) t
  | Tlet _ | Tcase _ | Teps _ | Tquant _ ->
      (* do not open unless necessary *)
      let mvs = Mvs.set_inter mvs t.t_vars in
      if Mvs.is_empty mvs then t else
      t_map (subst_at_now now mvs) t
  | _ ->
      t_map (subst_at_now now mvs) t

(* generic expansion of an algebraic type value *)
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let analyze_var fn_down fn_join lkm km vs ity =
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  let var_of_fd fd =
    create_vsymbol (id_fresh "y") (ty_of_ity fd.fd_ity) in
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  let branch (cs,fdl) =
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    let vl = List.map var_of_fd fdl in
    let pat = pat_app cs (List.map pat_var vl) vs.vs_ty in
    let t = fn_join cs (List.map2 fn_down vl fdl) vs.vs_ty in
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    t_close_branch pat t in
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  let csl = Mlw_decl.inst_constructors lkm km ity in
  t_case (t_var vs) (List.map branch csl)
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(* given a map of modified regions, construct the updated value of [vs] *)
let update_var env (mreg : vsymbol Mreg.t) vs =
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  let rec update vs { fd_ity = ity; fd_mut = mut } =
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    (* are we a mutable variable? *)
    let get_vs r = Mreg.find_def vs r mreg in
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    let vs = Opt.fold (fun _ -> get_vs) vs mut in
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    (* should we update our value further? *)
    let check_reg r _ = reg_occurs r ity.ity_vars in
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    if ity_immutable ity || not (Mreg.exists check_reg mreg) then t_var vs
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    else analyze_var update fs_app env.pure_known env.prog_known vs ity in
  update vs { fd_ity = ity_of_vs vs; fd_ghost = false; fd_mut = None }
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(* given a map of modified regions, update every affected variable in [f] *)
let update_term env (mreg : vsymbol Mreg.t) f =
  (* [vars] : modified variable -> updated value *)
  let update vs _ = match update_var env mreg vs with
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    | { t_node = Tvar nv } when vs_equal vs nv -> None
    | t -> Some t in
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  let vars = Mvs.mapi_filter update f.t_vars in
  (* [vv'] : modified variable -> fresh variable *)
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  let new_var vs _ = mk_var vs.vs_name model2_lab vs.vs_ty in
  let vv' = Mvs.mapi new_var vars in
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  (* update modified variables *)
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  let update v t f = wp_let (Mvs.find v vv') t f in
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  Mvs.fold update vars (subst_at_now true vv' f)

(* look for a variable with a single region equal to [reg] *)
let var_of_region reg f =
  let test vs _ _ = match (ity_of_vs vs).ity_node with
    | Ityapp (_,_,[r]) when reg_equal r reg -> Some vs
    | _ -> None in
  Mvs.fold test f.t_vars None

let quantify env regs f =
  (* mreg : modified region -> vs *)
  let get_var reg () =
    let ty = ty_of_ity reg.reg_ity in
    let id = match var_of_region reg f with
      | Some vs -> vs.vs_name
      | None -> reg.reg_name in
    mk_var id model1_lab ty in
  let mreg = Mreg.mapi get_var regs in
  (* quantify over the modified resions *)
  let f = update_term env mreg f in
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  wp_forall (List.rev (Mreg.values mreg)) f
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(** Invariants *)

let get_invariant km t =
  let ty = t_type t in
  let ts = match ty.ty_node with
    | Tyapp (ts,_) -> ts
    | _ -> assert false in
  let rec find_td = function
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    | (its,_,inv) :: _ when ts_equal ts its.its_ts -> inv
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    | _ :: tdl -> find_td tdl
    | [] -> assert false in
  let pd = Mid.find ts.ts_name km in
  let inv = match pd.Mlw_decl.pd_node with
    | Mlw_decl.PDdata tdl -> find_td tdl
    | _ -> assert false in
  let sbs = Ty.ty_match Mtv.empty (t_type inv) ty in
  let u, p = open_post (t_ty_subst sbs Mvs.empty inv) in
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  wp_expl expl_type_inv (t_subst_single u t p)
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let ps_inv = Term.create_psymbol (id_fresh "inv")
  [ty_var (create_tvsymbol (id_fresh "a"))]

let full_invariant lkm km vs ity =
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  let rec update vs { fd_ity = ity } =
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    if not (ity_has_inv ity) then t_true else
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    (* what is our current invariant? *)
    let f = match ity.ity_node with
      | Ityapp (its,_,_) when its.its_inv ->
          if Debug.test_flag no_track
          then get_invariant km (t_var vs)
          else ps_app ps_inv [t_var vs]
      | _ -> t_true in
    (* what are our sub-invariants? *)
    let join _ fl _ = wp_ands ~sym:true fl in
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    let g = analyze_var update join lkm km vs ity in
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    (* put everything together *)
    wp_and ~sym:true f g
  in
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  update vs { fd_ity = ity; fd_ghost = false; fd_mut = None }
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(** Value tracking *)
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type point = int
type value = point list Mls.t (* constructor -> field list *)
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type state = {
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  st_km   : Mlw_decl.known_map;
  st_lkm  : Decl.known_map;
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  st_mem  : value Hint.t;
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  st_next : point ref;
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}

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(* dead code
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type names = point Mvs.t  (* variable -> point *)
type condition = lsymbol Mint.t (* point -> constructor *)
type lesson = condition list Mint.t (* point -> conditions for invariant *)
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*)
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let empty_state lkm km = {
  st_km   = km;
  st_lkm  = lkm;
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  st_mem  = Hint.create 5;
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  st_next = ref 0;
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}

let next_point state =
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  let res = !(state.st_next) in incr state.st_next; res
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let make_value state ty =
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  let get_p _ = next_point state in
  let new_cs cs = List.map get_p cs.ls_args in
  let add_cs m (cs,_) = Mls.add cs (new_cs cs) m in
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  let csl = match ty.ty_node with
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    | Tyapp (ts,_) -> Decl.find_constructors state.st_lkm ts
    | _ -> [] in
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  List.fold_left add_cs Mls.empty csl

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let match_point state ty p =
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  try Hint.find state.st_mem p with Not_found ->
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  let value = make_value state ty in
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  if not (Mls.is_empty value) then
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    Hint.replace state.st_mem p value;
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  value

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let rec open_pattern state names value p pat = match pat.pat_node with
  | Pwild -> names
  | Pvar vs -> Mvs.add vs p names
  | Papp (cs,patl) ->
      let add_pat names p pat =
        let value = match_point state pat.pat_ty p in
        open_pattern state names value p pat in
      List.fold_left2 add_pat names (Mls.find cs value) patl
  | Por _ ->
      let add_vs vs s = Mvs.add vs (next_point state) s in
      Svs.fold add_vs pat.pat_vars names
  | Pas (pat,vs) ->
      open_pattern state (Mvs.add vs p names) value p pat

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let rec point_of_term state names t = match t.t_node with
  | Tvar vs ->
      Mvs.find vs names
  | Tapp (ls, tl) ->
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      begin match Mid.find ls.ls_name state.st_lkm with
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        | { Decl.d_node = Decl.Ddata tdl } ->
            let is_cs (cs,_) = ls_equal ls cs in
            let is_cs (_,csl) = List.exists is_cs csl in
            if List.exists is_cs tdl
            then point_of_constructor state names ls tl
            else point_of_projection state names ls (List.hd tl)
        | _ -> next_point state
      end
  | Tlet (t1, bt) ->
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      let p1 = point_of_term state names t1 in
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      let v, t2 = t_open_bound bt in
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      let names = Mvs.add v p1 names in
      point_of_term state names t2
  | Tcase (t1,[br]) ->
      let pat, t2 = t_open_branch br in
      let p1 = point_of_term state names t1 in
      let value = match_point state pat.pat_ty p1 in
      let names = open_pattern state names value p1 pat in
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      point_of_term state names t2
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  | Tcase (t1,bl) ->
      (* we treat here the case of a value update: the value
         of each branch must be a distinct constructor *)
      let p = next_point state in
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      let ty = Opt.get t.t_ty in
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      let p1 = point_of_term state names t1 in
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      let value = match_point state (Opt.get t1.t_ty) p1 in
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      let branch acc br =
        let pat, t2 = t_open_branch br in
        let ls = match t2.t_node with
          | Tapp (ls,_) -> ls | _ -> raise Exit in
        let names = open_pattern state names value p1 pat in
        let p2 = point_of_term state names t2 in
        let v2 = match_point state ty p2 in
        Mls.add_new Exit ls (Mls.find_exn Exit ls v2) acc
      in
      begin try
        let value = List.fold_left branch Mls.empty bl in
        let value = Mls.set_union value (make_value state ty) in
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        Hint.replace state.st_mem p value
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      with Exit -> () end;
      p
  | Tconst _ | Tif _ | Teps _ -> next_point state
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  | Tquant _ | Tbinop _ | Tnot _ | Ttrue | Tfalse -> assert false

and point_of_constructor state names ls tl =
  let p = next_point state in
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  let pl = List.map (point_of_term state names) tl in
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  let value = make_value state (Opt.get ls.ls_value) in
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  let value = Mls.add ls pl value in
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  Hint.replace state.st_mem p value;
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  p

and point_of_projection state names ls t1 =
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  let ty = Opt.get t1.t_ty in
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  let csl = match ty.ty_node with
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    | Tyapp (ts,_) -> Decl.find_constructors state.st_lkm ts
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    | _ -> assert false in
  match csl with
    | [cs,pjl] ->
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        let p1 = point_of_term state names t1 in
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        let value = match_point state ty p1 in
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        let rec find_p pjl pl = match pjl, pl with
          | Some pj::_, p::_ when ls_equal ls pj -> p
          | _::pjl, _::pl -> find_p pjl pl
          | _ -> assert false in
        find_p pjl (Mls.find cs value)
    | _ -> next_point state (* more than one, can't choose *)

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let rec track_values state names lesson cond f = match f.t_node with
  | Tapp (ls, [t1]) when ls_equal ls ps_inv ->
      let p1 = point_of_term state names t1 in
      let condl = Mint.find_def [] p1 lesson in
      let contains c1 c2 = Mint.submap (fun _ -> ls_equal) c2 c1 in
      if List.exists (contains cond) condl then
        lesson, t_true
      else
        let good c = not (contains c cond) in
        let condl = List.filter good condl in
        let l = Mint.add p1 (cond::condl) lesson in
        l, get_invariant state.st_km t1
  | Tbinop (Timplies, f1, f2) ->
      let l, f1 = track_values state names lesson cond f1 in
      let _, f2 = track_values state names l cond f2 in
      lesson, t_label_copy f (t_implies_simp f1 f2)
  | Tbinop (Tand, f1, f2) ->
      let l, f1 = track_values state names lesson cond f1 in
      let l, f2 = track_values state names l cond f2 in
      l, t_label_copy f (t_and_simp f1 f2)
  | Tif (fc, f1, f2) ->
      let _, f1 = track_values state names lesson cond f1 in
      let _, f2 = track_values state names lesson cond f2 in
      lesson, t_label_copy f (t_if_simp fc f1 f2)
  | Tcase (t1, bl) ->
      let p1 = point_of_term state names t1 in
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      let value = match_point state (Opt.get t1.t_ty) p1 in
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      let is_pat_var = function
        | { pat_node = Pvar _ } -> true | _ -> false in
      let branch l br =
        let pat, f1, cb = t_open_branch_cb br in
        let learn, cond = match bl, pat.pat_node with
          | [_], _ -> true, cond (* one branch, can learn *)
          | _, Papp (cs, pl) when List.for_all is_pat_var pl ->
              (try true, Mint.add_new Exit p1 cs cond (* can learn *)
              with Exit -> false, cond) (* contradiction, cannot learn *)
          | _, _ -> false, cond (* complex pattern, will not learn *)
        in
        let names = open_pattern state names value p1 pat in
        let m, f1 = track_values state names lesson cond f1 in
        let l = if learn then m else l in
        l, cb pat f1
      in
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      let l, bl = Lists.map_fold_left branch lesson bl in
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      l, t_label_copy f (t_case t1 bl)
  | Tlet (t1, bf) ->
      let p1 = point_of_term state names t1 in
      let v, f1, cb = t_open_bound_cb bf in
      let names = Mvs.add v p1 names in
      let l, f1 = track_values state names lesson cond f1 in
      l, t_label_copy f (t_let_simp t1 (cb v f1))
  | Tquant (Tforall, qf) ->
      let vl, trl, f1, cb = t_open_quant_cb qf in
      let add_vs s vs = Mvs.add vs (next_point state) s in
      let names = List.fold_left add_vs names vl in
      let l, f1 = track_values state names lesson cond f1 in
      l, t_label_copy f (t_forall_simp (cb vl trl f1))
  | Tbinop ((Tor|Tiff),_,_) | Tquant (Texists,_)
  | Tapp _ | Tnot _ | Ttrue | Tfalse -> lesson, f
  | Tvar _ | Tconst _ | Teps _ -> assert false

let track_values lkm km f =
  let state = empty_state lkm km in
  let _, f = track_values state Mvs.empty Mint.empty Mint.empty f in
  f
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(** Weakest preconditions *)

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let rec wp_expr env e q xq =
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  let f = wp_desc env e q xq in
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  if Debug.test_flag debug then begin
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    Format.eprintf "@[--------@\n@[<hov 2>e = %a@]@\n" Mlw_pretty.print_expr e;
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    Format.eprintf "@[<hov 2>q = %a@]@\n" Pretty.print_term q;
    Format.eprintf "@[<hov 2>f = %a@]@\n----@]@." Pretty.print_term f;
  end;
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  f
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and wp_desc env e q xq = match e.e_node with
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  | Elogic t ->
      let v, q = open_post q in
      let t = wp_label e t in
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      (* NOTE: if you replace this t_subst by t_let or anything else,
         you must handle separately the case "let mark = 'now in ...",
         which requires 'now to be substituted for mark in q *)
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      t_subst_single v (to_term t) q
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  | Evalue pv ->
      let v, q = open_post q in
      let t = wp_label e (t_var pv.pv_vs) in
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      t_subst_single v t q
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  | Earrow _ ->
      let q = open_unit_post q in
      (* wp_label e *) q (* FIXME? *)
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  | Elet ({ let_sym = LetV v; let_expr = e1 }, e2)
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    when Opt.equal Loc.equal v.pv_vs.vs_name.id_loc e1.e_loc ->
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    (* we push the label down, past the implicitly inserted "let" *)
      let w = wp_expr env (e_label_copy e e2) q xq in
      let q = create_post v.pv_vs w in
      wp_expr env e1 q xq
  | Elet ({ let_sym = LetV v; let_expr = e1 }, e2) ->
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      let w = wp_expr env e2 q xq in
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      let q = create_post v.pv_vs w in
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      wp_label e (wp_expr env e1 q xq)
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  | Elet ({ let_sym = LetA _; let_expr = e1 }, e2) ->
      let w = wp_expr env e2 q xq in
      let q = create_unit_post w in
      wp_label e (wp_expr env e1 q xq)
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  | Erec (fdl, e1) ->
      let fr = wp_rec_defn env fdl in
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      let fe = wp_expr env e1 q xq in
      let fr = wp_ands ~sym:true fr in
      wp_label e (wp_and ~sym:true fr fe)
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  | Eif (e1, e2, e3) ->
      let res = vs_result e1 in
      let test = t_equ (t_var res) t_bool_true in
      let test = t_label ?loc:e1.e_loc model3_lab test in
      (* if both branches are pure, do not split *)
      let w =
        let get_term e = match e.e_node with
          | Elogic t -> to_term t
          | Evalue v -> t_var v.pv_vs
          | _ -> raise Exit in
        try
          let r2 = get_term e2 in
          let r3 = get_term e3 in
          let v, q = open_post q in
          t_subst_single v (t_if_simp test r2 r3) q
        with Exit ->
          let w2 = wp_expr env e2 q xq in
          let w3 = wp_expr env e3 q xq in
          t_if_simp test w2 w3
      in
      let q = create_post res w in
      wp_label e (wp_expr env e1 q xq)
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  (* optimization for the particular case let _ = e1 in e2 *)
  | Ecase (e1, [{ ppat_pattern = { pat_node = Term.Pwild }}, e2]) ->
      let w = wp_expr env e2 q xq in
      let q = create_post (vs_result e1) w in
      wp_label e (wp_expr env e1 q xq)
  (* optimization for the particular case let () = e1 in e2 *)
  | Ecase (e1, [{ ppat_pattern = { pat_node = Term.Papp (cs,[]) }}, e2])
    when ls_equal cs fs_void ->
      let w = wp_expr env e2 q xq in
      let q = create_unit_post w in
      wp_label e (wp_expr env e1 q xq)
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  | Ecase (e1, bl) ->
      let res = vs_result e1 in
      let branch ({ ppat_pattern = pat }, e) =
        t_close_branch pat (wp_expr env e q xq) in
      let w = t_case (t_var res) (List.map branch bl) in
      let q = create_post res w in
      wp_label e (wp_expr env e1 q xq)
  | Eghost e1 ->
      wp_label e (wp_expr env e1 q xq)
  | Eraise (xs, e1) ->
      let q = try Mexn.find xs xq with
        Not_found -> assert false in
      wp_label e (wp_expr env e1 q xq)
  | Etry (e1, bl) ->
      let branch (xs,v,e) acc =
        let w = wp_expr env e q xq in
        let q = create_post v.pv_vs w in
        Mexn.add xs q acc in
      let xq = List.fold_right branch bl xq in
      wp_label e (wp_expr env e1 q xq)
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  | Eassert (Aassert, f) ->
      let q = open_unit_post q in
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      let f = wp_expl expl_assert f in
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      wp_and ~sym:false (wp_label e f) q
  | Eassert (Acheck, f) ->
      let q = open_unit_post q in
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      let f = wp_expl expl_check f in
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      wp_and ~sym:true (wp_label e f) q
  | Eassert (Aassume, f) ->
      let q = open_unit_post q in
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      let f = wp_expl expl_assume f in
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      wp_implies (wp_label e f) q
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  | Eabsurd ->
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      wp_label e (t_label_add expl_absurd t_absurd)
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  | Eany spec ->
      let p = wp_label e (wp_expl expl_pre spec.c_pre) in
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      let p = t_label ?loc:e.e_loc p.t_label p in
      (* TODO: propagate call labels into tyc.c_post *)
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      let w = wp_abstract env spec.c_effect spec.c_post spec.c_xpost q xq in
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      wp_and ~sym:false p w
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  | Eapp (e1,_,spec) ->
      let p = wp_label e (wp_expl expl_pre spec.c_pre) in
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      let p = t_label ?loc:e.e_loc p.t_label p in
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      let d =
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        if spec.c_letrec = 0 || spec.c_variant = [] then t_true else
        let olds = Mint.find_def [] spec.c_letrec env.letrec_var in
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        if olds = [] then t_true (* we are out of letrec *) else
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        decrease e.e_loc expl_variant env olds spec.c_variant in
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      (* TODO: propagate call labels into tyc.c_post *)
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      let w = wp_abstract env spec.c_effect spec.c_post spec.c_xpost q xq in
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      let w = wp_and ~sym:true d (wp_and ~sym:false p w) in
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      let q = create_unit_post w in
      wp_expr env e1 q xq (* FIXME? should (wp_label e) rather be here? *)
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  | Eabstr (e1, spec) ->
      let p = wp_label e (wp_expl expl_pre spec.c_pre) in
      let w1 = backstep (wp_expr env e1) spec.c_post spec.c_xpost in
      let w2 = wp_abstract env e1.e_effect spec.c_post spec.c_xpost q xq in
      wp_and ~sym:false p (wp_and ~sym:true (wp_label e w1) w2)
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  | Eassign (pl, e1, reg, pv) ->
      (* if we create an intermediate variable npv to represent e1
         in the post-condition of the assign, the call to wp_abstract
         will have to update this variable separately (in addition to
         all existing variables in q that require update), creating
         duplication.  To avoid it, we try to detect whether the value
         of e1 can be represented by an existing pure term that can
         be reused in the post-condition. *)
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      let rec get_term d = match d.e_node with
        | Eghost e | Elet (_,e) | Erec (_,e) -> get_term e
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        | Evalue v -> vs_result e1, t_var v.pv_vs
        | Elogic t -> vs_result e1, t
        | _ ->
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            let ity = ity_of_expr e1 in
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            let id = id_fresh ?loc:e1.e_loc "o" in
            (* must be a pvsymbol or restore_pv will fail *)
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            let npv = create_pvsymbol id ~ghost:e1.e_ghost ity in
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            npv.pv_vs, t_var npv.pv_vs
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      in
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      let res, t = get_term e1 in
      let t = fs_app pl.pl_ls [t] pv.pv_vs.vs_ty in
      let c_q = create_unit_post (t_equ t (t_var pv.pv_vs)) in
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      let eff = eff_write eff_empty reg in
      let w = wp_abstract env eff c_q Mexn.empty q xq in
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      let q = create_post res w in
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      wp_label e (wp_expr env e1 q xq)
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  | Eloop (inv, varl, e1) ->
      (* TODO: what do we do about well-foundness? *)
      let i = wp_expl expl_loop_keep inv in
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      let olds = List.map (fun (t,_) -> t_at_old t) varl in
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      let d = decrease e.e_loc expl_loopvar env olds varl in
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      let q = create_unit_post (wp_and ~sym:true i d) in
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      let w = backstep (wp_expr env e1) q xq in
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      let regs = regs_of_writes e1.e_effect in
      let w = quantify env regs (wp_implies inv w) in
      let i = wp_expl expl_loop_init inv in
      wp_label e (wp_and ~sym:true i w)
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  | Efor ({pv_vs = x}, ({pv_vs = v1}, d, {pv_vs = v2}), inv, e1) ->
      (* wp(for x = v1 to v2 do inv { I(x) } e1, Q, R) =
             v1 > v2  -> Q
         and v1 <= v2 ->     I(v1)
                         and forall S. forall i. v1 <= i <= v2 ->
                                                 I(i) -> wp(e1, I(i+1), R)
                                       and I(v2+1) -> Q *)
      let gt, le, incr = match d with
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        | Mlw_expr.To     -> env.ps_int_gt, env.ps_int_le, env.fs_int_pl
        | Mlw_expr.DownTo -> env.ps_int_lt, env.ps_int_ge, env.fs_int_mn
      in
      let one = t_nat_const 1 in
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      let v1_gt_v2 = ps_app gt [t_var v1; t_var v2] in
      let v1_le_v2 = ps_app le [t_var v1; t_var v2] in
      let q = open_unit_post q in
      let wp_init =
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        wp_expl expl_loop_init (t_subst_single x (t_var v1) inv) in
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      let wp_step =
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        let next = fs_app incr [t_var x; one] ty_int in
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        let post = wp_expl expl_loop_keep (t_subst_single x next inv) in
        wp_expr env e1 (create_unit_post post) xq in
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      let wp_last =
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        let v2pl1 = fs_app incr [t_var v2; one] ty_int in
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        wp_implies (t_subst_single x v2pl1 inv) q in
      let wp_good = wp_and ~sym:true
        wp_init
        (quantify env (regs_of_writes e1.e_effect)
           (wp_and ~sym:true
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              (wp_forall [x] (wp_implies
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                (wp_and ~sym:true (ps_app le [t_var v1; t_var x])
                                  (ps_app le [t_var x;  t_var v2]))
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                (wp_implies inv wp_step)))
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              wp_last))
      in
      let wp_full = wp_and ~sym:true
        (wp_implies v1_gt_v2 q)
        (wp_implies v1_le_v2 wp_good)
      in
      wp_label e wp_full
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and wp_abstract env c_eff c_q c_xq q xq =
  let regs = regs_of_writes c_eff in
  let exns = exns_of_raises c_eff in
  let quantify_post c_q q =
    let v, f = open_post q in
    let c_v, c_f = open_post c_q in
    let c_f = t_subst_single c_v (t_var v) c_f in
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    let f = wp_forall_post v c_f f in
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    quantify env regs f
  in
  let quantify_xpost _ c_xq xq =
    Some (quantify_post c_xq xq) in
  let proceed c_q c_xq =
    let f = quantify_post c_q q in
    (* every xs in exns is guaranteed to be in c_xq and xq *)
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    assert (Mexn.set_submap exns xq);
    assert (Mexn.set_submap exns c_xq);
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    let xq = Mexn.set_inter xq exns in
    let c_xq = Mexn.set_inter c_xq exns in
    let mexn = Mexn.inter quantify_xpost c_xq xq in
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    (* FIXME? This wp_ands is asymmetric in Pgm_wp *)
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    wp_ands ~sym:true (f :: Mexn.values mexn)
  in
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  backstep proceed c_q c_xq
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and wp_fun_defn env { fun_ps = ps ; fun_lambda = l } =
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  let lab = fresh_mark () and c = l.l_spec in
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  let add_arg sbs pv = ity_match sbs pv.pv_ity pv.pv_ity in
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  let subst = List.fold_left add_arg ps.ps_subst l.l_args in
  let regs = Mreg.map (fun _ -> ()) subst.ity_subst_reg in
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  let args = List.map (fun pv -> pv.pv_vs) l.l_args in
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  let env =
    if c.c_letrec = 0 || c.c_variant = [] then env else
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    let lab = t_var lab in
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    let t_at_lab (t,_) = t_app fs_at [t; lab] t.t_ty in
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    let tl = List.map t_at_lab c.c_variant in
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    let lrv = Mint.add c.c_letrec tl env.letrec_var in
    { env with letrec_var = lrv } in
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  let q = old_mark lab (wp_expl expl_post c.c_post) in
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  let conv p = old_mark lab (wp_expl expl_xpost p) in
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  let f = wp_expr env l.l_expr q (Mexn.map conv c.c_xpost) in
  let f = wp_implies c.c_pre (erase_mark lab f) in
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  wp_forall args (quantify env regs f)
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and wp_rec_defn env fdl = List.map (wp_fun_defn env) fdl
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(***
let bool_to_prop env f =
  let ts_bool  = find_ts ~pure:true env "bool" in
  let ls_andb  = find_ls ~pure:true env "andb" in
  let ls_orb   = find_ls ~pure:true env "orb" in
  let ls_notb  = find_ls ~pure:true env "notb" in
  let ls_True  = find_ls ~pure:true env "True" in
  let ls_False = find_ls ~pure:true env "False" in
  let t_True   = fs_app ls_True [] (ty_app ts_bool []) in
  let is_bool ls = ls_equal ls ls_True || ls_equal ls ls_False in
  let rec t_iff_bool f1 f2 = match f1.t_node, f2.t_node with
    | Tnot f1, _ -> t_not_simp (t_iff_bool f1 f2)
    | _, Tnot f2 -> t_not_simp (t_iff_bool f1 f2)
    | Tapp (ps1, [t1; { t_node = Tapp (ls1, []) }]),
      Tapp (ps2, [t2; { t_node = Tapp (ls2, []) }])
      when ls_equal ps1 ps_equ && ls_equal ps2 ps_equ &&
           is_bool ls1 && is_bool ls2 ->
        if ls_equal ls1 ls2 then t_equ t1 t2 else t_neq t1 t2
    | _ ->
        t_iff_simp f1 f2
  in
  let rec t_btop t = t_label ?loc:t.t_loc t.t_label (* t_label_copy? *)
    (match t.t_node with
    | Tif (f,t1,t2) ->
        t_if_simp (f_btop f) (t_btop t1) (t_btop t2)
    | Tapp (ls, [t1;t2]) when ls_equal ls ls_andb ->
        t_and_simp (t_btop t1) (t_btop t2)
    | Tapp (ls, [t1;t2]) when ls_equal ls ls_orb ->
        t_or_simp (t_btop t1) (t_btop t2)
    | Tapp (ls, [t1]) when ls_equal ls ls_notb ->
        t_not_simp (t_btop t1)
    | Tapp (ls, []) when ls_equal ls ls_True ->
        t_true
    | Tapp (ls, []) when ls_equal ls ls_False ->
        t_false
    | _ ->
        t_equ_simp (f_btop t) t_True)
  and f_btop f = match f.t_node with
    | Tapp (ls, [{t_ty = Some {ty_node = Tyapp (ts, [])}} as l; r])
      when ls_equal ls ps_equ && ts_equal ts ts_bool ->
        t_label ?loc:f.t_loc f.t_label (t_iff_bool (t_btop l) (t_btop r))
    | _ ->
        t_map_simp f_btop f
  in
  f_btop f
***)
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(* replace t_absurd with t_false *)
let rec unabsurd f = match f.t_node with
  | Tapp (ls, []) when ls_equal ls ls_absurd ->
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      t_label_copy f (t_label_add keep_on_simp_label t_false)
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  | _ ->
      t_map unabsurd f

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let add_wp_decl km name f uc =
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  (* prepare a proposition symbol *)
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  let s = "WP_parameter " ^ name.id_string in
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  let lab = Ident.create_label ("expl:VC for " ^ name.id_string) in
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  let label = Slab.add lab name.id_label in
  let id = id_fresh ~label ?loc:name.id_loc s in
  let pr = create_prsymbol id in
  (* prepare the VC formula *)
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  let f = remove_at f in
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  (* let f = bool_to_prop uc f in *)
  let f = unabsurd f in
  (* get a known map with tuples added *)
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  let lkm = Theory.get_known uc in
  (* remove redundant invariants *)
  let f = if Debug.test_flag no_track then f else track_values lkm km f in
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  (* simplify f *)
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  let f = if Debug.test_flag no_eval then f else
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    (* do preliminary checks on f to spare eval_match any surprises *)
    let _lkm = Decl.known_add_decl lkm (create_prop_decl Pgoal pr f) in
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    Eval_match.eval_match ~inline:Eval_match.inline_nonrec_linear lkm f in
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  (* printf "wp: f=%a@." print_term f; *)
  let d = create_prop_decl Pgoal pr f in
  Theory.add_decl uc d

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let mk_env env km th =
  let th_int = Env.find_theory env ["int"] "Int" in
  { prog_known = km;
    pure_known = Theory.get_known th;
    global_env = env;
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    ps_int_le  = Theory.ns_find_ls th_int.th_export ["infix <="];
    ps_int_ge  = Theory.ns_find_ls th_int.th_export ["infix >="];
    ps_int_lt  = Theory.ns_find_ls th_int.th_export ["infix <"];
    ps_int_gt  = Theory.ns_find_ls th_int.th_export ["infix >"];
    fs_int_pl  = Theory.ns_find_ls th_int.th_export ["infix +"];
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    fs_int_mn  = Theory.ns_find_ls th_int.th_export ["infix -"];
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    letrec_var = Mint.empty;
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  }
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let wp_let env km th { let_sym = lv; let_expr = e } =
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  let env = mk_env env km th in
  let q, xq = default_post e.e_vty e.e_effect in
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  let f = wp_expr env e q xq in
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  let f = wp_forall (Mvs.keys f.t_vars) f in
  let id = match lv with
    | LetV pv -> pv.pv_vs.vs_name
    | LetA ps -> ps.ps_name in
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  add_wp_decl km id f th
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let wp_rec env km th fdl =
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  let env = mk_env env km th in
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  let fl = wp_rec_defn env fdl in
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  let add_one th d f =
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    Debug.dprintf debug "wp %s = %a@\n----------------@."
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      d.fun_ps.ps_name.id_string Pretty.print_term f;
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    let f = wp_forall (Mvs.keys f.t_vars) f in
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    add_wp_decl km d.fun_ps.ps_name f th
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  in
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  List.fold_left2 add_one th fdl fl
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let wp_val _env _km th _lv = th
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(*****************************************************************************)

(* Efficient Weakest Preconditions

  Following Leino, see
  http://research.microsoft.com/apps/pubs/default.aspx?id=70052

  Roughly, the idea is the following. From a program expression e, we compute
  two formulas OK and N. Formula OK means ``the execution of e does not go
  wrong'' and formula N is an input-output relation between initial and
  final state of e's execution.

  Thus the weakest precondition of e is simply OK.
  N is involved in recursive computations, e.g.
  OK(fun x -> {p} e {q}) = forall x. p => OK(e) /\ (forall result. N(e) => q)
  And so on.

  In practice, this is a bit more involved, since execution of e may raise
  exceptions. So formula N comes with other formulas E(x), once for each
  exception x that is possibly raised by e. E(x) is the input-output relation
  that holds when exception x is raised.
*)

let fast_wp = Debug.register_flag "fast_wp"
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  ~desc:"Efficient@ Weakest@ Preconditions.@ \
    Work@ in@ progress,@ not@ ready@ for@ use."
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module Subst : sig
   (* A substitution, or state, represents the state at a given point in the
      program. It maps each region to the name that should be used to refer to
      the value of the region in the current state. *)

   type t
   (* the type of substitutions *)

   val init : effect -> t
   (* the initial substitution for a program with given effects *)

   val init_reg : Sreg.t -> t
   (* the initial substitution given a set of regions *)

   val empty : t
   (* the empty substitution *)

   val refresh : Sreg.t -> t -> t
   (* refresh the state, ie, generate new names for all variables in the region
      set *)

   val term : wp_env -> t -> term -> term
   (* In the given term, substitute the variables that refer to the current
      state by the symbols that stand for the value of these variables at this
      point in execution, as recorded in subtitution [t]. *)

   val merge_states : t -> Sreg.t * t -> Sreg.t * t -> t * term * term
   (* Given a base state and two branches (represented by written regions in
      each branch, and a state), return the state of the join point of the
      branches, and two formulas. The first formula links the first branch
      state with the join state, the second formula links the second branch
      state with the join state. *)

end = struct

  type t = vsymbol option ref Mreg.t
  (* Each region is mapped to a reference, which represents the name of the
     region. At beginning, this reference points to [None], meaning that the
     name of the region is not yet decided. When needed, the value under the
     reference is changed to [Some v], with [v] a fresh variable symbol, to fix
     the name of the region. This delayed choice of name allows getting better
     names for regions, so a region for variable [x] ends up called something
     like [x1] instead of the generic [rho1]. *)

  let name_from_region ?hint ?id reg =
    let id = match id with
      | Some x ->
          (* an id was provided, take it *)
          x
      | None ->
          (* none was provided, maybe there was a hint state *)
          begin match hint with
            | None ->
                (* no hint, return default *)
                reg.reg_name
            | Some s ->
                begin try match !(Mreg.find reg s) with
                  | None ->
                      (* the hint state also contains no id, return default *)
                      reg.reg_name
                  | Some x ->
                      (* the hint state contains something, take it *)
                      x.vs_name
                  with Not_found ->
                    (* the hint state has no entry for that region,
                       return default *)
                    reg.reg_name
                end
          end
    in
    mk_var id Slab.empty (ty_of_ity reg.reg_ity)

  let init_reg reg =
    Mreg.map (fun () -> ref None) reg

  let init effect =
    init_reg effect.eff_writes

  let empty = Mreg.empty

  let refresh regset s =
    Sreg.fold (fun reg acc -> Mreg.add reg (ref None) acc) regset s

  let term env sub t =
    let mreg = Mreg.mapi_filter (fun reg vr ->
      match !vr with
      | Some _ -> !vr
      | None ->
          match var_of_region reg t with
          | Some v ->
              let v' = name_from_region ~id:v.vs_name reg in
              vr := Some v';
              !vr
          | None -> None) sub in
    let r = update_term env mreg t in
    r

  let show_state fmt s =
    Format.fprintf fmt "{ ";
    Mreg.iter (fun k rv ->
      match !rv with
      | Some v ->
          Format.fprintf fmt " %a |-> %a; @ "
            Mlw_pretty.print_reg k Pretty.print_vs v;
      | None ->
          Format.fprintf fmt " %a |-> _; @  "
            Mlw_pretty.print_reg k) s;
    Format.fprintf fmt "}"

  (* Update the name of region [reg] in substitution [s], possibly based on
     the provided [hint], and return a variable of that name. *)
  let region_name ?hint reg s =
    let rv = Mreg.find reg s in
    match !rv with
      | Some x -> t_var x
      | None ->
          let new_name = name_from_region ?hint reg in
          rv := Some new_name;
          t_var (new_name)

  let merge_states base (reg1,s1) (reg2,s2) =
    let all_regs = Sreg.union reg1 reg2 in
    Sreg.fold
      (fun reg (s, f1, f2) ->
        (* If both branches modify region [reg], pick the name from [s2], and
           add the necessary equations to [f1]. Otherwise, pick the name from
           the branch that modifies [reg], and add the necessary equations to
           the formula for the other branch.*)
        if Sreg.mem reg reg1 && Sreg.mem reg reg2 then begin
          Mreg.add reg (Mreg.find reg s2) s,
          t_and_simp f1 (t_equ (region_name reg s2) (region_name reg s1)),
          f2
        end else if Sreg.mem reg reg2 then begin
          Mreg.add reg (Mreg.find reg s2) s,
          t_and_simp f1 (t_equ (region_name reg s2)
                          (region_name ~hint:s2 reg base)),
          f2
        end else begin
          Mreg.add reg (Mreg.find reg s1) s,
          f1,
          t_and_simp f2 (t_equ (region_name reg s1)
                          (region_name ~hint:s1 reg base))
        end) all_regs (base, t_true, t_true)
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end