mlw_wp.ml 32 KB
Newer Older
Andrei Paskevich's avatar
Andrei Paskevich committed
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
(**************************************************************************)
(*                                                                        *)
(*  Copyright (C) 2010-2012                                               *)
(*    François Bobot                                                      *)
(*    Jean-Christophe Filliâtre                                           *)
(*    Claude Marché                                                       *)
(*    Guillaume Melquiond                                                 *)
(*    Andrei Paskevich                                                    *)
(*                                                                        *)
(*  This software is free software; you can redistribute it and/or        *)
(*  modify it under the terms of the GNU Library General Public           *)
(*  License version 2.1, with the special exception on linking            *)
(*  described in file LICENSE.                                            *)
(*                                                                        *)
(*  This software is distributed in the hope that it will be useful,      *)
(*  but WITHOUT ANY WARRANTY; without even the implied warranty of        *)
(*  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.                  *)
(*                                                                        *)
(**************************************************************************)

open Why3
22
open Util
Andrei Paskevich's avatar
Andrei Paskevich committed
23 24 25
open Ident
open Ty
open Term
26
open Decl
Andrei Paskevich's avatar
Andrei Paskevich committed
27 28 29 30 31
open Theory
open Mlw_ty
open Mlw_ty.T
open Mlw_expr

32 33
let debug = Debug.register_flag "whyml_wp"

34
(** Marks *)
Andrei Paskevich's avatar
Andrei Paskevich committed
35 36 37 38

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

39 40 41
let vtv_mark = vty_value (ity_pur ts_mark [])

let fresh_mark () = create_vsymbol (id_fresh "'mark") ty_mark
42

Andrei Paskevich's avatar
Andrei Paskevich committed
43 44 45 46 47 48 49 50
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)

51 52
let th_mark_at =
  let uc = create_theory (id_fresh "WP builtins: at") in
Andrei Paskevich's avatar
Andrei Paskevich committed
53 54
  let uc = add_ty_decl uc ts_mark in
  let uc = add_param_decl uc fs_at in
55 56 57 58 59
  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
Andrei Paskevich's avatar
Andrei Paskevich committed
60 61 62
  let uc = add_param_decl uc fs_old in
  close_theory uc

Andrei Paskevich's avatar
Andrei Paskevich committed
63 64 65
let fs_now = create_lsymbol (id_fresh "'now") [] (Some ty_mark)
let t_now = fs_app fs_now [] ty_mark
let e_now = e_lapp fs_now [] (ity_pur ts_mark [])
Andrei Paskevich's avatar
Andrei Paskevich committed
66

67 68 69 70 71
(* [vs_old] appears in the postconditions given to the core API,
   which expects every vsymbol to be a pure part of a pvsymbol *)
let pv_old = create_pvsymbol (id_fresh "'old") vtv_mark
let vs_old = pv_old.pv_vs
let t_old  = t_var vs_old
72

Andrei Paskevich's avatar
Andrei Paskevich committed
73 74
let t_at_old t = t_app fs_at [t; t_old] t.t_ty

75 76
let ls_absurd = create_lsymbol (id_fresh "absurd") [] None
let t_absurd  = ps_app ls_absurd []
77

78
let mk_t_if f = t_if f t_bool_true t_bool_false
79
let to_term t = if t.t_ty = None then mk_t_if t else t
80

81 82
(* any vs in post/xpost is either a pvsymbol or a fresh mark *)
let vtv_of_vs vs =
83
  try (restore_pv vs).pv_vtv with Not_found -> vtv_mark
84 85 86

(* replace every occurrence of [old(t)] with [at(t,'old)] *)
let rec remove_old f = match f.t_node with
Andrei Paskevich's avatar
Andrei Paskevich committed
87
  | Tapp (ls,[t]) when ls_equal ls fs_old -> t_at_old (remove_old t)
88 89 90 91 92 93 94 95
  | _ -> 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

96 97 98 99
(* 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] *)
Andrei Paskevich's avatar
Andrei Paskevich committed
100 101
let erase_mark lab t = t_subst_single lab t_now t

Andrei Paskevich's avatar
Andrei Paskevich committed
102 103
(* retreat to the point of the current postcondition's ['old] *)
let backstep fn q xq =
Andrei Paskevich's avatar
Andrei Paskevich committed
104 105 106
  let lab = fresh_mark () in
  let f = fn (old_mark lab q) (Mexn.map (old_mark lab) xq) in
  erase_mark lab f
107

108
(** WP utilities *)
109

110 111 112
let fs_void = fs_tuple 0
let t_void = fs_app fs_void [] ty_unit

113 114 115 116 117 118
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
119
  create_post vs t_true, Mexn.mapi default_exn_post ef.eff_raises
120

121 122 123 124 125
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

Andrei Paskevich's avatar
Andrei Paskevich committed
126 127 128 129 130
let expl_pre       = Ident.create_label "expl:precondition"
let expl_post      = Ident.create_label "expl:normal postcondition"
let expl_xpost     = Ident.create_label "expl:exceptional postcondition"
let expl_assert    = Ident.create_label "expl:assertion"
let expl_check     = Ident.create_label "expl:check"
131
let expl_inv       = Ident.create_label "expl:type invariant"
132
let expl_variant   = Ident.create_label "expl:variant decreases"
Andrei Paskevich's avatar
Andrei Paskevich committed
133 134 135 136 137 138
let expl_loop_init = Ident.create_label "expl:loop invariant init"
let expl_loop_keep = Ident.create_label "expl:loop invariant preservation"
let expl_loop_var  = Ident.create_label "expl:loop variant decreases"
(* FIXME? couldn't we just reuse "loop invariant" explanations? *)
let expl_for_init  = Ident.create_label "expl:for loop initialization"
let expl_for_keep  = Ident.create_label "expl:for loop preservation"
139

140 141
let wp_expl l f =
  let lab = Slab.add Split_goal.stop_split f.t_label in
142
  t_label ?loc:f.t_loc (Slab.add l lab) f
143

144
let wp_and ~sym f1 f2 =
145 146
  if sym then t_and_simp f1 f2 else t_and_asym_simp f1 f2

147
let wp_ands ~sym fl =
148 149
  if sym then t_and_simp_l fl else t_and_asym_simp_l fl

150
let wp_implies f1 f2 = t_implies_simp f1 f2
151

152 153
let wp_let v t f = t_let_close_simp v t f

154 155
let wp_forall vl f = t_forall_close_simp vl [] f

156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178
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) ->
        begin match down l with
          | None, _ ->
              let t, r = down r in
              t, t_label_copy p (t_and_simp l r)
          | t, l ->
              t, t_label_copy p (t_and_simp l r)
        end
    | 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, t_true
    | _ ->
        None, p
  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)
179

180 181 182
let regs_of_reads  eff = Sreg.union eff.eff_reads eff.eff_ghostr
let regs_of_writes eff = Sreg.union eff.eff_writes eff.eff_ghostw
let regs_of_effect eff = Sreg.union (regs_of_reads eff) (regs_of_writes eff)
Andrei Paskevich's avatar
Andrei Paskevich committed
183
let exns_of_raises eff = Sexn.union eff.eff_raises eff.eff_ghostx
184

185 186 187 188
let open_post q =
  let v, f = open_post q in
  v, t_label_copy q f

189 190 191 192 193 194 195 196 197 198 199
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)

200 201 202 203 204 205
(** WP state *)

type wp_env = {
  prog_known : Mlw_decl.known_map;
  pure_known : Decl.known_map;
  global_env : Env.env;
Andrei Paskevich's avatar
Andrei Paskevich committed
206 207 208 209 210
  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;
211
  letrec_var : term list Mint.t;
212
}
213

214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241
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,_) =
    let f, pl = Util.map_fold_left add_arg t_false cs.ls_args in
    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
        (ps_app env.ps_int_le [t_int_const "0"; old_t])
        (ps_app env.ps_int_lt [t; old_t])
  | None -> decrease_alg ?loc env old_t t

242 243
let decrease ?loc env olds varl =
  let rec decr pr olds varl = match olds, varl with
244 245 246 247 248 249 250 251 252 253
    | [], [] -> (* 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)
254
    | _ -> assert false
Andrei Paskevich's avatar
Andrei Paskevich committed
255
  in
256
  decr t_true olds varl
Andrei Paskevich's avatar
Andrei Paskevich committed
257

258 259
(** Reconstruct pure values after writes *)

260
let find_constructors lkn kn sts ity = match ity.ity_node with
261 262 263
  | Itypur (ts,_) ->
      let base = ity_pur ts (List.map ity_var ts.ts_args) in
      let sbs = ity_match ity_subst_empty base ity in
264
      let csl = Decl.find_constructors lkn ts in
265 266 267 268 269 270 271 272
      if csl = [] || Sts.mem ts sts then Loc.errorm
        "Cannot update values of type %a" Mlw_pretty.print_ity base;
      let subst ty = ity_full_inst sbs (ity_of_ty ty), None in
      let cnstr (cs,_) = cs, List.map subst cs.ls_args in
      Sts.add ts sts, List.map cnstr csl
  | Ityapp (its,_,_) ->
      let base = ity_app its (List.map ity_var its.its_args) its.its_regs in
      let sbs = ity_match ity_subst_empty base ity in
273
      let csl = Mlw_decl.find_constructors kn its in
274 275 276 277 278 279 280 281 282
      if csl = [] || Sts.mem its.its_pure sts then Loc.errorm
        "Cannot update values of type %a" Mlw_pretty.print_ity base;
      let subst vtv =
        ity_full_inst sbs vtv.vtv_ity,
        Util.option_map (reg_full_inst sbs) vtv.vtv_mut in
      let cnstr (cs,_) = cs.pl_ls, List.map subst cs.pl_args in
      Sts.add its.its_pure sts, List.map cnstr csl
  | Ityvar _ -> assert false

283 284 285 286 287 288 289 290 291 292 293
let analyze_var fn_down fn_join lkn kn sts vs ity =
  let sts, csl = find_constructors lkn kn sts ity in
  let branch (cs,ityl) =
    let mk_var (ity,_) = create_vsymbol (id_fresh "y") (ty_of_ity ity) in
    let vars = List.map mk_var ityl in
    let mk_arg vs (ity, mut) = fn_down sts vs ity mut in
    let t = fn_join cs (List.map2 mk_arg vars ityl) vs.vs_ty in
    let pat = pat_app cs (List.map pat_var vars) vs.vs_ty in
    t_close_branch pat t in
  t_case (t_var vs) (List.map branch csl)

294
let update_var env mreg vs =
295 296 297 298 299 300
  let rec update sts vs ity mut =
    (* are we a mutable variable? *)
    let get_vs r = Mreg.find_def vs r mreg in
    let vs = Util.option_apply vs get_vs mut in
    (* should we update our value further? *)
    let check_reg r _ = reg_occurs r ity.ity_vars in
301 302
    if ity_pure ity || not (Mreg.exists check_reg mreg) then t_var vs
    else analyze_var update fs_app env.pure_known env.prog_known sts vs ity
303
  in
304
  let vtv = vtv_of_vs vs in
305 306
  update Sts.empty vs vtv.vtv_ity vtv.vtv_mut

Andrei Paskevich's avatar
Andrei Paskevich committed
307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325
(* substitute the updated values in the "contemporary" variables *)
let rec subst_at_now now m t = match t.t_node with
  | Tvar vs when now ->
      begin try t_var (Mvs.find vs m) 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 m) t
  | Tlet _ | Tcase _ | Teps _ | Tquant _ ->
      (* do not open unless necessary *)
      let m = Mvs.set_inter m t.t_vars in
      if Mvs.is_empty m then t else
      t_map (subst_at_now now m) t
  | _ ->
      t_map (subst_at_now now m) t

326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342
(* quantify over all references in eff
   eff : effect
   f   : formula

   let eff = { rho1, ..., rhon }
   we collect in vars all variables involving these regions
   let vars = { v1, ..., vm }

     forall r1:ty(rho1). ... forall rn:ty(rhon).
     let v'1 = update v1 r1...rn in
     ...
     let v'm = update vm r1...rn in
     f[vi <- v'i]
*)

let model1_lab = Slab.singleton (create_label "model:1")
let model2_lab = Slab.singleton (create_label "model:quantify(2)")
343 344
let model3_lab = Slab.singleton (create_label "model:cond")

345 346
let mk_var id label ty = create_vsymbol (id_clone ~label id) ty

347 348
let quantify env regs f =
  (* mreg : updated region -> vs *)
349
  let get_var reg () =
350
    let test vs _ id = match vtv_of_vs vs with
351 352 353 354 355 356
      | { vtv_ity = { ity_node = Ityapp (_,_,[r]) }}
      | { vtv_mut = Some r } when reg_equal r reg -> vs.vs_name
      | _ -> id in
    let id = Mvs.fold test f.t_vars reg.reg_name in
    mk_var id model1_lab (ty_of_ity reg.reg_ity)
  in
357
  let mreg = Mreg.mapi get_var regs in
358
  (* update all program variables involving these regions *)
359
  let update_var vs _ = match update_var env mreg vs with
360 361 362 363 364 365 366
    | { t_node = Tvar nv } when vs_equal vs nv -> None
    | t -> Some t in
  let vars = Mvs.mapi_filter update_var f.t_vars in
  (* vv' : old vs -> new vs *)
  let new_var vs _ = mk_var vs.vs_name model2_lab vs.vs_ty in
  let vv' = Mvs.mapi new_var vars in
  (* quantify *)
367
  let update v t f = wp_let (Mvs.find v vv') t f in
Andrei Paskevich's avatar
Andrei Paskevich committed
368
  let f = Mvs.fold update vars (subst_at_now true vv' f) in
369
  wp_forall (List.rev (Mreg.values mreg)) f
370

371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390
(* value tracking *)

type point = int
type value = point list Mls.t (* constructor -> field list *)
type state = {
  kn     : Decl.known_map;
  memory : (point, value) Hashtbl.t;
  next   : point ref;
}
type names = point Mvs.t

let empty_state kn = {
  kn     = kn;
  memory = Hashtbl.create 5;
  next   = ref 0;
}

let next_point state =
  let res = !(state.next) in incr state.next; res

391
let make_value state ty =
392 393 394
  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
395 396 397
  let csl = match ty.ty_node with
    | Tyapp (ts,_) -> Decl.find_constructors state.kn ts
    | _ -> assert false in
398 399
  List.fold_left add_cs Mls.empty csl

400
let match_point state ty p =
401
  try Hashtbl.find state.memory p with Not_found ->
402
  let value = make_value state ty in
403 404 405
  Hashtbl.replace state.memory p value;
  value

406 407 408 409 410 411 412 413 414 415 416 417 418 419
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

420 421 422 423 424 425 426 427 428 429 430 431 432 433
let rec point_of_term state names t = match t.t_node with
  | Tvar vs ->
      Mvs.find vs names
  | Tapp (ls, tl) ->
      begin match Mid.find ls.ls_name state.kn with
        | { 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) ->
434
      let p1 = point_of_term state names t1 in
435
      let v, t2 = t_open_bound bt in
436 437 438 439 440 441 442
      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
443
      point_of_term state names t2
444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466
  | 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
      let ty = of_option t.t_ty in
      let p1 = point_of_term state names t1 in
      let value = match_point state (of_option t1.t_ty) p1 in
      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
        Hashtbl.replace state.memory p value
      with Exit -> () end;
      p
  | Tconst _ | Tif _ | Teps _ -> next_point state
467 468 469 470
  | Tquant _ | Tbinop _ | Tnot _ | Ttrue | Tfalse -> assert false

and point_of_constructor state names ls tl =
  let p = next_point state in
471 472 473
  let pl = List.map (point_of_term state names) tl in
  let value = make_value state (of_option ls.ls_value) in
  let value = Mls.add ls pl value in
474 475 476 477
  Hashtbl.replace state.memory p value;
  p

and point_of_projection state names ls t1 =
478 479 480 481 482 483
  let ty = of_option t1.t_ty in
  let csl = match ty.ty_node with
    | Tyapp (ts,_) -> Decl.find_constructors state.kn ts
    | _ -> assert false in
  match csl with
    | [cs,pjl] ->
484
        let p1 = point_of_term state names t1 in
485
        let value = match_point state ty p1 in
486 487 488 489 490 491 492
        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 *)

493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530
(* invariants *)

let get_invariant kn v =
  let ts = match v.vs_ty.ty_node with
    | Tyapp (ts,_) -> ts
    | _ -> assert false in
  let rec find_td = function
    | (its,_,inv) :: _ when ts_equal ts its.its_pure -> inv
    | _ :: tdl -> find_td tdl
    | [] -> assert false in
  let pd = Mid.find ts.ts_name kn 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) v.vs_ty in
  let u, p = open_post (t_ty_subst sbs Mvs.empty inv) in
  wp_expl expl_inv (t_subst_single u (t_var v) p)

let ps_inv = Term.create_psymbol (id_fresh "inv")
  [ty_var (create_tvsymbol (id_fresh "a"))]

let full_invariant lkn kn vs ity =
  let rec update sts vs ity _ =
    if not (ity_inv ity) then t_true else
    (* what is our current invariant? *)
    let f = match ity.ity_node with
      | Ityapp (its,_,_) when its.its_inv ->
          get_invariant kn vs
        (* ps_app ps_inv [t_var vs] *)
      | _ -> t_true in
    (* what are our sub-invariants? *)
    let join _ fl _ = wp_ands ~sym:true fl in
    let g = analyze_var update join lkn kn sts vs ity in
    (* put everything together *)
    wp_and ~sym:true f g
  in
  update Sts.empty vs ity None

531 532
(** Weakest preconditions *)

533
let rec wp_expr env e q xq =
Andrei Paskevich's avatar
Andrei Paskevich committed
534
  let f = wp_desc env e q xq in
535
  if Debug.test_flag debug then begin
536
    Format.eprintf "@[--------@\n@[<hov 2>e = %a@]@\n" Mlw_pretty.print_expr e;
537 538 539
    Format.eprintf "@[<hov 2>q = %a@]@\n" Pretty.print_term q;
    Format.eprintf "@[<hov 2>f = %a@]@\n----@]@." Pretty.print_term f;
  end;
540
  f
541

542
and wp_desc env e q xq = match e.e_node with
543 544 545
  | Elogic t ->
      let v, q = open_post q in
      let t = wp_label e t in
546 547 548
      (* 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 *)
549
      t_subst_single v (to_term t) q
Andrei Paskevich's avatar
Andrei Paskevich committed
550 551 552
  | Evalue pv ->
      let v, q = open_post q in
      let t = wp_label e (t_var pv.pv_vs) in
553
      t_subst_single v t q
554 555 556
  | Earrow _ ->
      let q = open_unit_post q in
      (* wp_label e *) q (* FIXME? *)
557
  | Elet ({ let_sym = lv; let_expr = e1 }, e2) ->
558 559 560 561 562
      let w = wp_expr env e2 q xq in
      let q = match lv with
        | LetV v -> create_post v.pv_vs w
        | LetA _ -> create_unit_post w in
      wp_label e (wp_expr env e1 q xq)
Andrei Paskevich's avatar
Andrei Paskevich committed
563
  | Erec (rdl, e) ->
564 565 566
      let fr = wp_rec_defn env rdl in
      let fe = wp_expr env e q xq in
      wp_and ~sym:true (wp_ands ~sym:true fr) fe
567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588
  | 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)
589 590 591 592 593 594 595 596 597 598 599
  (* 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)
600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619
  | 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)
620 621
  | Eassert (Aassert, f) ->
      let q = open_unit_post q in
622
      let f = wp_expl expl_assert f in
623 624 625
      wp_and ~sym:false (wp_label e f) q
  | Eassert (Acheck, f) ->
      let q = open_unit_post q in
626
      let f = wp_expl expl_check f in
627 628 629 630
      wp_and ~sym:true (wp_label e f) q
  | Eassert (Aassume, f) ->
      let q = open_unit_post q in
      wp_implies (wp_label e f) q
Andrei Paskevich's avatar
Andrei Paskevich committed
631
  | Eabsurd ->
632
      wp_label e t_absurd
633 634
  | Eany spec ->
      let p = wp_label e (wp_expl expl_pre spec.c_pre) in
Andrei Paskevich's avatar
Andrei Paskevich committed
635 636
      let p = t_label ?loc:e.e_loc p.t_label p in
      (* TODO: propagate call labels into tyc.c_post *)
637
      let w = wp_abstract env spec.c_effect spec.c_post spec.c_xpost q xq in
Andrei Paskevich's avatar
Andrei Paskevich committed
638
      wp_and ~sym:false p w (* FIXME? do we need pre? *)
639 640
  | Eapp (e1,_,spec) ->
      let p = wp_label e (wp_expl expl_pre spec.c_pre) in
641
      let p = t_label ?loc:e.e_loc p.t_label p in
642 643 644 645 646
      let d = if spec.c_letrec = 0 then t_true else
        let olds = Mint.find_def [] spec.c_letrec env.letrec_var in
        if olds = [] then t_true (* we are out of letrec *) else
        let d = decrease ?loc:e.e_loc env olds spec.c_variant in
        wp_expl expl_variant (t_label ?loc:e.e_loc d.t_label d) in
647
      (* TODO: propagate call labels into tyc.c_post *)
648
      let w = wp_abstract env spec.c_effect spec.c_post spec.c_xpost q xq in
649
      let w = wp_and ~sym:false (wp_and ~sym:true d p) w in (* FIXME? ~sym? *)
650 651 652
      let q = create_unit_post w in
      wp_expr env e1 q xq (* FIXME? should (wp_label e) rather be here? *)
  | Eabstr (e1, c_q, c_xq) ->
Andrei Paskevich's avatar
Andrei Paskevich committed
653
      let w1 = backstep (wp_expr env e1) c_q c_xq in
654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669
      let w2 = wp_abstract env e1.e_effect c_q c_xq q xq in
      wp_and ~sym:true (wp_label e w1) w2
  | Eassign (e1, reg, pv) ->
      let rec get_term d = match d.e_node with
        | Elogic t -> t
        | Evalue v -> t_var v.pv_vs
        | Eghost e | Elet (_,e) | Erec (_,e) -> get_term e
        | _ -> Loc.errorm ?loc:e.e_loc
            "Cannot compute the WP for this assignment"
      in
      let f = t_equ (get_term e1) (t_var pv.pv_vs) in
      let c_q = create_unit_post f in
      let eff = eff_write eff_empty reg in
      let w = wp_abstract env eff c_q Mexn.empty q xq in
      let q = create_post (vs_result e1) w in
      wp_label e (wp_expr env e1 q xq)
Andrei Paskevich's avatar
Andrei Paskevich committed
670 671 672
  | Eloop (inv, varl, e1) ->
      (* TODO: what do we do about well-foundness? *)
      let i = wp_expl expl_loop_keep inv in
673 674
      let olds = List.map (fun (t,_) -> t_at_old t) varl in
      let d = decrease ?loc:e.e_loc env olds varl in
675
      let d = wp_expl expl_loop_var d in
Andrei Paskevich's avatar
Andrei Paskevich committed
676
      let q = create_unit_post (wp_and ~sym:true i d) in
Andrei Paskevich's avatar
Andrei Paskevich committed
677
      let w = backstep (wp_expr env e1) q xq in
Andrei Paskevich's avatar
Andrei Paskevich committed
678 679 680 681
      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)
Andrei Paskevich's avatar
Andrei Paskevich committed
682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718
  | 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
        | Mlw_expr.To     -> env.ps_int_gt, env.ps_int_le, t_int_const "1"
        | Mlw_expr.DownTo -> env.ps_int_lt, env.ps_int_ge, t_int_const "-1" in
      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 =
        wp_expl expl_for_init (t_subst_single x (t_var v1) inv) in
      let wp_step =
        let nextx = fs_app env.fs_int_pl [t_var x; incr] ty_int in
        let post = create_unit_post (t_subst_single x nextx inv) in
        wp_expr env e1 post xq in
      let wp_last =
        let v2pl1 = fs_app env.fs_int_pl [t_var v2; incr] ty_int in
        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
              (wp_expl expl_for_keep (wp_forall [x] (wp_implies
                (wp_and ~sym:true (ps_app le [t_var v1; t_var x])
                                  (ps_app le [t_var x;  t_var v2]))
                (wp_implies inv wp_step))))
              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
719

Andrei Paskevich's avatar
Andrei Paskevich committed
720 721 722 723 724 725 726
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
727
    let f = wp_forall_post v c_f f in
Andrei Paskevich's avatar
Andrei Paskevich committed
728 729 730 731 732 733 734
    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 *)
735 736
    assert (Mexn.set_submap exns xq);
    assert (Mexn.set_submap exns c_xq);
Andrei Paskevich's avatar
Andrei Paskevich committed
737 738 739
    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
740
    (* FIXME? This wp_ands is asymmetric in Pgm_wp *)
Andrei Paskevich's avatar
Andrei Paskevich committed
741 742
    wp_ands ~sym:true (f :: Mexn.values mexn)
  in
Andrei Paskevich's avatar
Andrei Paskevich committed
743
  backstep proceed c_q c_xq
Andrei Paskevich's avatar
Andrei Paskevich committed
744

745 746 747
and wp_lambda env lr l =
  let lab = fresh_mark () in
  let args = List.map (fun pv -> pv.pv_vs) l.l_args in
Andrei Paskevich's avatar
Andrei Paskevich committed
748
  let env = if lr = 0 || l.l_variant = [] then env else
749
    let lab = t_var lab in
Andrei Paskevich's avatar
Andrei Paskevich committed
750 751
    let t_at_lab (t,_) = t_app fs_at [t; lab] t.t_ty in
    let tl = List.map t_at_lab l.l_variant in
752 753 754 755 756 757
    { env with letrec_var = Mint.add lr tl env.letrec_var }
  in
  let q = old_mark lab (wp_expl expl_post l.l_post) in
  let conv p = old_mark lab (wp_expl expl_xpost p) in
  let f = wp_expr env l.l_expr q (Mexn.map conv l.l_xpost) in
  let f = wp_implies l.l_pre (erase_mark lab f) in
758
  let f = quantify env (regs_of_effect l.l_expr.e_effect) f in
759
  wp_forall args f
760

761 762
and wp_rec_defn env { rec_defn = rdl; rec_letrec = lr } =
  List.map (fun rd -> wp_lambda env lr rd.fun_lambda) rdl
763

764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809
(***
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
***)
810

811 812 813 814 815 816 817 818 819
(* replace t_absurd with t_false *)
let rec unabsurd f = match f.t_node with
  | Tapp (ls, []) when ls_equal ls ls_absurd ->
      t_label_copy f t_false
  | _ ->
      t_map unabsurd f

let add_wp_decl name f uc =
  (* prepare a proposition symbol *)
Andrei Paskevich's avatar
Andrei Paskevich committed
820
  let s = "WP_parameter " ^ name.id_string in
821
  let lab = Ident.create_label ("expl:parameter " ^ name.id_string) in
822 823 824 825
  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 *)
826
  let f = remove_at f in
827 828 829 830 831 832 833 834 835 836
  (* let f = bool_to_prop uc f in *)
  let f = unabsurd f in
  (* get a known map with tuples added *)
  let km = Theory.get_known uc in
  (* simplify f *)
  let f = Eval_match.eval_match ~inline:Eval_match.inline_nonrec_linear km f in
  (* printf "wp: f=%a@." print_term f; *)
  let d = create_prop_decl Pgoal pr f in
  Theory.add_decl uc d

Andrei Paskevich's avatar
Andrei Paskevich committed
837 838 839 840 841
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;
Andrei Paskevich's avatar
Andrei Paskevich committed
842 843 844 845 846
    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 +"];
847
    letrec_var = Mint.empty;
Andrei Paskevich's avatar
Andrei Paskevich committed
848
  }
849

850
let wp_let env km th { let_sym = lv; let_expr = e } =
851 852
  let env = mk_env env km th in
  let q, xq = default_post e.e_vty e.e_effect in
853
  let f = wp_expr env e q xq in
854 855 856 857
  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
858 859
  add_wp_decl id f th

860 861
let wp_rec env km th rdl =
  let env = mk_env env km th in
862 863
  let fl = wp_rec_defn env rdl in
  let add_one th d f =
864
    Debug.dprintf debug "wp %s = %a@\n----------------@."
865
      d.fun_ps.ps_name.id_string Pretty.print_term f;
866
    let f = wp_forall (Mvs.keys f.t_vars) f in
867
    add_wp_decl d.fun_ps.ps_name f th
868
  in
869
  List.fold_left2 add_one th rdl.rec_defn fl
870

871
let wp_val _env _km th _lv = th