blocking_semantics3.mlw 22.9 KB
Newer Older
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

(** {1 A certified WP calculus} *)

(** {2 A simple imperative language with expressions, syntax and semantics} *)

theory ImpExpr

use import int.Int
use import int.MinMax
use import bool.Bool
use export list.List
use map.Map as IdMap

(** types and values *)

type datatype = TYunit | TYint | TYbool
type value = Vvoid | Vint int | Vbool bool

(** terms and formulas *)

type operator = Oplus | Ominus | Omult | Ole

23
(** ident for mutable variables *)
24 25
type mident

26 27 28
axiom mident_decide :
  forall m1 m2: mident. m1 = m2 \/ m1 <> m2

29
(** ident for immutable variables *)
30 31 32
type ident = {| ident_index : int |}

(** Terms *)
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
33
type term =
34 35 36 37 38 39 40 41
  | Tvalue value
  | Tvar ident
  | Tderef mident
  | Tbin term operator term


predicate var_occurs_in_term (x:ident) (t:term) =
  match t with
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
42 43 44 45
  | Tvalue _  -> false
  |  Tvar i  -> x=i
  |  Tderef _  -> false
  |  Tbin t1 _ t2 -> var_occurs_in_term x t1 \/ var_occurs_in_term x t2
46 47
  end

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
48 49
(* predicate term_inv (t:term) = *)
(*   forall x:ident. var_occurs_in_term x t -> x.ident_index <= t.term_maxvar *)
50 51

function mk_tvalue (v:value) : term =
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
52
   Tvalue v
53 54

function mk_tvar (i:ident) : term =
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
55
   Tvar i
56 57

function mk_tderef (r:mident) : term =
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
58
   Tderef r
59 60

function mk_tbin (t1:term) (o:operator) (t2:term) : term =
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
61
    Tbin t1 o t2
62 63 64 65 66 67 68 69 70 71 72


(** Formulas *)
type fmla =
  | Fterm term
  | Fand fmla fmla
  | Fnot fmla
  | Fimplies fmla fmla
  | Flet ident term fmla         (* let id = term in fmla *)
  | Fforall ident datatype fmla  (* forall id : ty, fmla *)

73 74 75 76 77 78 79 80
(** Statements *)
type stmt =
  | Sskip
  | Sassign mident term
  | Sseq stmt stmt
  | Sif term stmt stmt
  | Sassert fmla
  | Swhile term fmla stmt  (* while cond invariant inv body *)
81

82 83 84
lemma decide_is_skip:
  forall s:stmt. s = Sskip \/ s <> Sskip

85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111
(** Typing *)

function type_value (v:value) : datatype =
    match v with
      | Vvoid  -> TYunit
      | Vint int ->  TYint
      | Vbool bool -> TYbool
end

inductive type_operator (op:operator) (ty1 ty2 ty: datatype) =
      | Type_plus : type_operator Oplus TYint TYint TYint
      | Type_minus : type_operator Ominus TYint TYint TYint
      | Type_mult : type_operator Omult TYint TYint TYint
      | Type_le : type_operator Ole TYint TYint TYbool

type type_stack = list (ident, datatype)  (* map local immutable variables to their type *)
function get_vartype (i:ident) (pi:type_stack) : datatype =
  match pi with
  | Nil -> TYunit
  | Cons (x,ty) r -> if x=i then ty else get_vartype i r
  end

type type_env = IdMap.map mident datatype  (* map global mutable variables to their type *)
function get_reftype (i:mident) (e:type_env) : datatype = IdMap.get e i

inductive type_term type_env type_stack term datatype =
  | Type_value :
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
112 113
      forall sigma: type_env, pi:type_stack, v:value.
	type_term sigma pi  (Tvalue v) (type_value v)
114
  | Type_var :
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
115
      forall sigma: type_env, pi:type_stack, v: ident, ty:datatype.
116
        (get_vartype v pi = ty) ->
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
117
        type_term sigma pi (Tvar v) ty
118
  | Type_deref :
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
119
      forall sigma: type_env, pi:type_stack, v: mident, ty:datatype.
120
        (get_reftype v sigma = ty) ->
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
121
        type_term sigma pi (Tderef v) ty
122 123
  | Type_bin :
      forall sigma: type_env, pi:type_stack, t1 t2 : term, op:operator,
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
124
        ty1 ty2 ty:datatype.
125 126 127
        type_term sigma pi t1 ty1 ->
	type_term sigma pi t2 ty2 ->
	type_operator op ty1 ty2 ty ->
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
128
        type_term sigma pi (Tbin t1 op t2) ty
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166

inductive type_fmla type_env type_stack fmla =
  | Type_term :
      forall sigma: type_env, pi:type_stack, t:term.
	type_term sigma pi t TYbool ->
	type_fmla sigma pi (Fterm t)
  | Type_conj :
      forall sigma: type_env, pi:type_stack, f1 f2:fmla.
	type_fmla sigma pi f1 ->
        type_fmla sigma pi f2 ->
        type_fmla sigma pi (Fand f1 f2)
  | Type_neg :
      forall sigma: type_env, pi:type_stack, f:fmla.
	type_fmla sigma pi f ->
        type_fmla sigma pi (Fnot f)
  | Type_implies :
      forall sigma: type_env, pi:type_stack, f1 f2:fmla.
	type_fmla sigma pi f1 ->
        type_fmla sigma pi f2 ->
        type_fmla sigma pi (Fimplies f1 f2)
  | Type_let :
      forall sigma: type_env, pi:type_stack, x:ident, t:term, f:fmla, ty:datatype.
	type_term sigma pi t ty ->
        type_fmla sigma (Cons (x,ty) pi) f ->
        type_fmla sigma pi (Flet x t f)
  | Type_forall1 :
      forall sigma: type_env, pi:type_stack, x:ident, f:fmla.
        type_fmla sigma (Cons (x,TYint) pi) f ->
  	type_fmla sigma pi (Fforall x TYint f)
  | Type_forall2 :
      forall sigma: type_env, pi:type_stack, x:ident, f:fmla.
        type_fmla sigma (Cons (x,TYbool) pi) f ->
  	type_fmla sigma pi (Fforall x TYbool f)
  | Type_forall3:
      forall sigma: type_env, pi:type_stack, x:ident, f:fmla.
        type_fmla sigma (Cons (x,TYunit) pi) f ->
  	type_fmla sigma pi (Fforall x TYunit f)

167 168 169 170 171 172 173 174 175 176 177
inductive type_stmt type_env type_stack stmt =
  | Type_skip :
      forall sigma: type_env, pi:type_stack.
	type_stmt sigma pi Sskip
  | Type_seq :
      forall sigma: type_env, pi:type_stack, s1 s2:stmt.
        type_stmt sigma pi s1 ->
	type_stmt sigma pi s2 ->
	type_stmt sigma pi (Sseq s1 s2)
  | Type_assigns :
      forall sigma: type_env, pi:type_stack, x:mident, t:term, ty:datatype.
178
	(get_reftype x sigma = ty) ->
179 180 181 182 183 184 185 186 187 188
        type_term sigma pi t ty ->
        type_stmt sigma pi (Sassign x t)
  | Type_if :
      forall sigma: type_env, pi:type_stack, t:term, s1 s2:stmt.
	type_term sigma pi t TYbool ->
	type_stmt sigma pi s1 ->
	type_stmt sigma pi s2 ->
    	type_stmt sigma pi (Sif t s1 s2)
  | Type_assert :
      forall sigma: type_env, pi:type_stack, p:fmla.
189
	type_fmla sigma pi p ->
190 191 192
    	type_stmt sigma pi (Sassert p)
  | Type_while :
      forall sigma: type_env, pi:type_stack, guard:term, body:stmt, inv:fmla.
193
	type_fmla sigma pi inv ->
194 195 196
        type_term sigma pi guard TYbool ->
        type_stmt sigma pi body ->
        type_stmt sigma pi (Swhile guard inv body) 
197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232

(** Operational semantic *)
type env = IdMap.map mident value  (* map global mutable variables to their value *)
function get_env (i:mident) (e:env) : value = IdMap.get e i

type stack = list (ident, value)  (* map local immutable variables to their value *)
function get_stack (i:ident) (pi:stack) : value =
  match pi with
  | Nil -> Vvoid
  | Cons (x,v) r -> if x=i then v else get_stack i r
  end

lemma get_stack_eq:
  forall x:ident, v:value, r:stack.
    get_stack x (Cons (x,v) r) = v

lemma get_stack_neq:
  forall x i:ident, v:value, r:stack.
    x <> i -> get_stack i (Cons (x,v) r) = get_stack i r

(** semantics of formulas *)

function eval_bin (x:value) (op:operator) (y:value) : value =
  match x,y with
  | Vint x,Vint y ->
     match op with
     | Oplus -> Vint (x+y)
     | Ominus -> Vint (x-y)
     | Omult -> Vint (x*y)
     | Ole -> Vbool (if x <= y then True else False)
     end
  | _,_ -> Vvoid
  end

function eval_term (sigma:env) (pi:stack) (t:term) : value =
  match t with
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
233 234 235 236
  | Tvalue v -> v
  |  Tvar id  -> get_stack id pi
  |  Tderef id  -> get_env id sigma
  |  Tbin t1 op t2  ->
237 238 239
     eval_bin (eval_term sigma pi t1) op (eval_term sigma pi t2)
  end

240 241 242 243 244 245 246 247 248

lemma eval_bool_term:
  forall sigma:env, pi:stack, sigmat:type_env, pit:type_stack, t:term.
    type_term sigmat pit t TYbool ->
    (* TODO: compatibility sigma, sigmat and pi,pit *)
    exists b:bool.
      eval_term sigma pi t = Vbool b


249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269
predicate eval_fmla (sigma:env) (pi:stack) (f:fmla) =
  match f with
  | Fterm t -> eval_term sigma pi t = Vbool True
  | Fand f1 f2 -> eval_fmla sigma pi f1 /\ eval_fmla sigma pi f2
  | Fnot f -> not (eval_fmla sigma pi f)
  | Fimplies f1 f2 -> eval_fmla sigma pi f1 -> eval_fmla sigma pi f2
  | Flet x t f ->
      eval_fmla sigma (Cons (x,eval_term sigma pi t) pi) f
  | Fforall x TYint f ->
     forall n:int. eval_fmla sigma (Cons (x,Vint n) pi) f
  | Fforall x TYbool f ->
     forall b:bool. eval_fmla sigma (Cons (x,Vbool b) pi) f
  | Fforall x TYunit f ->  eval_fmla sigma (Cons (x,Vvoid) pi) f
  end

(** substitution of a reference [r] by a logic variable [v]
   warning: proper behavior only guaranted if [v] is "fresh",
   i.e index(v) > term_maxvar(t) *)

function msubst_term (t:term) (r:mident) (v:ident) : term =
  match t with
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
270 271 272
  | Tvalue _ | Tvar _  -> t
  | Tderef x -> if r = x then mk_tvar v else t
  | Tbin t1 op t2  ->
273 274 275 276 277
      mk_tbin (msubst_term t1 r v) op (msubst_term t2 r v) 
  end

function subst_term (t:term) (r:ident) (v:ident) : term =
  match t with
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
278 279
  | Tvalue _ | Tderef _  -> t
  | Tvar x  ->
280
      if r = x then mk_tvar v else t
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
281
  | Tbin t1 op t2  ->
282 283 284 285 286
     mk_tbin (subst_term t1 r v) op (subst_term t2 r v)
  end

(** [fresh_in_term id t] is true when [id] does not occur in [t] *)
predicate fresh_in_term (id:ident) (t:term) =
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
287
    not (var_occurs_in_term id t)
288

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
289 290 291 292 293
lemma fresh_in_binop:
  forall t t':term, op:operator, v:ident.
    fresh_in_term v (mk_tbin t op t') ->
      fresh_in_term v t  /\ fresh_in_term v t'
	  
294
lemma eval_msubst_term:
295
  forall e:term, sigma:env, pi:stack, x:mident, v:ident.
296 297 298 299
    fresh_in_term v e ->
    eval_term sigma pi (msubst_term e x v) =
    eval_term (IdMap.set sigma x (get_stack v pi)) pi e

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
300 301 302 303 304
(* lemma eval_subst_term: *)
(*   forall sigma:env, pi:stack, e:term, x:ident, v:ident. *)
(*     fresh_in_term v e -> *)
(*     eval_term sigma pi (subst_term e x v) = *)
(*     eval_term sigma (Cons (x, (get_stack v pi)) pi) e *)
305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344

lemma eval_term_change_free :
  forall t:term, sigma:env, pi:stack, id:ident, v:value.
    fresh_in_term id t ->
    eval_term sigma (Cons (id,v) pi) t = eval_term sigma pi t

predicate fresh_in_fmla (id:ident) (f:fmla) =
  match f with
  | Fterm e -> fresh_in_term id e
  | Fand f1 f2   | Fimplies f1 f2 ->
      fresh_in_fmla id f1 /\ fresh_in_fmla id f2
  | Fnot f -> fresh_in_fmla id f
  | Flet y t f -> id <> y /\ fresh_in_term id t /\ fresh_in_fmla id f
  | Fforall y ty f -> id <> y /\ fresh_in_fmla id f
  end

function subst (f:fmla) (x:ident) (v:ident) : fmla =
  match f with
  | Fterm e -> Fterm (subst_term e x v)
  | Fand f1 f2 -> Fand (subst f1 x v) (subst f2 x v)
  | Fnot f -> Fnot (subst f x v)
  | Fimplies f1 f2 -> Fimplies (subst f1 x v) (subst f2 x v)
  | Flet y t f -> Flet y (subst_term t x v) (subst f x v)
  | Fforall y ty f -> Fforall y ty (subst f x v)
  end

function msubst (f:fmla) (x:mident) (v:ident) : fmla =
  match f with
  | Fterm e -> Fterm (msubst_term e x v)
  | Fand f1 f2 -> Fand (msubst f1 x v) (msubst f2 x v)
  | Fnot f -> Fnot (msubst f x v)
  | Fimplies f1 f2 -> Fimplies (msubst f1 x v) (msubst f2 x v)
  | Flet y t f -> Flet y (msubst_term t x v) (msubst f x v)
  | Fforall y ty f -> Fforall y ty (msubst f x v)
  end

lemma subst_fresh :
  forall f:fmla, x:ident, v:ident.
   fresh_in_fmla x f -> subst f x v = f

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
345 346 347 348
(* Not needed *)
(* lemma let_subst: *)
(*     forall t:term, f:fmla, x id':ident, id :mident. *)
(*     msubst (Flet x t f) id id' = Flet x (msubst_term t id id') (msubst f id id') *)
349

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
350
(* Need it for monotonicity and wp_reduction *)
351 352 353 354 355 356
lemma eval_msubst:
  forall f:fmla, sigma:env, pi:stack, x:mident, v:ident.
    fresh_in_fmla v f ->
    (eval_fmla sigma pi (msubst f x v) <->
     eval_fmla (IdMap.set sigma x (get_stack v pi)) pi f)

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
357 358 359 360 361
(* lemma eval_subst: *)
(*   forall f:fmla, sigma:env, pi:stack, x:ident, v:ident. *)
(*     fresh_in_fmla v f -> *)
(*     (eval_fmla sigma pi (subst f x v) <-> *)
(*      eval_fmla sigma (Cons(x, (get_stack v pi)) pi) f) *)
362

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
363 364 365 366 367 368 369 370 371 372
lemma eval_same_var_term:
  forall t:term, sigma:env, pi:stack, id:ident, v1 v2:value.
    eval_term sigma (Cons (id,v1) (Cons (id,v2) pi)) t =
    eval_term sigma (Cons (id,v1) pi) t

lemma eval_same_var:
  forall f:fmla, sigma:env, pi:stack, id:ident, v1 v2:value.
    eval_fmla sigma (Cons (id,v1) (Cons (id,v2) pi)) f <->
    eval_fmla sigma (Cons (id,v1) pi) f

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
373 374 375 376 377 378 379 380 381 382 383
lemma eval_swap_term:
  forall t:term, sigma:env, pi:stack, id1 id2:ident, v1 v2:value.
    id1 <> id2 ->
    (eval_term sigma (Cons (id1,v1) (Cons (id2,v2) pi)) t =
    eval_term sigma (Cons (id2,v2) (Cons (id1,v1) pi)) t)

lemma eval_swap:
  forall f:fmla, sigma:env, pi:stack, id1 id2:ident, v1 v2:value.
    id1 <> id2 ->
    (eval_fmla sigma (Cons (id1,v1) (Cons (id2,v2) pi)) f <->
    eval_fmla sigma (Cons (id2,v2) (Cons (id1,v1) pi)) f)
384

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
385
 (* Need it for monotonicity*)
386 387 388 389 390
lemma eval_change_free :
  forall f:fmla, sigma:env, pi:stack, id:ident, v:value.
    fresh_in_fmla id f ->
    (eval_fmla sigma (Cons (id,v) pi) f <-> eval_fmla sigma pi f)

atafat's avatar
atafat committed
391
(** [valid_fmla f] is true when [f] is valid in any environment *)
392 393
  predicate valid_fmla (p:fmla) = forall sigma:env, pi:stack. eval_fmla sigma pi p

394
(* Not needed *)
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
395 396 397 398 399 400
(* axiom msubst_implies : *)
(* forall p q:fmla. *)
(*   valid_fmla (Fimplies p q) -> *)
(*   forall sigma:env, pi:stack, x:mident, id:ident. *)
(*     fresh_in_fmla id (Fand p q) ->  *)
(*     eval_fmla sigma (Cons (id, (get_env x sigma)) pi) (Fimplies (msubst p x id) (msubst q x id))  *)
atafat's avatar
atafat committed
401

402
(** let id' = t in f[id <- id'] <=> let id = t in f*)
Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
403 404 405 406 407 408 409 410 411 412 413 414
(* Not needed *)
(* lemma let_equiv : *)
(*   forall id:ident, id':ident, t:term, f:fmla. *)
(*     forall sigma:env, pi:stack. *)
(*       fresh_in_fmla id' f -> *)
(* 	eval_fmla sigma pi (Flet id' t (subst f id id')) *)
(* 	 -> eval_fmla sigma pi (Flet id t f) *)

(* lemma let_implies : *)
(*   forall id:ident, t:term, p q:fmla. *)
(*     valid_fmla (Fimplies p q) -> *)
(*     valid_fmla (Fimplies (Flet id t p) (Flet id t q)) *)
415

416 417 418 419 420 421 422 423
predicate fresh_in_stmt (id:ident) (s:stmt) =
  match s with
  | Sskip -> true
  | Sseq s1 s2 -> fresh_in_stmt id s1 /\ fresh_in_stmt id s2
  | Sassign _ t -> fresh_in_term id t
  | Sif t s1 s2 -> fresh_in_term id t /\ fresh_in_stmt id s1 /\ fresh_in_stmt id s2
  | Sassert f -> fresh_in_fmla id f
  | Swhile cond inv body -> fresh_in_term id cond /\ fresh_in_fmla id inv /\ fresh_in_stmt id body
424 425 426 427 428
  end


(** small-steps semantics for expressions *)

429 430 431 432 433 434 435 436 437 438 439 440 441 442 443
inductive one_step env stack stmt env stack stmt =

  | one_step_assign :
      forall sigma sigma':env, pi:stack, x:mident, t:term.
        sigma' = IdMap.set sigma x (eval_term sigma pi t) ->
        one_step sigma pi (Sassign x t) sigma' pi Sskip

  | one_step_seq_noskip:
      forall sigma sigma':env, pi pi':stack, s1 s1' s2:stmt.
        one_step sigma pi s1 sigma' pi' s1' ->
          one_step sigma pi (Sseq s1 s2) sigma' pi' (Sseq s1' s2)

  | one_step_seq_skip:
      forall sigma:env, pi:stack, s:stmt.
        one_step sigma pi (Sseq Sskip s) sigma pi s
444 445

  | one_step_if_true:
446 447 448
      forall sigma:env, pi:stack, t:term, s1 s2:stmt.
        eval_term sigma pi t = Vbool True ->
        one_step sigma pi (Sif t s1 s2) sigma pi s1
449 450

  | one_step_if_false:
451 452 453
      forall sigma:env, pi:stack, t:term, s1 s2:stmt.
        eval_term sigma pi t = Vbool False ->
        one_step sigma pi (Sif t s1 s2) sigma pi s2
454 455 456 457 458

  | one_step_assert:
      forall sigma:env, pi:stack, f:fmla.
        (* blocking semantics *)
        eval_fmla sigma pi f ->
459
          one_step sigma pi (Sassert f) sigma pi Sskip
460

461 462
  | one_step_while_true:
      forall sigma:env, pi:stack, cond:term, inv:fmla, body:stmt.
463 464
        (* blocking semantics *)
        eval_fmla sigma pi inv ->
465 466 467 468
        eval_term sigma pi cond = Vbool True ->
        one_step sigma pi (Swhile cond inv body) sigma pi
        (Sseq body (Swhile cond inv body))

MARCHE Claude's avatar
MARCHE Claude committed
469
  | one_step_while_false:
470 471 472 473 474
      forall sigma:env, pi:stack, cond:term, inv:fmla, body:stmt.
        (* blocking semantics *)
        eval_fmla sigma pi inv ->
        eval_term sigma pi cond = Vbool False ->
        one_step sigma pi (Swhile cond inv body) sigma pi Sskip
475 476 477

 (** many steps of execution *)

478
 inductive many_steps env stack stmt env stack stmt int =
479
   | many_steps_refl:
480
     forall sigma:env, pi:stack, s:stmt. many_steps sigma pi s sigma pi s 0
481
   | many_steps_trans:
482 483 484 485
     forall sigma1 sigma2 sigma3:env, pi1 pi2 pi3:stack, s1 s2 s3:stmt, n:int.
       one_step sigma1 pi1 s1 sigma2 pi2 s2 ->
       many_steps sigma2 pi2 s2 sigma3 pi3 s3 n ->
       many_steps sigma1 pi1 s1 sigma3 pi3 s3 (n+1)
486

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
487 488 489
lemma steps_non_neg:
  forall sigma1 sigma2:env, pi1 pi2:stack, s1 s2:stmt, n:int.
    many_steps sigma1 pi1 s1 sigma2 pi2 s2 n -> n >= 0
490

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
491
(* Used by Hoare_logic/seq_rule*)
492
  lemma many_steps_seq:
493 494
    forall sigma1 sigma3:env, pi1 pi3:stack, s1 s2:stmt, n:int.
      many_steps sigma1 pi1 (Sseq s1 s2) sigma3 pi3 Sskip n ->
495
      exists sigma2:env, pi2:stack, n1 n2:int.
496 497
        many_steps sigma1 pi1 s1 sigma2 pi2 Sskip n1 /\
        many_steps sigma2 pi2 s2 sigma3 pi3 Sskip n2 /\
498 499
        n = 1 + n1 + n2

Asma Tafat-Bouzid's avatar
Asma Tafat-Bouzid committed
500 501 502 503 504
 (* lemma one_step_change_free : *)
 (*  forall s s':stmt, sigma sigma':env, pi pi':stack, id:ident, v:value. *)
 (*    fresh_in_stmt id s -> *)
 (*    one_step sigma (Cons (id,v) pi) s sigma' pi' s' -> *)
 (*    one_step sigma pi s sigma' pi' s' *)
505 506 507 508 509


(** {3 Hoare triples} *)

(** partial correctness *)
510
predicate valid_triple (p:fmla) (s:stmt) (q:fmla) =
511
    forall sigma:env, pi:stack. eval_fmla sigma pi p ->
512 513 514
      forall sigma':env, pi':stack, n:int.
        many_steps sigma pi s sigma' pi' Sskip n ->
          eval_fmla sigma' pi' q
515 516

(*** total correctness *)
517
predicate total_valid_triple (p:fmla) (s:stmt) (q:fmla) =
518
    forall sigma:env, pi:stack. eval_fmla sigma pi p ->
519 520 521
      exists sigma':env, pi':stack, n:int.
        many_steps sigma pi s sigma' pi' Sskip n /\
        eval_fmla sigma' pi' q
522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549

end


theory TestSemantics

use import ImpExpr

function my_sigma : env = IdMap.const (Vint 0)
constant x : ident
constant y : mident

function my_pi : stack = Cons (x, Vint 42) Nil

goal Test13 :
  eval_term my_sigma my_pi (mk_tvalue (Vint 13)) = Vint 13

goal Test42 :
  eval_term my_sigma my_pi (mk_tvar x) = Vint 42

goal Test0 :
  eval_term my_sigma my_pi (mk_tderef y) = Vint 0

goal Test55 :
  eval_term my_sigma my_pi (mk_tbin (mk_tvar x) Oplus (mk_tvalue (Vint 13))) = Vint 55

goal Ass42 :
  forall sigma':env, pi':stack.
550
    one_step my_sigma my_pi (Sassign y (mk_tvalue (Vint 42))) sigma' pi' Sskip ->
551 552 553
      IdMap.get sigma' y = Vint 42

goal If42 :
554
    forall sigma1 sigma2:env, pi1 pi2:stack, s:stmt.
555
      one_step my_sigma my_pi
556 557 558 559 560
        (Sif (mk_tbin (mk_tderef y) Ole (mk_tvalue (Vint 10)))
             (Sassign y (mk_tvalue (Vint 13)))
             (Sassign y (mk_tvalue (Vint 42))))
        sigma1 pi1 s ->
      one_step sigma1 pi1 s sigma2 pi2 Sskip ->
561 562 563 564 565 566 567 568 569 570 571 572 573 574
        IdMap.get sigma2 y = Vint 13

end

(** {2 Hoare logic} *)

theory HoareLogic

use import ImpExpr


(** Hoare logic rules (partial correctness) *)

lemma consequence_rule:
575
  forall p p' q q':fmla, s:stmt.
576
  valid_fmla (Fimplies p' p) ->
577
  valid_triple p s q ->
578
  valid_fmla (Fimplies q q') ->
579
  valid_triple p' s q'
580

581 582
lemma skip_rule:
  forall q:fmla. valid_triple q Sskip q
583 584

lemma assign_rule:
585 586 587
  forall p:fmla, x:mident, id:ident, t:term.
  fresh_in_fmla id p ->
  valid_triple (Flet id t (msubst p x id)) (Sassign x t) p
588 589

lemma seq_rule:
590 591 592
  forall p q r:fmla, s1 s2:stmt.
  valid_triple p s1 r /\ valid_triple r s2 q ->
  valid_triple p (Sseq s1 s2) q
593 594

lemma if_rule:
595 596 597 598
  forall t:term, p q:fmla, s1 s2:stmt.
  valid_triple (Fand p (Fterm t)) s1 q /\
  valid_triple (Fand p (Fnot (Fterm t))) s2 q ->
  valid_triple p (Sif t s1 s2) q
599 600 601

lemma assert_rule:
  forall f p:fmla. valid_fmla (Fimplies p f) ->
602
  valid_triple p (Sassert f) p
603 604 605

lemma assert_rule_ext:
  forall f p:fmla.
606
  valid_triple (Fimplies f p) (Sassert f) p
607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657

(*
lemma while_rule:
  forall e:term, inv:fmla, i:expr.
  valid_triple (Fand (Fterm e) inv) i inv ->
  valid_triple inv (Swhile e inv i) (Fand (Fnot (Fterm e)) inv)

lemma while_rule_ext:
  forall e:term, inv inv':fmla, i:expr.
  valid_fmla (Fimplies inv' inv) ->
  valid_triple (Fand (Fterm e) inv') i inv' ->
  valid_triple inv' (Swhile e inv i) (Fand (Fnot (Fterm e)) inv')
*)

(*** frame rule ? *)

end

(** {2 WP calculus} *)

theory WP

use import ImpExpr
use import bool.Bool

use set.Set

(** [assigns sigma W sigma'] is true when the only differences between
    [sigma] and [sigma'] are the value of references in [W] *)

predicate assigns (sigma:env) (a:Set.set mident) (sigma':env) =
  forall i:mident. not (Set.mem i a) ->
    IdMap.get sigma i = IdMap.get sigma' i

lemma assigns_refl:
  forall sigma:env, a:Set.set mident. assigns sigma a sigma

lemma assigns_trans:
  forall sigma1 sigma2 sigma3:env, a:Set.set mident.
    assigns sigma1 a sigma2 /\ assigns sigma2 a sigma3 ->
    assigns sigma1 a sigma3

lemma assigns_union_left:
  forall sigma sigma':env, s1 s2:Set.set mident.
    assigns sigma s1 sigma' -> assigns sigma (Set.union s1 s2) sigma'

lemma assigns_union_right:
  forall sigma sigma':env, s1 s2:Set.set mident.
    assigns sigma s2 sigma' -> assigns sigma (Set.union s1 s2) sigma'

(** [expr_writes e W] is true when the only references modified by [e] are in [W] *)
658 659 660 661 662 663 664
predicate stmt_writes (s:stmt) (w:Set.set mident) =
  match s with
  | Sskip | Sassert _ -> true
  | Sassign id _ -> Set.mem id w
  | Sseq s1 s2 -> stmt_writes s1 w /\ stmt_writes s2 w
  | Sif t s1 s2 -> stmt_writes s1 w /\ stmt_writes s2 w
  | Swhile _ _ body -> stmt_writes body w
665 666
  end

667
  function fresh_from (f:fmla) (s:stmt) : ident
668

669
  (* Need it for monotonicity*)
670 671
  axiom fresh_from_fmla: forall s:stmt, f:fmla.
     fresh_in_fmla (fresh_from f s) f
672

673 674
  axiom fresh_from_stmt: forall s:stmt, f:fmla.
     fresh_in_stmt (fresh_from f s) s
675

676
  function abstract_effects (s:stmt) (f:fmla) : fmla
677

MARCHE Claude's avatar
MARCHE Claude committed
678 679 680 681 682
  axiom abstract_effects_generalize :
     forall sigma:env, pi:stack, s:stmt, f:fmla.
        eval_fmla sigma pi (abstract_effects s f) ->
        eval_fmla sigma pi f

atafat's avatar
atafat committed
683 684
  axiom abstract_effects_monotonic :
     forall s:stmt, f:fmla.
atafat's avatar
atafat committed
685 686
        forall sigma:env, pi:stack. eval_fmla sigma pi f ->
        forall sigma:env, pi:stack. eval_fmla sigma pi (abstract_effects s f)
atafat's avatar
atafat committed
687

688 689 690 691
  function wp (s:stmt) (q:fmla) : fmla =
    match s with
    | Sskip -> q
    | Sassert f ->
692
        (* asymmetric and *)
693 694 695 696 697 698 699 700 701
        Fand f (Fimplies f q)
    | Sseq s1 s2 -> wp s1 (wp s2 q)
    | Sassign x t ->
        let id = fresh_from q s in
        Flet id t (msubst q x id)
    | Sif t s1 s2 ->
        Fand (Fimplies (Fterm t) (wp s1 q))
             (Fimplies (Fnot (Fterm t)) (wp s2 q))
    | Swhile cond inv body ->
702 703
        Fand inv
          (abstract_effects body
704 705 706
            (Fand
              (Fimplies (Fand (Fterm cond) inv) (wp body inv))
              (Fimplies (Fand (Fnot (Fterm cond)) inv) q)))
707 708 709

    end

MARCHE Claude's avatar
MARCHE Claude committed
710 711 712 713 714 715
  axiom abstract_effects_writes :
     forall sigma:env, pi:stack, s:stmt, q:fmla.
        eval_fmla sigma pi (abstract_effects s q) ->
        eval_fmla sigma pi (wp s (abstract_effects s q))


716 717
  (* lemma wp_subst: *)
  (*   forall e:expr, q:fmla, id :mident, id':ident. *)
718
  (*   fresh_in_stmt id e -> *)
719 720 721
  (*     subst (wp e q) id id' = wp e (subst q id id') *)

  lemma monotonicity:
722
    forall s:stmt, p q:fmla.
723
      valid_fmla (Fimplies p q)
724
     ->	valid_fmla (Fimplies (wp s p) (wp s q) )
atafat's avatar
atafat committed
725 726 727 728 729 730

  lemma distrib_conj:
    forall s:stmt, sigma:env, pi:stack, p q:fmla.
     (eval_fmla sigma pi (wp s p)) /\
     (eval_fmla sigma pi (wp s q)) ->
     eval_fmla sigma pi (wp s (Fand p q)) 
731 732

  lemma wp_reduction:
733 734
    forall sigma sigma':env, pi pi':stack, s s':stmt.
    one_step sigma pi s sigma' pi' s' ->
735
    forall q:fmla.
736 737
      eval_fmla sigma pi (wp s q) ->
      eval_fmla sigma' pi' (wp s' q)
738 739

  lemma progress:
740 741 742
    forall s:stmt, sigma:env, pi:stack,
      sigmat: type_env, pit: type_stack, q:fmla.
      type_stmt sigmat pit s ->
743
(* useful ?
744
      type_fmla sigmat pit q ->
745
*)
746 747 748 749
      eval_fmla sigma pi (wp s q) -> 
      s <> Sskip ->
      exists sigma':env, pi':stack, s':stmt.
      one_step sigma pi s sigma' pi' s'
750 751 752 753 754 755 756 757 758

end


(***
Local Variables:
compile-command: "why3ide blocking_semantics3.mlw"
End:
*)