Expressions.v 12.1 KB
Newer Older
1
(**
2
  Formalization of the base expression language for the daisy framework
3
 **)
4
Require Import Coq.Reals.Reals Coq.micromega.Psatz Coq.QArith.QArith Coq.QArith.Qreals.
5
Require Import Daisy.Infra.RealRationalProps Daisy.Infra.RationalSimps.
6
Require Export Daisy.Infra.Abbrevs Daisy.Infra.RealSimps Daisy.Infra.NatSet Daisy.IntervalArithQ Daisy.IntervalArith Daisy.Infra.MachineType.
7

8 9 10 11 12
(**
  Expressions will use binary operators.
  Define them first
**)
Inductive binop : Type := Plus | Sub | Mult | Div.
13

14
Definition binopEqBool (b1:binop) (b2:binop) :=
='s avatar
= committed
15 16 17 18 19 20
  match b1, b2 with
  | Plus, Plus => true
  | Sub,  Sub  => true
  | Mult, Mult => true
  | Div,  Div  => true
  | _,_ => false
21 22
  end.

23 24 25 26
(**
  Next define an evaluation function for binary operators on reals.
  Errors are added on the expression evaluation level later.
 **)
27
Definition evalBinop (o:binop) (v1:R) (v2:R) :=
28 29 30 31 32 33
  match o with
  | Plus => Rplus v1 v2
  | Sub => Rminus v1 v2
  | Mult => Rmult v1 v2
  | Div => Rdiv v1 v2
  end.
34

35 36 37 38 39 40
Lemma binopEqBool_refl b:
  binopEqBool b b = true.
Proof.
  case b; auto.
Qed.

41 42 43 44 45 46
(**
   Expressions will use unary operators.
   Define them first
 **)
Inductive unop: Type := Neg | Inv.

47
Definition unopEqBool (o1:unop) (o2:unop) :=
='s avatar
= committed
48 49 50 51
  match o1, o2 with
  | Neg, Neg => true
  | Inv, Inv => true
  | _ , _ => false
52 53
  end.

54 55 56 57 58 59
Lemma unopEqBool_refl b:
  unopEqBool b b = true.
Proof.
  case b; auto.
Qed.

60 61
(**
   Define evaluation for unary operators on reals.
62
   Errors are added in the expression evaluation level later.
63
 **)
64
Definition evalUnop (o:unop) (v:R):=
65 66 67 68 69
  match o with
  |Neg => (- v)%R
  |Inv => (/ v)%R
  end .

70 71


72
(**
73 74
  Define expressions parametric over some value type V.
  Will ease reasoning about different instantiations later.
75
**)
76
Inductive exp (V:Type): Type :=
77
  Var: nat -> exp V
78
| Const: mType -> V -> exp V
79
| Unop: unop -> exp V -> exp V
80 81
| Binop: binop -> exp V -> exp V -> exp V
| Downcast: mType -> exp V -> exp V.
82

83 84 85 86
(**
  Boolean equality function on expressions.
  Used in certificates to define the analysis result as function
**)
87
Fixpoint expEqBool (e1:exp Q) (e2:exp Q) :=
='s avatar
= committed
88
  match e1, e2 with
89
  | Var _ v1, Var _ v2 => (v1 =? v2)
='s avatar
= committed
90 91 92 93 94
  | Const m1 n1, Const m2 n2 => andb (mTypeEqBool m1 m2) (Qeq_bool n1 n2)
  | Unop o1 e11, Unop o2 e22 => andb (unopEqBool o1 o2) (expEqBool e11 e22)
  | Binop o1 e11 e12, Binop o2 e21 e22 => andb (binopEqBool o1 o2) (andb (expEqBool e11 e21) (expEqBool e12 e22))
  | Downcast m1 f1, Downcast m2 f2 => andb (mTypeEqBool m1 m2) (expEqBool f1 f2)
  | _, _ => false
95 96
  end.

97

98
Lemma expEqBool_refl e:
99 100
  expEqBool e e = true.
Proof.
101
  induction e; try (apply andb_true_iff; split); simpl in *; auto; try (apply EquivEqBoolEq; auto). 
102 103 104 105 106 107 108
  - symmetry; apply beq_nat_refl.
  - apply Qeq_bool_iff; lra.
  - case u; auto.
  - case b; auto.
  - apply andb_true_iff; split.
    apply IHe1. apply IHe2.
Qed.
109

110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129
Lemma beq_nat_sym a b:
  beq_nat a b = beq_nat b a.
Proof.
  case_eq (a =? b); intros.
  - apply beq_nat_true in H.
    rewrite H.
    apply beq_nat_refl. 
  - apply beq_nat_false in H.
    case_eq (b =? a); intros.
    + apply beq_nat_true in H0.
      rewrite H0 in H.
      auto.
    + auto.
Qed.      

Lemma expEqBool_sym e e':
  expEqBool e e' = expEqBool e' e.
Proof.
  revert e'.
  induction e; intros e'; destruct e'; simpl; try auto.
130
  - apply beq_nat_sym.
131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146
  - f_equal.
    + apply mTypeEqBool_sym; auto.
    + apply Qeq_bool_sym.
  - f_equal.
    + destruct u; auto.
    + apply IHe.
  - f_equal.      
    + destruct b; auto.
    + f_equal.
      * apply IHe1.
      * apply IHe2.
  - f_equal.
    + apply mTypeEqBool_sym; auto.
    + apply IHe.
Qed.

='s avatar
= committed
147 148 149 150 151
Lemma expEqBool_trans e f g:
  expEqBool e f = true ->
  expEqBool f g = true ->
  expEqBool e g = true.
Proof.
152 153 154
  revert e f g; induction e; destruct f; intros; simpl in H; inversion H; rewrite H; clear H; destruct g; simpl in H0; inversion H0; rewrite H0; clear H0; try (apply andb_true_iff in H1; destruct H1; apply andb_true_iff in H2; destruct H2; simpl).
  - apply beq_nat_true in H2.
    apply beq_nat_true in H1.
='s avatar
= committed
155
    subst.
156 157
    unfold expEqBool.
    rewrite <- beq_nat_refl. 
='s avatar
= committed
158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186
    auto.
  - apply EquivEqBoolEq in H1.
    apply EquivEqBoolEq in H.
    subst.
    rewrite mTypeEqBool_refl; simpl.
    apply Qeq_bool_iff in H2.
    apply Qeq_bool_iff in H0.
    apply Qeq_bool_iff.
    lra.
  - assert (u = u0) by (destruct u; destruct u0; inversion H1; auto).
    assert (u0 = u1) by (destruct u0; destruct u1; inversion H; auto).
    subst.
    assert (unopEqBool u1 u1 = true) by (destruct u1; auto).
    apply andb_true_iff; split; try auto.
    eapply IHe; eauto.
  - apply andb_true_iff; split.
    + destruct b; destruct b0; destruct b1; auto.
    + apply andb_true_iff in H2; destruct H2.
      apply andb_true_iff in H0; destruct H0.
      apply andb_true_iff; split.
      eapply IHe1; eauto.
      eapply IHe2; eauto.
  - apply EquivEqBoolEq in H1.
    apply EquivEqBoolEq in H.
    subst.
    rewrite mTypeEqBool_refl; simpl.
    eapply IHe; eauto.
Qed.

='s avatar
= committed
187
   
='s avatar
= committed
188

189 190
Fixpoint toRExp (e:exp Q) :=
  match e with
191
  |Var _ v => Var R v
192
  |Const m n => Const m (Q2R n)
193 194 195
  |Unop o e1 => Unop o (toRExp e1)
  |Binop o e1 e2 => Binop o (toRExp e1) (toRExp e2)
  |Downcast m e1 => Downcast m (toRExp e1)
196
  end.
197

198 199
Fixpoint toREval (e:exp R) :=
  match e with
200
  | Var _ v => Var R v
201
  | Const _ n => Const M0 n
202 203
  | Unop o e1 => Unop o (toREval e1)
  | Binop o e1 e2 => Binop o (toREval e1) (toREval e2)
204
  | Downcast _ e1 =>  (toREval e1)
205
  end.
206

207
(* TODO: put to REValVars here? *)
208 209


210 211 212 213
(**
  Define a perturbation function to ease writing of basic definitions
**)
Definition perturb (r:R) (e:R) :=
214
  (r * (1 + e))%R.
Heiko Becker's avatar
Heiko Becker committed
215

216
(**
217
Define expression evaluation relation parametric by an "error" epsilon.
218 219 220
The result value expresses float computations according to the IEEE standard,
using a perturbation of the real valued computation by (1 + delta), where
|delta| <= machine epsilon.
221
**)
222
Inductive eval_exp (E:env) (defVars: nat -> option mType) :(exp R) -> R -> mType -> Prop :=
223
| Var_load m x v:
224
    defVars x = Some m ->
225
    E x = Some v ->
226
    eval_exp E defVars (Var R x) v m
227 228
| Const_dist m n delta:
    Rle (Rabs delta) (Q2R (meps m)) ->
229
    eval_exp E defVars (Const m n) (perturb n delta) m
230
| Unop_neg m f1 v1:
231 232
    eval_exp E defVars f1 v1 m ->
    eval_exp E defVars (Unop Neg f1) (evalUnop Neg v1) m
233 234
| Unop_inv m f1 v1 delta:
    Rle (Rabs delta) (Q2R (meps m)) ->
235 236
    eval_exp E defVars  f1 v1 m ->
    eval_exp E defVars (Unop Inv f1) (perturb (evalUnop Inv v1) delta) m
237 238
| Binop_dist m1 m2 op f1 f2 v1 v2 delta:
    Rle (Rabs delta) (Q2R (meps (computeJoin m1 m2))) ->
239 240
    eval_exp E defVars f1 v1 m1 ->
    eval_exp E defVars f2 v2 m2 ->
241
    ((op = Div) -> (~ v2 = 0)%R) ->
242
    eval_exp E defVars (Binop op f1 f2) (perturb (evalBinop op v1 v2) delta)  (computeJoin m1 m2)
243
| Downcast_dist m m1 f1 v1 delta:
244
    (* Downcast expression f1 (evaluating to machine type m1), to a machine type m, less precise than m1.*)
245 246
    isMorePrecise m1 m = true ->
    Rle (Rabs delta) (Q2R (meps m)) ->
247 248
    eval_exp E defVars f1 v1 m1 ->
    eval_exp E defVars (Downcast m f1) (perturb v1 delta) m.
249 250


251 252 253 254 255
(**
  Define the set of "used" variables of an expression to be the set of variables
  occuring in it
**)
Fixpoint usedVars (V:Type) (e:exp V) :NatSet.t :=
256
  match e with
257
  | Var _ x => NatSet.singleton x
258 259
  | Unop u e1 => usedVars e1
  | Binop b e1 e2 => NatSet.union (usedVars e1) (usedVars e2)
260
  | Downcast _ e1 => usedVars e1
261 262
  | _ => NatSet.empty
  end.
263

264
(**
265
  If |delta| <= 0 then perturb v delta is exactly v.
266
**)
267
Lemma delta_0_deterministic (v:R) (delta:R):
Heiko Becker's avatar
Heiko Becker committed
268 269 270 271 272
  (Rabs delta <= 0)%R ->
  perturb v delta = v.
Proof.
  intros abs_0; apply Rabs_0_impl_eq in abs_0; subst.
  unfold perturb.
273
  lra.
Heiko Becker's avatar
Heiko Becker committed
274 275
Qed.

276
(* TODO: need of `general` case? *)
277
Lemma general_meps_0_deterministic (f:exp R) (E:env) defVars:
278 279
  forall v1 v2 m1,
    m1 = M0 ->
280 281
    eval_exp E defVars (toREval f) v1 m1 ->
    eval_exp E defVars (toREval f) v2 M0 ->
282 283
    v1 = v2.
Proof.
284
  induction f; intros * m10_eq eval_v1 eval_v2.
285 286
  - inversion eval_v1; inversion eval_v2; subst; auto;
      try repeat (repeat rewrite delta_0_deterministic; simpl in *; rewrite Q2R0_is_0 in *; subst; auto); simpl.
287
    rewrite H6 in H1; inversion H1; subst; auto.
288 289 290 291
  - inversion eval_v1; inversion eval_v2; subst; auto;
      try repeat (repeat rewrite delta_0_deterministic; simpl in *; rewrite Q2R0_is_0 in *; subst; auto); simpl.
  - inversion eval_v1; inversion eval_v2; subst; auto;
      try repeat (repeat rewrite delta_0_deterministic; simpl in *; rewrite Q2R0_is_0 in *; subst; auto); simpl.
292
    + apply Ropp_eq_compat. apply (IHf v0 v3 M0); auto.     
293 294
    + inversion H4.
    + inversion H5.
295
    + rewrite (IHf v0 v3 M0); auto.
296 297
  - inversion eval_v1; inversion eval_v2; subst; auto;
      try repeat (repeat rewrite delta_0_deterministic; simpl in *; rewrite Q2R0_is_0 in *; subst; auto); simpl.
298 299
    destruct m0; destruct m2; inversion H5.
    destruct m3; destruct m4; inversion H11.
300
    simpl in *.
301 302
    rewrite (IHf1 v0 v4 M0); auto.
    rewrite (IHf2 v5 v3 M0); auto.
303 304 305
    rewrite Q2R0_is_0 in H2,H12.
    rewrite delta_0_deterministic; auto.
    rewrite delta_0_deterministic; auto.
306 307
  - simpl toREval in eval_v1.
    simpl toREval in eval_v2.
308
    apply (IHf v1 v2 m1); auto.
309 310
Qed.

311 312 313 314 315 316 317 318
(* Lemma rnd_0_deterministic f E m v: *)
(*   eval_exp E (toREval (Downcast m f)) v M0 <-> *)
(*   eval_exp E (toREval f) v M0. *)
(* Proof. *)
(*   split; intros. *)
(*   - simpl in H. auto. *)
(*   - simpl; auto. *)
(* Qed. *)
319 320

  
321
(**
322
Evaluation with 0 as machine epsilon is deterministic
323
**)
324
Lemma meps_0_deterministic (f:exp R) (E:env) defVars:
325
  forall v1 v2,
326 327
  eval_exp E defVars (toREval f) v1 M0 ->
  eval_exp E defVars (toREval f) v2 M0 ->
328 329
  v1 = v2.
Proof.
330
  intros v1 v2 ev1 ev2.
331 332
  assert (M0 = M0) by auto.
  apply (general_meps_0_deterministic f H ev1 ev2). 
333 334
Qed.

335

336 337 338 339 340 341 342
Fixpoint toREvalVars (d:nat -> option mType) (n:nat) :=
  match d n with
  | Some m => Some M0
  | None => None
  end.


343 344 345 346
(**
Helping lemma. Needed in soundness proof.
For each evaluation of using an arbitrary epsilon, we can replace it by
evaluating the subexpressions and then binding the result values to different
347
variables in the Environment.
348 349
 **)

350 351
Lemma binary_unfolding b f1 f2 m E vF defVars:
  eval_exp E defVars (Binop b f1 f2) vF m ->
352
  exists vF1 vF2 m1 m2,
353
    m = computeJoin m1 m2 /\
354 355
    eval_exp E defVars f1 vF1 m1 /\
    eval_exp E defVars f2 vF2 m2 /\
356
    eval_exp (updEnv 2 vF2 (updEnv 1 vF1 emptyEnv))
357 358
             (fun n => if (n =? 2) then Some m2 else if (n =? 1) then Some m1 else defVars n)
             (Binop b (Var R 1) (Var R 2)) vF m.
359
Proof.
360 361
  intros eval_float.
  inversion eval_float; subst.
362 363
  exists v1 ; exists v2; exists m1; exists m2; repeat split; try auto.
  eapply Binop_dist; eauto.
364 365 366 367
  - pose proof (isMorePrecise_refl m1).
    eapply Var_load; eauto.
  - pose proof (isMorePrecise_refl m2).
    eapply Var_load; eauto.
368 369
Qed.

370 371 372 373 374 375 376 377 378 379 380 381 382 383 384
(* (* Analogous lemma for unary expressions. *) *)
(* Lemma unary_unfolding (e:exp R) (m:mType) (E:env) (v:R) defVars: *)
(*   (eval_exp E defVars (Unop Inv e) v m -> *)
(*    exists v1 m1, *)
(*      eval_exp E defVars e v1 m1 /\ *)
(*      eval_exp (updEnv 1 v1 E) (fun n => if (n =? 1) then Some m1 else defVars n) (Unop Inv (Var R 1)) v m). *)
(* Proof. *)
(*   intros eval_un. *)
(*     inversion eval_un; subst. *)
(*     exists v1; exists m. *)
(*     repeat split; try auto. *)
(*     econstructor; try auto. *)
(*     pose proof (isMorePrecise_refl m). *)
(*     econstructor; eauto. *)
(* Qed. *)
385

386 387 388 389 390 391
(* (** *)
(*   Using the parametric expressions, define boolean expressions for conditionals *)
(* **) *)
(* Inductive bexp (V:Type) : Type := *)
(*   leq: exp V -> exp V -> bexp V *)
(* | less: exp V -> exp V -> bexp V. *)
392

393
(**
394
  Define evaluation of boolean expressions
395
 **)
396 397 398 399 400 401 402 403 404 405 406 407 408 409
(* Inductive bval (E:env): (bexp R) -> Prop -> Prop := *)
(*   leq_eval (f1:exp R) (f2:exp R) (v1:R) (v2:R): *)
(*     eval_exp E f1 v1 -> *)
(*     eval_exp E f2 v2 -> *)
(*     bval E (leq f1 f2) (Rle v1 v2) *)
(* |less_eval (f1:exp R) (f2:exp R) (v1:R) (v2:R): *)
(*     eval_exp E f1 v1 -> *)
(*     eval_exp E f2 v2 -> *)
(*     bval E (less f1 f2) (Rlt v1 v2). *)
(* (** *)
(*  Simplify arithmetic later by making > >= only abbreviations *)
(* **) *)
(* Definition gr := fun (V:Type) (f1: exp V) (f2: exp V) => less f2 f1. *)
(* Definition greq := fun (V:Type) (f1:exp V) (f2: exp V) => leq f2 f1. *)