Expressions.v 11.7 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
  Define expressions parametric over some value type V.
  Will ease reasoning about different instantiations later.
73
**)
74 75
Inductive exp (V:Type): Type :=
  Var: nat -> exp V
76
| Const: mType -> V -> exp V
77
| Unop: unop -> exp V -> exp V
78 79
| Binop: binop -> exp V -> exp V -> exp V
| Downcast: mType -> exp V -> exp V.
80

81 82 83 84
(**
  Boolean equality function on expressions.
  Used in certificates to define the analysis result as function
**)
85
Fixpoint expEqBool (e1:exp Q) (e2:exp Q) :=
='s avatar
= committed
86
  match e1, e2 with
87
  | Var _ v1, Var _ v2 => (v1 =? v2)
='s avatar
= committed
88 89 90 91 92
  | 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
93 94
  end.

95
Lemma expEqBool_refl e:
96 97
  expEqBool e e = true.
Proof.
='s avatar
= committed
98
  induction e; try (apply andb_true_iff; split); simpl in *; auto; try (apply EquivEqBoolEq; auto).
99 100 101 102 103 104 105
  - 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.
106

107 108 109 110 111
Lemma expEqBool_sym e e':
  expEqBool e e' = expEqBool e' e.
Proof.
  revert e'.
  induction e; intros e'; destruct e'; simpl; try auto.
112
  - apply beq_nat_sym.
113 114 115 116 117 118
  - f_equal.
    + apply mTypeEqBool_sym; auto.
    + apply Qeq_bool_sym.
  - f_equal.
    + destruct u; auto.
    + apply IHe.
='s avatar
= committed
119
  - f_equal.
120 121 122 123 124 125 126 127 128
    + destruct b; auto.
    + f_equal.
      * apply IHe1.
      * apply IHe2.
  - f_equal.
    + apply mTypeEqBool_sym; auto.
    + apply IHe.
Qed.

='s avatar
= committed
129 130 131 132 133
Lemma expEqBool_trans e f g:
  expEqBool e f = true ->
  expEqBool f g = true ->
  expEqBool e g = true.
Proof.
134 135 136 137 138
  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).
139 140
  - apply beq_nat_true in H2.
    apply beq_nat_true in H1.
='s avatar
= committed
141
    subst.
142
    unfold expEqBool.
='s avatar
= committed
143
    rewrite <- beq_nat_refl.
='s avatar
= committed
144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172
    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.

173 174
Fixpoint toRExp (e:exp Q) :=
  match e with
175
  |Var _ v => Var R v
176
  |Const m n => Const m (Q2R n)
177 178 179
  |Unop o e1 => Unop o (toRExp e1)
  |Binop o e1 e2 => Binop o (toRExp e1) (toRExp e2)
  |Downcast m e1 => Downcast m (toRExp e1)
180
  end.
181

182 183
Fixpoint toREval (e:exp R) :=
  match e with
184
  | Var _ v => Var R v
185
  | Const _ n => Const M0 n
186 187
  | Unop o e1 => Unop o (toREval e1)
  | Binop o e1 e2 => Binop o (toREval e1) (toREval e2)
188
  | Downcast _ e1 =>  (toREval e1)
189
  end.
190

='s avatar
= committed
191 192 193 194
Fixpoint toREvalVars (d:nat -> option mType) (n:nat) :=
  match d n with
  | Some m => Some M0
  | None => None
195
  end.
196

197 198 199 200
(**
  Define a perturbation function to ease writing of basic definitions
**)
Definition perturb (r:R) (e:R) :=
201
  (r * (1 + e))%R.
Heiko Becker's avatar
Heiko Becker committed
202

203
(**
204
Define expression evaluation relation parametric by an "error" epsilon.
205 206 207
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.
208
**)
209
Inductive eval_exp (E:env) (defVars: nat -> option mType) :(exp R) -> R -> mType -> Prop :=
210
| Var_load m x v:
211
    defVars x = Some m ->
212
    E x = Some v ->
213
    eval_exp E defVars (Var R x) v m
214 215
| Const_dist m n delta:
    Rle (Rabs delta) (Q2R (meps m)) ->
216
    eval_exp E defVars (Const m n) (perturb n delta) m
217
| Unop_neg m f1 v1:
218 219
    eval_exp E defVars f1 v1 m ->
    eval_exp E defVars (Unop Neg f1) (evalUnop Neg v1) m
220 221
| Unop_inv m f1 v1 delta:
    Rle (Rabs delta) (Q2R (meps m)) ->
222 223
    eval_exp E defVars  f1 v1 m ->
    eval_exp E defVars (Unop Inv f1) (perturb (evalUnop Inv v1) delta) m
224
| Downcast_dist m m1 f1 v1 delta:
225
    (* Downcast expression f1 (evaluating to machine type m1), to a machine type m, less precise than m1.*)
226 227
    isMorePrecise m1 m = true ->
    Rle (Rabs delta) (Q2R (meps m)) ->
228
    eval_exp E defVars f1 v1 m1 ->
229 230 231 232 233
    eval_exp E defVars (Downcast m f1) (perturb v1 delta) m
| Binop_dist m1 m2 op f1 f2 v1 v2 delta:
    Rle (Rabs delta) (Q2R (meps (computeJoin m1 m2))) ->
    eval_exp E defVars f1 v1 m1 ->
    eval_exp E defVars f2 v2 m2 ->
234
    ((op = Div) -> (~ v2 = 0)%R) ->
235
    eval_exp E defVars (Binop op f1 f2) (perturb (evalBinop op v1 v2) delta)  (computeJoin m1 m2).
236

237 238 239 240 241
(**
  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 :=
242 243
  match e with
  | Var _ x => NatSet.singleton x
244 245
  | Unop u e1 => usedVars e1
  | Binop b e1 e2 => NatSet.union (usedVars e1) (usedVars e2)
246
  | Downcast _ e1 => usedVars e1
247 248
  | _ => NatSet.empty
  end.
249

250
(**
251
  If |delta| <= 0 then perturb v delta is exactly v.
252
**)
253
Lemma delta_0_deterministic (v:R) (delta:R):
Heiko Becker's avatar
Heiko Becker committed
254 255 256 257 258
  (Rabs delta <= 0)%R ->
  perturb v delta = v.
Proof.
  intros abs_0; apply Rabs_0_impl_eq in abs_0; subst.
  unfold perturb.
259
  lra.
Heiko Becker's avatar
Heiko Becker committed
260 261
Qed.

262
(* TODO: need of `general` case? *)
263
Lemma general_meps_0_deterministic (f:exp R) (E:env) defVars:
264 265
  forall v1 v2 m1,
    m1 = M0 ->
266 267
    eval_exp E defVars (toREval f) v1 m1 ->
    eval_exp E defVars (toREval f) v2 M0 ->
268 269
    v1 = v2.
Proof.
270
  induction f; intros * m10_eq eval_v1 eval_v2.
271 272
  - 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.
273
    rewrite H6 in H1; inversion H1; subst; auto.
274 275 276 277
  - 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.
='s avatar
= committed
278
    + apply Ropp_eq_compat. apply (IHf v0 v3 M0); auto.
279 280
    + inversion H4.
    + inversion H5.
281
    + rewrite (IHf v0 v3 M0); auto.
282 283
  - 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.
284 285
    destruct m0; destruct m2; inversion H5.
    destruct m3; destruct m4; inversion H11.
286
    simpl in *.
287 288
    rewrite (IHf1 v0 v4 M0); auto.
    rewrite (IHf2 v5 v3 M0); auto.
289 290 291
    rewrite Q2R0_is_0 in H2,H12.
    rewrite delta_0_deterministic; auto.
    rewrite delta_0_deterministic; auto.
292 293
  - simpl toREval in eval_v1.
    simpl toREval in eval_v2.
294
    apply (IHf v1 v2 m1); auto.
295 296
Qed.

297 298 299 300 301 302 303 304
(* 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. *)
305

='s avatar
= committed
306

307
(**
308
Evaluation with 0 as machine epsilon is deterministic
309
**)
310
Lemma meps_0_deterministic (f:exp R) (E:env) defVars:
311
  forall v1 v2,
312 313
  eval_exp E defVars (toREval f) v1 M0 ->
  eval_exp E defVars (toREval f) v2 M0 ->
314 315
  v1 = v2.
Proof.
316
  intros v1 v2 ev1 ev2.
317
  assert (M0 = M0) by auto.
='s avatar
= committed
318
  apply (general_meps_0_deterministic f H ev1 ev2).
319 320
Qed.

321 322 323 324
(**
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
325
variables in the Environment.
326
 **)
327 328
Lemma binary_unfolding b f1 f2 m E vF defVars:
  eval_exp E defVars (Binop b f1 f2) vF m ->
329
  exists vF1 vF2 m1 m2,
330
    m = computeJoin m1 m2 /\
331 332
    eval_exp E defVars f1 vF1 m1 /\
    eval_exp E defVars f2 vF2 m2 /\
333
    eval_exp (updEnv 2 vF2 (updEnv 1 vF1 emptyEnv))
='s avatar
= committed
334
             (updDefVars 2 m2 (updDefVars 1 m1 defVars))
335
             (Binop b (Var R 1) (Var R 2)) vF m.
336
Proof.
337 338
  intros eval_float.
  inversion eval_float; subst.
339 340
  exists v1 ; exists v2; exists m1; exists m2; repeat split; try auto.
  eapply Binop_dist; eauto.
341 342 343 344
  - pose proof (isMorePrecise_refl m1).
    eapply Var_load; eauto.
  - pose proof (isMorePrecise_refl m2).
    eapply Var_load; eauto.
345 346
Qed.

347
(*
348 349 350
(**
Analogous lemma for unary expressions.
**)
351 352
Lemma unary_unfolding (e:exp R) (eps:R) (E:env) (v:R):
  (eval_exp eps E (Unop Inv e) v <->
353
   exists v1,
354 355
     eval_exp eps E e v1 /\
     eval_exp eps (updEnv 1 v1 E) (Unop Inv (Var R 1)) v).
356 357 358 359 360 361 362 363 364
Proof.
  split.
  - intros eval_un.
    inversion eval_un; subst.
    exists v1.
    repeat split; try auto.
    constructor; try auto.
    constructor; auto.
  - intros exists_val.
365 366
    destruct exists_val as [v1 [eval_f1 eval_e_E]].
    inversion eval_e_E; subst.
367 368 369
    inversion H1; subst.
    unfold updEnv in *; simpl in *.
    constructor; auto.
370
    inversion H3; subst; auto.
371
Qed. *)
372

373
(*   Using the parametric expressions, define boolean expressions for conditionals *)
374
(* **)
375 376 377
(* Inductive bexp (V:Type) : Type := *)
(*   leq: exp V -> exp V -> bexp V *)
(* | less: exp V -> exp V -> bexp V. *)
378

379
(**
380
  Define evaluation of boolean expressions
381
 **)
382 383 384 385 386 387 388 389 390 391 392 393 394 395
(* 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. *)