IntervalValidation.v 28.7 KB
Newer Older
Heiko Becker's avatar
Heiko Becker committed
1
(**
Heiko Becker's avatar
Heiko Becker committed
2 3 4 5 6
    Interval arithmetic checker and its soundness proof.
    The function validIntervalbounds checks wether the given analysis result is
    a valid range arithmetic for each sub term of the given expression e.
    The computation is done using our formalized interval arithmetic.
    The function is used in CertificateChecker.v to build the full checker.
Heiko Becker's avatar
Heiko Becker committed
7
**)
8
Require Import Coq.QArith.QArith Coq.QArith.Qreals QArith.Qminmax Coq.Lists.List Coq.micromega.Psatz.
9
Require Import Daisy.Infra.Abbrevs Daisy.Infra.RationalSimps Daisy.Infra.RealRationalProps.
10 11
Require Import Daisy.Infra.Ltacs Daisy.Infra.RealSimps.
Require Export Daisy.IntervalArithQ Daisy.IntervalArith Daisy.ssaPrgs.
12

13
Fixpoint validIntervalbounds (e:exp Q) (absenv:analysisResult) (P:precond) (validVars:NatSet.t) :=
14 15
  let (intv, _) := absenv e in
    match e with
16
    | Var _ m v =>
17 18 19
      if NatSet.mem v validVars
      then true
      else isSupersetIntv (P v) intv && (Qleb (ivlo (P v)) (ivhi (P v)))
Heiko Becker's avatar
Heiko Becker committed
20 21
    | Const n => isSupersetIntv (n,n) intv
    | Unop o f =>
22 23 24
      if validIntervalbounds f absenv P validVars
      then
        let (iv, _) := absenv f in
Heiko Becker's avatar
Heiko Becker committed
25
        match o with
26
        | Neg =>
Heiko Becker's avatar
Heiko Becker committed
27
          let new_iv := negateIntv iv in
28 29
          isSupersetIntv new_iv intv
        | Inv =>
30 31 32 33 34 35
          if (((Qleb (ivhi iv) 0) && (negb (Qeq_bool (ivhi iv) 0))) ||
              ((Qleb 0 (ivlo iv)) && (negb (Qeq_bool (ivlo iv) 0))))
          then
            let new_iv := invertIntv iv in
            isSupersetIntv new_iv intv
          else false
Heiko Becker's avatar
Heiko Becker committed
36
        end
37
      else false
Heiko Becker's avatar
Heiko Becker committed
38
    | Binop op f1 f2 =>
39 40 41 42 43
      if ((validIntervalbounds f1 absenv P validVars) &&
          (validIntervalbounds f2 absenv P validVars))
      then
        let (iv1,_) := absenv f1 in
        let (iv2,_) := absenv f2 in
Heiko Becker's avatar
Heiko Becker committed
44
          match op with
45 46 47 48 49 50 51 52 53
          | Plus =>
            let new_iv := addIntv iv1 iv2 in
            isSupersetIntv new_iv intv
          | Sub =>
            let new_iv := subtractIntv iv1 iv2 in
            isSupersetIntv new_iv intv
          | Mult =>
            let new_iv := multIntv iv1 iv2 in
            isSupersetIntv new_iv intv
Heiko Becker's avatar
Heiko Becker committed
54
          | Div =>
55 56 57 58 59 60
            if (((Qleb (ivhi iv2) 0) && (negb (Qeq_bool (ivhi iv2) 0))) ||
                ((Qleb 0 (ivlo iv2)) && (negb (Qeq_bool (ivlo iv2) 0))))
            then
              let new_iv := divideIntv iv1 iv2 in
              isSupersetIntv new_iv intv
            else false
61
          end
62
      else false
63 64 65 66
    | Downcast m f1 =>
      let (iv1, _) := absenv f1 in
      andb (validIntervalbounds f1 absenv P validVars) (andb (isSupersetIntv intv iv1) (isSupersetIntv iv1 intv))
           (* TODO: intv = iv1 might be a hard constraint... *)
67 68
    end.

69
Fixpoint validIntervalboundsCmd (f:cmd Q) (absenv:analysisResult) (P:precond) (validVars:NatSet.t) :bool:=
70
  match f with
71
  | Let m x e g =>
72
    if (validIntervalbounds e absenv P validVars &&
73 74
        Qeq_bool (fst (fst (absenv e))) (fst (fst (absenv (Var Q m x)))) &&
        Qeq_bool (snd (fst (absenv e))) (snd (fst (absenv (Var Q m x)))))
75 76
    then validIntervalboundsCmd g absenv P (NatSet.add x validVars)
    else false
77 78
  |Ret e =>
   validIntervalbounds e absenv P validVars
79 80
  end.

81 82 83 84 85 86 87 88 89
Fixpoint erasure (e:exp Q) :exp Q :=
  match e with
  |Var _ m x => Var Q M0 x
  |Unop u e => Unop u (erasure e)
  |Binop b e1 e2 => Binop b (erasure e1) (erasure e2)
  |Downcast _ e => Downcast M0 (erasure e)
  |_ => e
  end.

90 91 92 93 94 95
Fixpoint erasureCmd (c:cmd Q) :cmd Q :=
  match c with
  | Let m x e g => Let M0 x (erasure e) (erasureCmd g)
  | Ret e => Ret (erasure e)
  end.

96
Theorem ivbounds_approximatesPrecond_sound f absenv P V:
97
  validIntervalbounds (erasure f) absenv P V = true ->
Raphaël Monat's avatar
Raphaël Monat committed
98
  forall v, NatSet.In v (NatSet.diff (Expressions.usedVars f) V) ->
99
       Is_true(isSupersetIntv (P v) (fst (absenv (Var Q M0 v)))).
100
Proof.
Heiko Becker's avatar
Heiko Becker committed
101
  induction f; unfold validIntervalbounds.
102
  - simpl. intros approx_true v v_in_fV; simpl in *.
103 104 105
    rewrite NatSet.diff_spec in v_in_fV.
    rewrite NatSet.singleton_spec in v_in_fV;
      destruct v_in_fV; subst.
106
    destruct (absenv (Var Q M0 n)); simpl in *.
107 108 109 110
    case_eq (NatSet.mem n V); intros case_mem;
      rewrite case_mem in approx_true; simpl in *.
    + rewrite NatSet.mem_spec in case_mem.
      contradiction.
111 112 113
    + apply Is_true_eq_left in approx_true.
      apply andb_prop_elim in approx_true.
      destruct approx_true; auto.
114 115
  - intros approx_true v0 v_in_fV; simpl in *.
    inversion v_in_fV.
Heiko Becker's avatar
Heiko Becker committed
116
  - intros approx_unary_true v v_in_fV.
Raphaël Monat's avatar
Raphaël Monat committed
117
    unfold usedVars in v_in_fV.
Heiko Becker's avatar
Heiko Becker committed
118
    apply Is_true_eq_left in approx_unary_true.
119 120
    simpl in *.
    destruct (absenv (Unop u (erasure f))); destruct (absenv (erasure f)); simpl in *.
Heiko Becker's avatar
Heiko Becker committed
121 122 123 124
    apply andb_prop_elim in approx_unary_true.
    destruct approx_unary_true.
    apply IHf; try auto.
    apply Is_true_eq_true; auto.
125
  - intros approx_bin_true v v_in_fV.
126 127 128 129
    simpl in v_in_fV.
    rewrite NatSet.diff_spec in v_in_fV.
    destruct v_in_fV as [ v_in_fV v_not_in_V].
    rewrite NatSet.union_spec in v_in_fV.
130
    apply Is_true_eq_left in approx_bin_true.
131 132 133
    simpl in *.
    destruct (absenv (Binop b (erasure f1) (erasure f2))); destruct (absenv (erasure f1));
      destruct (absenv (erasure f2)); simpl in *.
134 135 136 137
    apply andb_prop_elim in approx_bin_true.
    destruct approx_bin_true.
    apply andb_prop_elim in H.
    destruct H.
138
    destruct v_in_fV.
Heiko Becker's avatar
Heiko Becker committed
139
    + apply IHf1; auto.
140
      apply Is_true_eq_true; auto.
141
      rewrite NatSet.diff_spec; split; auto.
Heiko Becker's avatar
Heiko Becker committed
142
    + apply IHf2; auto.
143
      apply Is_true_eq_true; auto.
144
      rewrite NatSet.diff_spec; split; auto.
145 146 147 148 149 150 151 152
  - intros approx_rnd_true v v_in_fV.
    simpl in *; destruct (absenv (Downcast M0 (erasure f))); destruct (absenv (erasure f)).
    apply Is_true_eq_left in approx_rnd_true.
    apply andb_prop_elim in approx_rnd_true.
    destruct approx_rnd_true.
    apply IHf; auto.
    apply Is_true_eq_true in H; auto.
Qed.
153

Heiko Becker's avatar
Heiko Becker committed
154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
Corollary Q2R_max4 a b c d:
  Q2R (IntervalArithQ.max4 a b c d) = max4 (Q2R a) (Q2R b) (Q2R c) (Q2R d).
Proof.
  unfold IntervalArithQ.max4, max4; repeat rewrite Q2R_max; auto.
Qed.

Corollary Q2R_min4 a b c d:
  Q2R (IntervalArithQ.min4 a b c d) = min4 (Q2R a) (Q2R b) (Q2R c) (Q2R d).
Proof.
  unfold IntervalArith.min4, min4; repeat rewrite Q2R_min; auto.
Qed.

Ltac env_assert absenv e name :=
  assert (exists iv err, absenv e = (iv,err)) as name by (destruct (absenv e); repeat eexists; auto).

169
Lemma validBoundsDiv_uneq_zero e1 e2 absenv P V ivlo_e2 ivhi_e2 err:
170
  absenv e2 = ((ivlo_e2,ivhi_e2), err) ->
171
  validIntervalbounds (Binop Div e1 e2) absenv P V = true ->
172 173 174 175 176 177 178
  (ivhi_e2 < 0) \/ (0 < ivlo_e2).
Proof.
  intros absenv_eq validBounds.
  unfold validIntervalbounds in validBounds.
  env_assert absenv (Binop Div e1 e2) abs_div; destruct abs_div as [iv_div [err_div abs_div]].
  env_assert absenv e1 abs_e1; destruct abs_e1 as [iv_e1 [err_e1 abs_e1]].
  rewrite abs_div, abs_e1, absenv_eq in validBounds.
179
  repeat (rewrite <- andb_lazy_alt in validBounds).
180 181 182
  apply Is_true_eq_left in validBounds.
  apply andb_prop_elim in validBounds.
  destruct validBounds as [_ validBounds]; apply andb_prop_elim in validBounds.
183
  destruct validBounds as [nodiv0 _].
184
  apply Is_true_eq_true in nodiv0.
185
  unfold isSupersetIntv in *; simpl in *.
186
  apply le_neq_bool_to_lt_prop; auto.
187 188
Qed.

189 190
Fixpoint getRetExp (V:Type) (f:cmd V) :=
  match f with
191
  |Let m x e g => getRetExp g
192 193 194
  | Ret e => e
  end.

195
Theorem validIntervalbounds_sound (f:exp Q) (absenv:analysisResult) (P:precond) fVars dVars (E:env):
Heiko Becker's avatar
Heiko Becker committed
196
  forall vR,
197
    validIntervalbounds (erasure f) absenv P dVars = true ->
198
    (forall v, NatSet.mem v dVars = true ->
199 200
          exists vR, E v = Some (vR, M0) /\
                (Q2R (fst (fst (absenv (Var Q M0 v)))) <= vR <= Q2R (snd (fst (absenv (Var Q M0 v)))))%R) ->
Raphaël Monat's avatar
Raphaël Monat committed
201
    NatSet.Subset (NatSet.diff (Expressions.usedVars f) dVars) fVars ->
202
    (forall v, NatSet.mem v fVars = true ->
203
          exists vR, E v = Some (vR, M0) /\
204
                (Q2R (fst (P v)) <= vR <= Q2R (snd (P v)))%R) ->
205 206
    eval_exp E (toREval (toRExp f)) vR M0 ->
  (Q2R (fst (fst (absenv (erasure f)))) <= vR <= Q2R (snd (fst (absenv (erasure f)))))%R.
207
Proof.
Raphaël Monat's avatar
Raphaël Monat committed
208
  induction f; intros vR valid_bounds valid_definedVars usedVars_subset valid_usedVars eval_f.
209
  - unfold validIntervalbounds in valid_bounds.
210
    env_assert absenv (Var Q M0 n) absenv_var.
211
    destruct absenv_var as [ iv [err absenv_var]].
Raphaël Monat's avatar
Raphaël Monat committed
212
    specialize (valid_usedVars n).
213
    simpl; rewrite absenv_var in *; simpl in *.
214
    inversion eval_f; subst.
215
    case_eq (NatSet.mem n dVars); intros case_mem; rewrite case_mem in *; simpl in *.
216 217
    + specialize (valid_definedVars n case_mem).
      destruct valid_definedVars as [vR' [E_n_eq precond_sound]].
218
      rewrite E_n_eq in *.
219
      inversion H4; subst.
220
      rewrite absenv_var in *; auto.
221
    + repeat (rewrite delta_0_deterministic in *; try auto).
222
      rewrite absenv_var in valid_bounds.
223 224
      unfold isSupersetIntv in valid_bounds.
      andb_to_prop valid_bounds.
225 226 227 228
      apply Qle_bool_iff in L0;
        apply Qle_bool_iff in R0.
      apply Qle_Rle in L0;
        apply Qle_Rle in R0.
229
      simpl in *.
230 231 232 233
      assert (NatSet.mem n fVars = true) as in_fVars.
      * assert (NatSet.In n (NatSet.singleton n))
          as in_singleton by (rewrite NatSet.singleton_spec; auto).
        rewrite NatSet.mem_spec.
Raphaël Monat's avatar
Raphaël Monat committed
234 235
        hnf in usedVars_subset.
        apply usedVars_subset.
236
        rewrite NatSet.diff_spec, NatSet.singleton_spec.
237
        split; try auto.
238 239 240
        hnf; intros in_dVars.
        rewrite <- NatSet.mem_spec in in_dVars.
        rewrite in_dVars in case_mem; congruence.
Raphaël Monat's avatar
Raphaël Monat committed
241 242
      * specialize (valid_usedVars in_fVars);
          destruct valid_usedVars as [vR' [vR_def P_valid]].
243
        rewrite vR_def in H4; inversion H4; subst.
244
        lra.
245
  - unfold validIntervalbounds in valid_bounds.
246 247
    simpl erasure in valid_bounds.
    simpl in *;  destruct (absenv (Const v)) as [intv err]; simpl in *.
248 249
    apply Is_true_eq_left in valid_bounds.
    apply andb_prop_elim in valid_bounds.
Heiko Becker's avatar
Heiko Becker committed
250
    destruct valid_bounds as [valid_lo valid_hi].
Heiko Becker's avatar
Heiko Becker committed
251
    inversion eval_f; subst.
252
    rewrite delta_0_deterministic; auto.
253 254
    unfold contained; simpl.
    split.
Heiko Becker's avatar
Heiko Becker committed
255
    + apply Is_true_eq_true in valid_lo.
256
      unfold Qleb in *.
Heiko Becker's avatar
Heiko Becker committed
257 258 259 260 261 262
      apply Qle_bool_iff in valid_lo.
      apply Qle_Rle in valid_lo; auto.
    + apply Is_true_eq_true in valid_hi.
      unfold Qleb in *.
      apply Qle_bool_iff in valid_hi.
      apply Qle_Rle in valid_hi; auto.
263 264
    + simpl in H0; rewrite Q2R0_is_0 in H0; auto.
  - case_eq (absenv (Unop u (erasure f))); intros intv err absenv_unop.
Heiko Becker's avatar
Heiko Becker committed
265 266
    destruct intv as [unop_lo unop_hi]; simpl.
    unfold validIntervalbounds in valid_bounds.
267 268
    simpl in valid_bounds; rewrite absenv_unop in valid_bounds.
    case_eq (absenv (erasure f)); intros intv_f err_f absenv_f.
Heiko Becker's avatar
Heiko Becker committed
269 270 271 272 273 274
    rewrite absenv_f in valid_bounds.
    apply Is_true_eq_left in valid_bounds.
    apply andb_prop_elim in valid_bounds.
    destruct valid_bounds as [valid_rec valid_unop].
    apply Is_true_eq_true in valid_rec.
    inversion eval_f; subst.
275
    + specialize (IHf v1 valid_rec valid_definedVars usedVars_subset valid_usedVars H3).
Heiko Becker's avatar
Heiko Becker committed
276
      rewrite absenv_f in IHf; simpl in IHf.
277 278 279 280 281 282
      (* TODO: Make lemma *)
      unfold isSupersetIntv in valid_unop.
      apply andb_prop_elim in valid_unop.
      destruct valid_unop as [valid_lo valid_hi].
      apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
      apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
283
      pose proof (interval_negation_valid (iv :=(Q2R (fst intv_f),(Q2R (snd intv_f)))) (a :=v1)) as negation_valid.
284 285 286
      unfold contained, negateInterval in negation_valid; simpl in *.
      apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
      destruct IHf.
287
      split.
288
      * eapply Rle_trans. rewrite absenv_unop; simpl; apply valid_lo.
289 290
        rewrite Q2R_opp; lra.
      * eapply Rle_trans.
291
        Focus 2. rewrite absenv_unop; simpl; apply valid_hi.
292
        rewrite Q2R_opp; lra.
293
    + specialize (IHf v1 valid_rec valid_definedVars usedVars_subset valid_usedVars H4).
Heiko Becker's avatar
Heiko Becker committed
294
      rewrite absenv_f in IHf; simpl in IHf.
295
      apply andb_prop_elim in valid_unop.
296
      destruct valid_unop as [nodiv0 valid_unop].
297 298 299 300 301 302 303 304 305 306 307 308
      (* TODO: Make lemma *)
      unfold isSupersetIntv in valid_unop.
      apply andb_prop_elim in valid_unop.
      destruct valid_unop as [valid_lo valid_hi].
      apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
      apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
      assert ((Q2R (ivhi intv_f) < 0)%R \/ (0 < Q2R (ivlo intv_f))%R) as nodiv0_prop.
       * apply Is_true_eq_true in nodiv0.
         apply le_neq_bool_to_lt_prop in nodiv0.
         destruct nodiv0.
         { left; rewrite <- Q2R0_is_0; apply Qlt_Rlt; auto. }
         { right; rewrite <- Q2R0_is_0; apply Qlt_Rlt; auto. }
309
       * pose proof (interval_inversion_valid (iv :=(Q2R (fst intv_f),(Q2R (snd intv_f)))) (a :=v1)) as inv_valid.
310 311 312
         unfold contained, invertInterval in inv_valid; simpl in *.
         apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
         destruct IHf.
313
         rewrite delta_0_deterministic; auto.
314
         unfold perturb; split.
315
         { eapply Rle_trans. rewrite absenv_unop; simpl; apply valid_lo.
316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336
           destruct nodiv0_prop as [nodiv0_neg | nodiv0_pos].
           (* TODO: Extract lemma maybe *)
           - assert (0 < - (Q2R (snd intv_f)))%R as negation_pos by lra.
             assert (- (Q2R (snd intv_f)) <= - v1)%R as negation_flipped_hi by lra.
             apply Rinv_le_contravar in negation_flipped_hi; try auto.
             rewrite <- Ropp_inv_permute in negation_flipped_hi; try lra.
             rewrite <- Ropp_inv_permute in negation_flipped_hi; try lra.
             apply Ropp_le_contravar in negation_flipped_hi.
             repeat rewrite Ropp_involutive in negation_flipped_hi;
               rewrite Q2R_inv; auto.
             hnf; intros is_0.
             rewrite <- Q2R0_is_0 in nodiv0_neg.
             apply Rlt_Qlt in nodiv0_neg; lra.
           - rewrite Q2R_inv.
             apply Rinv_le_contravar; try lra.
             hnf; intros is_0.
             assert (Q2R (fst intv_f) <= Q2R (snd intv_f))%R by lra.
             rewrite <- Q2R0_is_0 in nodiv0_pos.
             apply Rlt_Qlt in nodiv0_pos; apply Rle_Qle in H2; lra.
         }
         { eapply Rle_trans.
337
           Focus 2. rewrite absenv_unop; simpl; apply valid_hi.
338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
           destruct nodiv0_prop as [nodiv0_neg | nodiv0_pos].
           - assert (Q2R (fst intv_f) < 0)%R as fst_lt_0 by lra.
             assert (0 < - (Q2R (fst intv_f)))%R as negation_pos by lra.
             assert (- v1 <= - (Q2R (fst intv_f)))%R as negation_flipped_lo by lra.
             apply Rinv_le_contravar in negation_flipped_lo; try auto.
             rewrite <- Ropp_inv_permute in negation_flipped_lo; try lra.
             rewrite <- Ropp_inv_permute in negation_flipped_lo; try lra.
             apply Ropp_le_contravar in negation_flipped_lo.
             repeat rewrite Ropp_involutive in negation_flipped_lo;
               rewrite Q2R_inv; auto.
             hnf; intros is_0.
             rewrite <- Q2R0_is_0 in negation_pos.
             rewrite <- Q2R_opp in negation_pos.
             apply Rlt_Qlt in negation_pos; lra.
             assert (0 < - (Q2R (snd intv_f)))%R by lra.
             lra.
           - rewrite Q2R_inv.
             apply Rinv_le_contravar; try lra.
             hnf; intros is_0.
             assert (Q2R (fst intv_f) <= Q2R (snd intv_f))%R by lra.
             rewrite <- Q2R0_is_0 in nodiv0_pos.
             apply Rlt_Qlt in nodiv0_pos; apply Rle_Qle in H2; lra.
         }
361
         { rewrite Q2R0_is_0 in H1; auto. }
362
  - inversion eval_f; subst.
363 364
    rewrite delta_0_deterministic in eval_f; auto.
    rewrite delta_0_deterministic; auto.
Heiko Becker's avatar
Heiko Becker committed
365
    simpl in valid_bounds.
366 367 368 369
    case_eq (absenv (Binop b (erasure f1) (erasure f2))); intros iv err absenv_bin.
    case_eq (absenv (erasure f1)); intros iv1 err1 absenv_f1.
    case_eq (absenv (erasure f2)); intros iv2 err2 absenv_f2.
    simpl.
Heiko Becker's avatar
Heiko Becker committed
370
    rewrite absenv_bin, absenv_f1, absenv_f2 in valid_bounds.
Heiko Becker's avatar
Heiko Becker committed
371 372 373 374 375 376
    apply Is_true_eq_left in valid_bounds.
    apply andb_prop_elim in valid_bounds.
    destruct valid_bounds as [valid_rec valid_bin].
    apply andb_prop_elim in valid_rec.
    destruct valid_rec as [valid_e1 valid_e2].
    apply Is_true_eq_true in valid_e1; apply Is_true_eq_true in  valid_e2.
377 378
    specialize (IHf1 v1 valid_e1 valid_definedVars);
      specialize (IHf2 v2 valid_e2 valid_definedVars).
Heiko Becker's avatar
Heiko Becker committed
379 380
    rewrite absenv_f1 in IHf1.
    rewrite absenv_f2 in IHf2.
381 382 383
    assert ((Q2R (fst (fst (iv1, err1))) <= v1 <= Q2R (snd (fst (iv1, err1))))%R) as valid_bounds_e1.
    + apply IHf1; try auto.
      intros v in_diff_e1.
Raphaël Monat's avatar
Raphaël Monat committed
384
      apply usedVars_subset.
385 386
      simpl. rewrite NatSet.diff_spec,NatSet.union_spec.
      rewrite NatSet.diff_spec in in_diff_e1.
Raphaël Monat's avatar
Raphaël Monat committed
387
      destruct in_diff_e1 as [ in_usedVars not_dVar].
388
      split; try auto.
389
      assert (m1 = M0) by (apply (ifM0isJoin_l M0 m1 m2); auto); subst; auto.
390 391 392
    + assert (Q2R (fst (fst (iv2, err2))) <= v2 <= Q2R (snd (fst (iv2, err2))))%R as valid_bounds_e2.
      * apply IHf2; try auto.
        intros v in_diff_e2.
Raphaël Monat's avatar
Raphaël Monat committed
393
        apply usedVars_subset.
394 395 396
        simpl. rewrite NatSet.diff_spec, NatSet.union_spec.
        rewrite NatSet.diff_spec in in_diff_e2.
        destruct in_diff_e2; split; auto.
397
        assert (m2 = M0) by (apply (ifM0isJoin_r M0 m1 m2); auto); subst; auto.
398 399 400 401 402 403 404 405 406 407 408 409
      * destruct b; simpl in *.
        { pose proof (interval_addition_valid (iv1 :=(Q2R (fst iv1),Q2R (snd iv1))) (iv2 :=(Q2R (fst iv2), Q2R (snd iv2)))) as valid_add.
          unfold validIntervalAdd in valid_add.
          specialize (valid_add v1 v2 valid_bounds_e1 valid_bounds_e2).
          unfold contained in valid_add.
          unfold isSupersetIntv in valid_bin.
          apply andb_prop_elim in valid_bin; destruct valid_bin as [valid_lo valid_hi].
          apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
          apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
          apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
          destruct valid_add as [valid_add_lo valid_add_hi].
          split.
410
          - eapply Rle_trans. rewrite absenv_bin; apply valid_lo.
411 412 413 414 415 416 417
            unfold ivlo. unfold addIntv.
            simpl in valid_add_lo.
            repeat rewrite <- Q2R_plus in valid_add_lo.
            rewrite <- Q2R_min4 in valid_add_lo.
            unfold absIvUpd; auto.
          - eapply Rle_trans.
            Focus 2.
418
            rewrite absenv_bin; apply valid_hi.
419 420 421 422 423 424 425 426 427 428 429 430 431 432 433
            unfold ivlo, addIntv.
            simpl in valid_add_hi.
            repeat rewrite <- Q2R_plus in valid_add_hi.
            rewrite <- Q2R_max4 in valid_add_hi.
            unfold absIvUpd; auto. }
        { pose proof (interval_subtraction_valid (iv1 := (Q2R (fst iv1),Q2R (snd iv1))) (iv2 :=(Q2R (fst iv2), Q2R (snd iv2)))) as valid_sub.
          specialize (valid_sub v1 v2 valid_bounds_e1 valid_bounds_e2).
          unfold contained in valid_sub.
          unfold isSupersetIntv in valid_bin.
          apply andb_prop_elim in valid_bin; destruct valid_bin as [valid_lo valid_hi].
          apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
          apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
          apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
          destruct valid_sub as [valid_sub_lo valid_sub_hi].
          split.
434
          - eapply Rle_trans. rewrite absenv_bin; apply valid_lo.
435 436 437 438 439 440 441 442
            unfold ivlo. unfold subtractIntv.
            simpl in valid_sub_lo.
            repeat rewrite <- Rsub_eq_Ropp_Rplus in valid_sub_lo.
            repeat rewrite <- Q2R_minus in valid_sub_lo.
            rewrite <- Q2R_min4 in valid_sub_lo.
            unfold absIvUpd; auto.
          - eapply Rle_trans.
            Focus 2.
443
            rewrite absenv_bin; apply valid_hi.
444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459
            unfold ivlo, addIntv.
            simpl in valid_sub_hi.
            repeat rewrite <- Rsub_eq_Ropp_Rplus in valid_sub_hi.
            repeat rewrite <- Q2R_minus in valid_sub_hi.
            rewrite <- Q2R_max4 in valid_sub_hi.
            unfold absIvUpd; auto. }
        { pose proof (interval_multiplication_valid (iv1 :=(Q2R (fst iv1),Q2R (snd iv1))) (iv2:=(Q2R (fst iv2), Q2R (snd iv2)))) as valid_mul.
          specialize (valid_mul v1 v2 valid_bounds_e1 valid_bounds_e2).
          unfold contained in valid_mul.
          unfold isSupersetIntv in valid_bin.
          apply andb_prop_elim in valid_bin; destruct valid_bin as [valid_lo valid_hi].
          apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
          apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
          apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
          destruct valid_mul as [valid_mul_lo valid_mul_hi].
          split.
460
          - eapply Rle_trans. rewrite absenv_bin; apply valid_lo.
461 462 463 464 465 466 467
            unfold ivlo. unfold multIntv.
            simpl in valid_mul_lo.
            repeat rewrite <- Q2R_mult in valid_mul_lo.
            rewrite <- Q2R_min4 in valid_mul_lo.
            unfold absIvUpd; auto.
          - eapply Rle_trans.
            Focus 2.
468
            rewrite absenv_bin; apply valid_hi.
469 470 471 472 473 474
            unfold ivlo, addIntv.
            simpl in valid_mul_hi.
            repeat rewrite <- Q2R_mult in valid_mul_hi.
            rewrite <- Q2R_max4 in valid_mul_hi.
            unfold absIvUpd; auto. }
        { pose proof (interval_division_valid (a:=v1) (b:=v2) (iv1:=(Q2R (fst iv1), Q2R (snd iv1))) (iv2:=(Q2R (fst iv2),Q2R (snd iv2)))) as valid_div.
475
          rewrite <- andb_lazy_alt in valid_bin.
476 477
          unfold contained in valid_div.
          unfold isSupersetIntv in valid_bin.
478 479
          apply andb_prop_elim in valid_bin; destruct valid_bin as [nodiv0 valid_bin].
          (** CONTINUE **)
480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511
          apply andb_prop_elim in valid_bin; destruct valid_bin as [valid_lo valid_hi].
          apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
          apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
          apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
          apply orb_prop_elim in nodiv0.
          assert (snd iv2 < 0 \/ 0 < fst iv2).
          - destruct nodiv0 as [lt_0 | lt_0];
              apply andb_prop_elim in lt_0; destruct lt_0 as [le_0 neq_0];
                apply Is_true_eq_true in le_0; apply Is_true_eq_true in neq_0;
                  apply negb_true_iff in neq_0; apply Qeq_bool_neq in neq_0;
                    rewrite Qle_bool_iff in le_0;
                    rewrite Qle_lteq in le_0; destruct le_0 as [lt_0 | eq_0];
                      [ | exfalso; apply neq_0; auto | | exfalso; apply neq_0; symmetry; auto]; auto.
          - destruct valid_div as [valid_div_lo valid_div_hi]; simpl; try auto.
            + rewrite <- Q2R0_is_0.
              destruct H; [left | right]; apply Qlt_Rlt; auto.
            + unfold divideInterval, IVlo, IVhi in valid_div_lo, valid_div_hi.
              simpl in *.
              assert (Q2R (fst iv2) <= (Q2R (snd iv2)))%R by lra.
              assert (~ snd iv2 == 0).
              * destruct H; try lra.
                hnf; intros ivhi2_0.
                apply Rle_Qle in H0.
                rewrite ivhi2_0 in H0.
                lra.
              * assert (~ fst iv2 == 0).
                { destruct H; try lra.
                  hnf; intros ivlo2_0.
                  apply Rle_Qle in H0.
                  rewrite ivlo2_0 in H0.
                  lra. }
                { split.
512
                  - eapply Rle_trans. rewrite absenv_bin; apply valid_lo.
513 514 515 516 517 518 519 520
                    unfold ivlo. unfold multIntv.
                    simpl in valid_div_lo.
                    rewrite <- Q2R_inv in valid_div_lo; [ | auto].
                    rewrite <- Q2R_inv in valid_div_lo; [ | auto].
                    repeat rewrite <- Q2R_mult in valid_div_lo.
                    rewrite <- Q2R_min4 in valid_div_lo; auto.
                  - eapply Rle_trans.
                    Focus 2.
521
                    rewrite absenv_bin; apply valid_hi.
522 523 524 525 526
                    simpl in valid_div_hi.
                    rewrite <- Q2R_inv in valid_div_hi; [ | auto].
                    rewrite <- Q2R_inv in valid_div_hi; [ | auto].
                    repeat rewrite <- Q2R_mult in valid_div_hi.
                    rewrite <- Q2R_max4 in valid_div_hi; auto. } }
527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545
    + simpl in H3; rewrite Q2R0_is_0 in H3; auto.
    + simpl in H3; rewrite Q2R0_is_0 in H3; auto.
  - unfold validIntervalbounds in valid_bounds.
    simpl erasure in valid_bounds.
    simpl in *; destruct (absenv (Downcast M0 (erasure f))); destruct (absenv (erasure f)); simpl in *.
    apply Is_true_eq_left in valid_bounds.
    apply andb_prop_elim in valid_bounds.
    destruct valid_bounds as [vI1 vI2].
    apply andb_prop_elim in vI2.
    destruct vI2 as [vI2 vI2']. 
    apply Is_true_eq_true in vI2.
    apply Is_true_eq_true in vI2'.
    assert (isEqIntv i i0) as Eq by (apply supIntvAntisym; auto).
    destruct Eq as [Eqlo Eqhi].
    simpl in *.
    apply Qeq_eqR in Eqlo; rewrite Eqlo.
    apply Qeq_eqR in Eqhi; rewrite Eqhi.
    apply IHf; auto.
    apply Is_true_eq_true in vI1; apply vI1.
546
Qed.
547 548

Theorem validIntervalboundsCmd_sound (f:cmd Q) (absenv:analysisResult):
549 550
  forall E vR fVars dVars outVars elo ehi err P,
    ssaPrg f (NatSet.union fVars dVars) outVars ->
551
    bstep (toREvalCmd (toRCmd f)) E vR M0  ->
552
    (forall v, NatSet.mem v dVars = true ->
553
          exists vR,
554 555
            E v = Some (vR, M0) /\
            (Q2R (fst (fst (absenv (Var Q M0 v)))) <= vR <= Q2R (snd (fst (absenv (Var Q M0 v)))))%R) ->
556 557
    (forall v, NatSet.mem v fVars = true ->
          exists vR,
558
            E v = Some (vR, M0) /\
559
            (Q2R (fst (P v)) <= vR <= Q2R (snd (P v)))%R) ->
560 561 562
    NatSet.Subset (NatSet.diff (Commands.freeVars f) dVars) fVars ->
    validIntervalboundsCmd (erasureCmd f) absenv P dVars = true ->
    absenv (getRetExp (erasureCmd f)) = ((elo, ehi), err) ->
Heiko Becker's avatar
Heiko Becker committed
563
    (Q2R elo <=  vR <= Q2R ehi)%R.
564 565
Proof.
  induction f;
566
    intros E vR fVars dVars outVars elo ehi err P ssa_f eval_f dVars_sound
Raphaël Monat's avatar
Raphaël Monat committed
567
           fVars_valid usedVars_subset valid_bounds_f absenv_f.
568 569 570 571
  - inversion ssa_f; subst.
    inversion eval_f; subst.
    unfold validIntervalboundsCmd in valid_bounds_f.
    andb_to_prop valid_bounds_f.
572
    inversion ssa_f; subst.
573
    specialize (IHf (updEnv n M0 v E) vR fVars (NatSet.add n dVars)).
574
    eapply IHf; eauto.
575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597
    + assert (NatSet.Equal (NatSet.add n (NatSet.union fVars dVars)) (NatSet.union fVars (NatSet.add n dVars))).
      * hnf. intros a; split; intros in_set.
        { rewrite NatSet.add_spec, NatSet.union_spec in in_set.
          rewrite NatSet.union_spec, NatSet.add_spec.
          destruct in_set as [P1 | [ P2 | P3]]; auto. }
        { rewrite NatSet.add_spec, NatSet.union_spec.
          rewrite NatSet.union_spec, NatSet.add_spec in in_set.
          destruct in_set as [P1 | [ P2 | P3]]; auto. }
      * eapply ssa_equal_set; eauto.
        symmetry; eauto.
    + intros v0 mem_v0.
      unfold updEnv.
      case_eq (v0 =? n); intros v0_eq.
      * rename R1 into eq_lo;
          rename R0 into eq_hi.
        apply Qeq_bool_iff in eq_lo;
          apply Qeq_eqR in eq_lo.
        apply Qeq_bool_iff in eq_hi;
          apply Qeq_eqR in eq_hi.
        rewrite Nat.eqb_eq in v0_eq; subst.
        rewrite <- eq_lo, <- eq_hi.
        exists v; split; auto.
        eapply validIntervalbounds_sound; eauto.
Raphaël Monat's avatar
Raphaël Monat committed
598 599 600
        simpl in usedVars_subset.
        hnf. intros a in_usedVars.
        apply usedVars_subset.
601
        rewrite NatSet.diff_spec, NatSet.remove_spec, NatSet.union_spec.
Raphaël Monat's avatar
Raphaël Monat committed
602 603
        rewrite NatSet.diff_spec in in_usedVars.
        destruct in_usedVars as [ in_usedVars not_dVar].
604 605
        repeat split; try auto.
        { hnf; intros; subst.
606 607 608
          specialize (H5 n in_usedVars).
          rewrite <- NatSet.mem_spec in H5.
          rewrite H5 in H6; congruence. }
609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625
      * apply dVars_sound. rewrite NatSet.mem_spec.
        rewrite NatSet.mem_spec in mem_v0.
        rewrite NatSet.add_spec in mem_v0.
        destruct mem_v0; try auto.
        rewrite Nat.eqb_neq in v0_eq.
        exfalso; apply v0_eq; auto.
    + intros v0 mem_fVars.
      unfold updEnv.
      case_eq (v0 =? n); intros case_v0; auto.
      rewrite Nat.eqb_eq in case_v0; subst.
      assert (NatSet.mem n (NatSet.union fVars dVars) = true) as in_union.
      * rewrite NatSet.mem_spec, NatSet.union_spec; rewrite <- NatSet.mem_spec; auto.
      * rewrite in_union in *; congruence.
    + clear L R1 R0 R IHf.
      hnf. intros a a_freeVar.
      rewrite NatSet.diff_spec in a_freeVar.
      destruct a_freeVar as [a_freeVar a_no_dVar].
Raphaël Monat's avatar
Raphaël Monat committed
626
      apply usedVars_subset.
627 628 629 630 631 632 633 634 635
      simpl.
      rewrite NatSet.diff_spec, NatSet.remove_spec, NatSet.union_spec.
      repeat split; try auto.
      * hnf; intros; subst.
        apply a_no_dVar.
        rewrite NatSet.add_spec; auto.
      * hnf; intros a_dVar.
        apply a_no_dVar.
        rewrite NatSet.add_spec; auto.
636 637 638
  - unfold validIntervalboundsCmd in valid_bounds_f.
    inversion eval_f; subst.
    unfold updEnv.
639 640
    assert (Q2R (fst (fst (absenv (erasure e)))) <= vR <= Q2R (snd (fst (absenv (erasure e)))))%R.
    + simpl in valid_bounds_f; eapply validIntervalbounds_sound; eauto.
641
    + simpl in *. rewrite absenv_f in *; auto.
642
Qed.