Expressions.v 9.56 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 6
Require Import Daisy.Infra.RealRationalProps.
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) :=
15 16 17 18 19 20 21
  match b1 with
    Plus => match b2 with Plus => true |_ => false end
  | Sub => match b2 with Sub => true |_ => false end
  | Mult => match b2 with Mult => true |_ => false end
  | Div => match b2 with Div => true |_ => false end
  end.

22 23 24 25
(**
  Next define an evaluation function for binary operators on reals.
  Errors are added on the expression evaluation level later.
 **)
26
Definition evalBinop (o:binop) (v1:R) (v2:R) :=
27 28 29 30 31 32
  match o with
  | Plus => Rplus v1 v2
  | Sub => Rminus v1 v2
  | Mult => Rmult v1 v2
  | Div => Rdiv v1 v2
  end.
33 34 35 36 37 38 39

(**
   Expressions will use unary operators.
   Define them first
 **)
Inductive unop: Type := Neg | Inv.

40
Definition unopEqBool (o1:unop) (o2:unop) :=
41 42 43 44 45 46 47
  match o1 with
  |Neg => match o2 with |Neg => true |_=> false end
  |Inv => match o2 with |Inv => true |_ => false end
  end.

(**
   Define evaluation for unary operators on reals.
48
   Errors are added in the expression evaluation level later.
49
 **)
50
Definition evalUnop (o:unop) (v:R):=
51 52 53 54 55
  match o with
  |Neg => (- v)%R
  |Inv => (/ v)%R
  end .

56 57


58
(**
59 60
  Define expressions parametric over some value type V.
  Will ease reasoning about different instantiations later.
61
**)
62
Inductive exp (V:Type): Type :=
63
  Var: mType -> nat -> exp V
64
| Const: V -> exp V
65
| Unop: unop -> exp V -> exp V
66 67
| Binop: binop -> exp V -> exp V -> exp V
| Downcast: mType -> exp V -> exp V.
68

69 70 71 72
(**
  Boolean equality function on expressions.
  Used in certificates to define the analysis result as function
**)
73
Fixpoint expEqBool (e1:exp Q) (e2:exp Q) :=
74
  match e1 with
75
  |Var _ m1 v1 =>
76
   match e2 with
77
   |Var _ m2 v2 => andb (mTypeEqBool m1 m2) (v1 =? v2)
78 79 80 81
   | _=> false
   end
  |Const n1 =>
   match e2 with
82
   |Const n2 => (Qeq_bool n1 n2)
83 84
   | _=> false
   end
85 86
  |Unop o1 e11 =>
   match e2 with
87
   |Unop o2 e22 => andb (unopEqBool o1 o2) (expEqBool e11 e22)
88 89 90
   |_ => false
   end
  |Binop o1 e11 e12 =>
91
   match e2 with
92
   |Binop o2 e21 e22 => andb (binopEqBool o1 o2) (andb (expEqBool e11 e21) (expEqBool e12 e22))
93 94
   |_ => false
   end
95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117
  |Downcast m1 f1 =>
   match e2 with
   |Downcast m2 f2 => andb (mTypeEqBool m1 m2) (expEqBool f1 f2)
   |_ => false                   
   end
  end.


Fixpoint toRExp (e:exp Q) :=
  match e with
  |Var _ m v => Var R m v
  |Const n => Const (Q2R n)
  |Unop o e1 => Unop o (toRExp e1)
  |Binop o e1 e2 => Binop o (toRExp e1) (toRExp e2)
  |Downcast m e1 => Downcast m (toRExp e1)
  end.

Fixpoint toREval (e:exp R) :=
  match e with
  | Var _ _ v => Var R M0 v
  | Const n => Const n
  | Unop o e1 => Unop o (toREval e1)
  | Binop o e1 e2 => Binop o (toREval e1) (toREval e2)
118
  | Downcast _ e1 =>  (toREval e1)
119
  end.
120

121 122 123 124 125 126 127 128 129
Definition toREvalEnv (E:env) : env :=
  fun (n:nat) =>
    let s := (E n) in
    match s with
    | None => None
    | Some (r, _) => Some (r, M0)
    end.


130 131 132 133
(**
  Define a perturbation function to ease writing of basic definitions
**)
Definition perturb (r:R) (e:R) :=
134
  (r * (1 + e))%R.
Heiko Becker's avatar
Heiko Becker committed
135

136
(**
137
Define expression evaluation relation parametric by an "error" epsilon.
138 139 140
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.
141
**)
142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169
Inductive eval_exp (E:env) :(exp R) -> R -> mType -> Prop :=
| Var_load m m1 x v:
    isMorePrecise m m1 = true ->
    (**mTypeEqBool m m1 = true ->*)
    E x = Some (v, m1) ->
    eval_exp E (Var R m1 x) v m
| Const_dist m n delta:
    Rle (Rabs delta) (Q2R (meps m)) ->
    eval_exp E (Const n) (perturb n delta) m
| Unop_neg m f1 v1:
    eval_exp E f1 v1 m ->
    eval_exp E (Unop Neg f1) (evalUnop Neg v1) m
| Unop_inv m f1 v1 delta:
    Rle (Rabs delta) (Q2R (meps m)) ->
    eval_exp E f1 v1 m ->
    eval_exp E (Unop Inv f1) (perturb (evalUnop Inv v1) delta) m
| Binop_dist m m1 m2 op f1 f2 v1 v2 delta:
    isJoinOf m m1 m2 = true ->
    Rle (Rabs delta) (Q2R (meps m)) ->
    eval_exp E f1 v1 m1 ->
    eval_exp E f2 v2 m2 ->
    eval_exp E (Binop op f1 f2) (perturb (evalBinop op v1 v2) delta) m
| Downcast_dist m m1 f1 v1 delta:
    (*    Qle_bool (meps m1) (meps m) = true ->*)
    isMorePrecise m1 m = true ->
    Rle (Rabs delta) (Q2R (meps m)) ->
    eval_exp E f1 v1 m1 ->
    eval_exp E (Downcast m f1) (perturb v1 delta) m.
170

171 172 173 174 175
(**
  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 :=
176
  match e with
177
  | Var _ _ x => NatSet.singleton x
178 179
  | Unop u e1 => usedVars e1
  | Binop b e1 e2 => NatSet.union (usedVars e1) (usedVars e2)
180
  | Downcast _ e1 => usedVars e1
181 182
  | _ => NatSet.empty
  end.
183

184
(**
185
  If |delta| <= 0 then perturb v delta is exactly v.
186
**)
187
Lemma delta_0_deterministic (v:R) (delta:R):
Heiko Becker's avatar
Heiko Becker committed
188 189 190 191 192
  (Rabs delta <= 0)%R ->
  perturb v delta = v.
Proof.
  intros abs_0; apply Rabs_0_impl_eq in abs_0; subst.
  unfold perturb.
193
  lra.
Heiko Becker's avatar
Heiko Becker committed
194 195
Qed.

196 197
    
Lemma general_meps_0_deterministic (f:exp R) (E:env):
198 199
  forall v1 v2 m1,
    m1 = M0 ->
200
    eval_exp E (toREval f) v1 m1 ->
201
    eval_exp E (toREval f) v2 M0 ->
202 203
    v1 = v2.
Proof.
204
  induction f; intros v1 v2 m1 m10_eq eval_v1 eval_v2.
205 206 207 208 209 210 211
  - 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.
    rewrite H4 in H10; inversion H10; subst; auto.
  - 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.
212
    + apply Ropp_eq_compat. apply (IHf v0 v3 M0); auto.     
213 214
    + inversion H4.
    + inversion H5.
215
    + rewrite (IHf v0 v3 M0); auto.
216 217
  - 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.
218 219 220 221 222 223 224 225
    assert (M0 = M0) as M00 by auto.
    pose proof (ifM0isJoin_l M0 m0 m2 M00 H2); auto.
    pose proof (ifM0isJoin_r M0 m0 m2 M00 H2); auto.
    pose proof (ifM0isJoin_l M0 m4 m5 M00 H11); auto.
    pose proof (ifM0isJoin_r M0 m4 m5 M00 H11); auto.
    subst.
    rewrite (IHf1 v0 v4 M0); auto.
    rewrite (IHf2 v5 v3 M0); auto.
226 227
  - simpl toREval in eval_v1.
    simpl toREval in eval_v2.
228
    apply (IHf v1 v2 m1); auto.
229 230 231 232
Qed.


  
233
(**
234
Evaluation with 0 as machine epsilon is deterministic
235
**)
236
Lemma meps_0_deterministic (f:exp R) (E:env):
237
  forall v1 v2,
238 239
  eval_exp E (toREval f) v1 M0 ->
  eval_exp E (toREval f) v2 M0 ->
240 241
  v1 = v2.
Proof.
242
  intros v1 v2 ev1 ev2.
243 244
  assert (M0 = M0) by auto.
  apply (general_meps_0_deterministic f H ev1 ev2). 
245 246
Qed.

247

248 249 250 251
(**
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
252
variables in the Environment.
253
**)
254 255 256 257 258 259 260
Lemma binary_unfolding b f1 f2 m E vF:
  eval_exp E (Binop b f1 f2) vF m ->
  exists vF1 vF2 m1 m2,
  eval_exp E f1 vF1 m1 /\
  eval_exp E f2 vF2 m2 /\
  eval_exp  (updEnv 2 m2 vF2 (updEnv 1 m1 vF1 emptyEnv))
           (Binop b (Var R m1 1) (Var R m2 2)) vF m.
261
Proof.
262 263
  intros eval_float.
  inversion eval_float; subst.
264 265 266 267 268 269 270 271
  exists v1 ; exists v2; exists m1; exists m2; repeat split; try auto.
  eapply Binop_dist; eauto.
  pose proof (isMorePrecise_refl m1).
  eapply Var_load; eauto.
  pose proof (isMorePrecise_refl m2).
  (* unfold mTypeEqBool; apply Qeq_bool_iff; apply Qeq_refl. *)
  eapply Var_load; eauto.
  (* unfold mTypeEqBool; apply Qeq_bool_iff; apply Qeq_refl. *)
272 273
Qed.

274 275 276 277 278 279 280 281
(* (** *)
(* Analogous lemma for unary expressions. *)
(* **) *)
Lemma unary_unfolding (e:exp R) (m:mType) (E:env) (v:R):
  (eval_exp E (Unop Inv e) v m ->
   exists v1 m1,
     eval_exp E e v1 m1 /\
     eval_exp (updEnv 1 m1 v1 E) (Unop Inv (Var R m1 1)) v m).
282
Proof.
283
  intros eval_un.
284
    inversion eval_un; subst.
285
    exists v1; exists m.
286
    repeat split; try auto.
287 288 289 290 291 292 293 294 295 296 297 298 299 300 301
    econstructor; try auto.
    pose proof (isMorePrecise_refl m).
    econstructor; eauto.
  (* - intros exists_val. *)
  (*   destruct exists_val as [v1 [m1 [eval_f1 eval_e_E]]]. *)
  (*   inversion eval_e_E; subst. *)
  (*   inversion H1; subst. *)
  (*   econstructor; eauto. *)
  (*   unfold updEnv in H6. *)
  (*   simpl in H6. *)
  (*   inversion H6. *)
  (*   rewrite <- H2. *)
    
  (*   rewrite <- H1. *)
  (*   auto. *)
302
Qed.
303

304 305 306 307 308 309
(**
  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.
310

311
(**
312
  Define evaluation of boolean expressions
313
 **)
314 315 316 317 318 319 320 321 322 323 324 325 326 327
(* 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. *)