Expressions.v 6.56 KB
Newer Older
1

2
(**
3
  Formalization of the base expression language for the daisy framework
4
 **)
5
Require Import Coq.Reals.Reals Coq.micromega.Psatz Coq.QArith.QArith Coq.QArith.Qreals.
6
Require Export Daisy.Infra.Abbrevs Daisy.Infra.RealSimps Daisy.Infra.NatSet.
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
  Define expressions parametric over some value type V.
  Will ease reasoning about different instantiations later.
59
**)
60 61 62
Inductive exp (V:Type): Type :=
  Var: nat -> exp V
| Const: V -> exp V
63
| Unop: unop -> exp V -> exp V
64
| Binop: binop -> exp V -> exp V -> exp V.
65

66 67 68 69
(**
  Boolean equality function on expressions.
  Used in certificates to define the analysis result as function
**)
70
Fixpoint expEqBool (e1:exp Q) (e2:exp Q) :=
71 72 73 74 75 76 77 78 79 80 81
  match e1 with
  |Var _ v1 =>
   match e2 with
   |Var _ v2 => v1 =? v2
   | _=> false
   end
  |Const n1 =>
   match e2 with
   |Const n2 => Qeq_bool n1 n2
   | _=> false
   end
82 83
  |Unop o1 e11 =>
   match e2 with
84
   |Unop o2 e22 => andb (unopEqBool o1 o2) (expEqBool e11 e22)
85 86 87
   |_ => false
   end
  |Binop o1 e11 e12 =>
88
   match e2 with
89
   |Binop o2 e21 e22 => andb (binopEqBool o1 o2) (andb (expEqBool e11 e21) (expEqBool e12 e22))
90 91 92
   |_ => false
   end
  end.
93

94 95 96 97
(**
  Define a perturbation function to ease writing of basic definitions
**)
Definition perturb (r:R) (e:R) :=
98
  (r * (1 + e))%R.
Heiko Becker's avatar
Heiko Becker committed
99

100
(**
101
Define expression evaluation relation parametric by an "error" epsilon.
102 103 104
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.
105
**)
106 107
Inductive eval_exp (eps:R) (E:env) :(exp R) -> R -> Prop :=
| Var_load x v:
108
    E x = Some v ->
109
    eval_exp eps E (Var R x) v
110 111
| Const_dist n delta:
    Rle (Rabs delta) eps ->
112
    eval_exp eps E (Const n) (perturb n delta)
113
| Unop_neg f1 v1:
114 115
    eval_exp eps E f1 v1 ->
    eval_exp eps E (Unop Neg f1) (evalUnop Neg v1)
116
| Unop_inv f1 v1 delta:
117
    Rle (Rabs delta) eps ->
118 119
    eval_exp eps E f1 v1 ->
    eval_exp eps E (Unop Inv f1) (perturb (evalUnop Inv v1) delta)
120 121
| Binop_dist op f1 f2 v1 v2 delta:
    Rle (Rabs delta) eps ->
122 123
    eval_exp eps E f1 v1 ->
    eval_exp eps E f2 v2 ->
124
    ((op = Div) -> (~ v2 = 0)%R) ->
125
    eval_exp eps E (Binop op f1 f2) (perturb (evalBinop op v1 v2) delta).
126

127 128 129 130 131
(**
  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 :=
132 133
  match e with
  | Var _ x => NatSet.singleton x
134 135
  | Unop u e1 => usedVars e1
  | Binop b e1 e2 => NatSet.union (usedVars e1) (usedVars e2)
136 137
  | _ => NatSet.empty
  end.
138

139
(**
140
  If |delta| <= 0 then perturb v delta is exactly v.
141
**)
142
Lemma delta_0_deterministic (v:R) (delta:R):
Heiko Becker's avatar
Heiko Becker committed
143 144 145 146 147
  (Rabs delta <= 0)%R ->
  perturb v delta = v.
Proof.
  intros abs_0; apply Rabs_0_impl_eq in abs_0; subst.
  unfold perturb.
148
  lra.
Heiko Becker's avatar
Heiko Becker committed
149 150
Qed.

151
(**
152
Evaluation with 0 as machine epsilon is deterministic
153
**)
154
Lemma meps_0_deterministic (f:exp R) (E:env):
155
  forall v1 v2,
156 157
  eval_exp 0 E f v1 ->
  eval_exp 0 E f v2 ->
158 159
  v1 = v2.
Proof.
160 161
  induction f; intros v1 v2 eval_v1 eval_v2;
    inversion eval_v1; inversion eval_v2;
162
      repeat try rewrite delta_0_deterministic; subst; auto.
163
  - rewrite H3 in H0; inversion H0;
164
      subst; auto.
165
  - rewrite (IHf v0 v3); auto.
166 167
  - inversion H3.
  - inversion H4.
168 169 170
  - rewrite (IHf v0 v3); auto.
  - rewrite (IHf1 v0 v4); auto.
    rewrite (IHf2 v3 v5); auto.
171 172
Qed.

173 174 175 176
(**
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
177
variables in the Environment.
178
**)
179 180 181 182 183
Lemma binary_unfolding b f1 f2 eps E vF:
  eval_exp eps E (Binop b f1 f2) vF ->
  exists vF1 vF2,
  eval_exp eps E f1 vF1 /\
  eval_exp eps E f2 vF2 /\
184
  eval_exp eps (updEnv 2 vF2 (updEnv 1 vF1 emptyEnv))
185
           (Binop b (Var R 1) (Var R 2)) vF.
186
Proof.
187 188 189 190 191 192 193 194
  intros eval_float.
  inversion eval_float; subst.
  exists v1 ; exists v2; repeat split; try auto.
  constructor; try auto.
  - constructor.
    unfold updEnv; cbv; auto.
  - constructor.
    unfold updEnv; cbv; auto.
195 196
Qed.

197 198 199
(**
Analogous lemma for unary expressions.
**)
200 201
Lemma unary_unfolding (e:exp R) (eps:R) (E:env) (v:R):
  (eval_exp eps E (Unop Inv e) v <->
202
   exists v1,
203 204
     eval_exp eps E e v1 /\
     eval_exp eps (updEnv 1 v1 E) (Unop Inv (Var R 1)) v).
205 206 207 208 209 210 211 212 213
Proof.
  split.
  - intros eval_un.
    inversion eval_un; subst.
    exists v1.
    repeat split; try auto.
    constructor; try auto.
    constructor; auto.
  - intros exists_val.
214 215
    destruct exists_val as [v1 [eval_f1 eval_e_E]].
    inversion eval_e_E; subst.
216 217 218
    inversion H1; subst.
    unfold updEnv in *; simpl in *.
    constructor; auto.
219
    inversion H2; subst; auto.
220
Qed.
221

222 223 224 225 226 227
(**
  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.
228

229
(**
230
  Define evaluation of boolean expressions
231
 **)
232
Inductive bval (eps:R) (E:env): (bexp R) -> Prop -> Prop :=
233
  leq_eval (f1:exp R) (f2:exp R) (v1:R) (v2:R):
234 235 236
    eval_exp eps E f1 v1 ->
    eval_exp eps E f2 v2 ->
    bval eps E (leq f1 f2) (Rle v1 v2)
237
|less_eval (f1:exp R) (f2:exp R) (v1:R) (v2:R):
238 239 240
    eval_exp eps E f1 v1 ->
    eval_exp eps E f2 v2 ->
    bval eps E (less f1 f2) (Rlt v1 v2).
241 242 243
(**
 Simplify arithmetic later by making > >= only abbreviations
**)
244 245
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.