blob: 8fcab9e123d5f3b98d9d1ca800557b275b1ed698 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
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
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
|
(* *********************************************************************)
(* *)
(* The Compcert verified compiler *)
(* *)
(* Xavier Leroy, Collège de France and INRIA Paris *)
(* *)
(* Copyright Institut National de Recherche en Informatique et en *)
(* Automatique. All rights reserved. This file is distributed *)
(* under the terms of the INRIA Non-Commercial License Agreement. *)
(* *)
(* *********************************************************************)
Require Import Coqlib.
Require Import AST Integers Floats.
Require Import Values Memory Globalenvs.
Require Import Op RTL.
Require Import NeedDomain.
(** Neededness analysis for AArch64 operators *)
Definition needs_of_shift (s: shift) (a: amount32) (nv: nval) :=
match s with
| Slsl => shlimm nv a
| Sasr => shrimm nv a
| Slsr => shruimm nv a
| Sror => ror nv a
end.
Definition zero_ext' (s: Z) (nv: nval) :=
if zle 0 s then zero_ext s nv else default nv.
Definition sign_ext' (s: Z) (nv: nval) :=
if zlt 0 s then sign_ext s nv else default nv.
Definition op1 (nv: nval) := nv :: nil.
Definition op2 (nv: nval) := nv :: nv :: nil.
Definition op1shift (s: shift) (a: amount32) (nv: nval) :=
needs_of_shift s a nv :: nil.
Definition op2shift (s: shift) (a: amount32) (nv: nval) :=
nv :: needs_of_shift s a nv :: nil.
Definition needs_of_condition (cond: condition): list nval := nil.
Definition needs_of_operation (op: operation) (nv: nval): list nval :=
match op with
| Omove => nv :: nil
| Ointconst _ => nil
| Olongconst _ => nil
| Ofloatconst _ => nil
| Osingleconst _ => nil
| Oaddrsymbol _ _ => nil
| Oaddrstack _ => nil
| Oshift s a => op1shift s a nv
| Oadd | Osub | Omul => op2 (modarith nv)
| Oaddshift s a | Osubshift s a => op2shift s a (modarith nv)
| Oaddimm _ => op1 (modarith nv)
| Oneg => op1 (modarith nv)
| Onegshift s a => op1shift s a (modarith nv)
| Omuladd | Omulsub =>
let n := modarith nv in n :: n :: n :: nil
| Odiv | Odivu => op2 (default nv)
| Oand | Oor | Oxor => op2 (bitwise nv)
| Oandshift s a | Oorshift s a | Oxorshift s a => op2shift s a (bitwise nv)
| Oandimm n => op1 (andimm nv n)
| Oorimm n => op1 (orimm nv n)
| Oxorimm n => op1 (bitwise nv)
| Onot => op1 (bitwise nv)
| Onotshift s a => needs_of_shift s a (bitwise nv) :: nil
| Obic | Oorn | Oeqv =>
let n := bitwise nv in n :: bitwise n :: nil
| Obicshift s a | Oornshift s a | Oeqvshift s a =>
let n := bitwise nv in n :: needs_of_shift s a (bitwise n) :: nil
| Oshl | Oshr | Oshru => op2 (default nv)
| Oshrximm _ => op1 (default nv)
| Ozext s => op1 (zero_ext' s nv)
| Osext s => op1 (sign_ext' s nv)
| Oshlzext s a => op1 (zero_ext' s (shlimm nv a))
| Oshlsext s a => op1 (sign_ext' s (shlimm nv a))
| Ozextshr a s => op1 (shruimm (zero_ext' s nv) a)
| Osextshr a s => op1 (shrimm (sign_ext' s nv) a)
| Oshiftl _ _ | Oextend _ _ => op1 (default nv)
| Omakelong | Olowlong | Ohighlong => op1 (default nv)
| Oaddl | Osubl | Omull => op2 (default nv)
| Oaddlshift _ _ | Oaddlext _ _ | Osublshift _ _ | Osublext _ _ => op2 (default nv)
| Oaddlimm _ => op1 (default nv)
| Onegl => op1 (default nv)
| Oneglshift _ _ => op1 (default nv)
| Omulladd | Omullsub => let n := default nv in n :: n :: n :: nil
| Omullhs | Omullhu | Odivl | Odivlu => op2 (default nv)
| Oandl | Oorl | Oxorl | Obicl | Oornl | Oeqvl => op2 (default nv)
| Oandlshift _ _ | Oorlshift _ _ | Oxorlshift _ _
| Obiclshift _ _ | Oornlshift _ _ | Oeqvlshift _ _ => op2 (default nv)
| Oandlimm _ | Oorlimm _ | Oxorlimm _ => op1 (default nv)
| Onotl => op1 (default nv)
| Onotlshift _ _ => op1 (default nv)
| Oshll | Oshrl | Oshrlu => op2 (default nv)
| Oshrlximm _ => op1 (default nv)
| Ozextl _ | Osextl _
| Oshllzext _ _ | Oshllsext _ _ | Ozextshrl _ _ | Osextshrl _ _ => op1 (default nv)
| Onegf | Oabsf => op1 (default nv)
| Oaddf | Osubf | Omulf | Odivf => op2 (default nv)
| Onegfs | Oabsfs => op1 (default nv)
| Oaddfs | Osubfs | Omulfs | Odivfs => op2 (default nv)
| Ofloatofsingle | Osingleoffloat => op1 (default nv)
| Ointoffloat | Ointuoffloat | Ofloatofint | Ofloatofintu => op1 (default nv)
| Olongoffloat | Olonguoffloat | Ofloatoflong | Ofloatoflongu => op1 (default nv)
| Ointofsingle | Ointuofsingle | Osingleofint | Osingleofintu => op1 (default nv)
| Olongofsingle | Olonguofsingle | Osingleoflong | Osingleoflongu => op1 (default nv)
| Ocmp c => needs_of_condition c
| Osel c ty => nv :: nv :: needs_of_condition c
end.
Definition operation_is_redundant (op: operation) (nv: nval): bool :=
match op with
| Ozext s => zle 0 s && zero_ext_redundant s nv
| Osext s => zlt 0 s && sign_ext_redundant s nv
| Oandimm n => andimm_redundant nv n
| Oorimm n => orimm_redundant nv n
| _ => false
end.
Ltac InvAgree :=
match goal with
| [H: vagree_list nil _ _ |- _ ] => inv H; InvAgree
| [H: vagree_list (_::_) _ _ |- _ ] => inv H; InvAgree
| _ => idtac
end.
Ltac TrivialExists :=
match goal with
| [ |- exists v, Some ?x = Some v /\ _ ] => exists x; split; auto
| _ => idtac
end.
Lemma shift_sound:
forall v w s a x,
vagree v w (needs_of_shift s a x) ->
vagree (eval_shift s v a) (eval_shift s w a) x.
Proof.
intros until x; destruct s; simpl; intros.
- apply shlimm_sound; auto.
- apply shruimm_sound; auto.
- apply shrimm_sound; auto.
- apply ror_sound; auto.
Qed.
Lemma zero_ext'_sound:
forall v w x n,
vagree v w (zero_ext' n x) ->
vagree (Val.zero_ext n v) (Val.zero_ext n w) x.
Proof.
unfold zero_ext'; intros. destruct (zle 0 n).
- apply zero_ext_sound; auto.
- assert (E: x = Nothing \/ Val.lessdef v w) by (destruct x; auto).
destruct E. subst x; simpl; auto. apply vagree_lessdef; apply Val.zero_ext_lessdef; auto.
Qed.
Lemma sign_ext'_sound:
forall v w x n,
vagree v w (sign_ext' n x) ->
vagree (Val.sign_ext n v) (Val.sign_ext n w) x.
Proof.
unfold sign_ext'; intros. destruct (zlt 0 n).
- apply sign_ext_sound; auto.
- assert (E: x = Nothing \/ Val.lessdef v w) by (destruct x; auto).
destruct E. subst x; simpl; auto. apply vagree_lessdef; apply Val.sign_ext_lessdef; auto.
Qed.
Section SOUNDNESS.
Variable ge: genv.
Variable sp: block.
Variables m m': mem.
Hypothesis PERM: forall b ofs k p, Mem.perm m b ofs k p -> Mem.perm m' b ofs k p.
Lemma needs_of_condition_sound:
forall cond args b args',
eval_condition cond args m = Some b ->
vagree_list args args' (needs_of_condition cond) ->
eval_condition cond args' m' = Some b.
Proof.
intros. unfold needs_of_condition in H0.
eapply default_needs_of_condition_sound; eauto.
Qed.
Lemma needs_of_operation_sound:
forall op args v nv args',
eval_operation ge (Vptr sp Ptrofs.zero) op args m = Some v ->
vagree_list args args' (needs_of_operation op nv) ->
nv <> Nothing ->
exists v',
eval_operation ge (Vptr sp Ptrofs.zero) op args' m' = Some v'
/\ vagree v v' nv.
Proof.
unfold needs_of_operation; intros; destruct op; try (eapply default_needs_of_operation_sound; eauto; fail);
simpl in *; FuncInv; InvAgree; TrivialExists.
- apply shift_sound; auto.
- apply add_sound; auto.
- apply add_sound; auto using shift_sound.
- apply add_sound; auto with na.
- apply neg_sound; auto.
- apply neg_sound; auto using shift_sound.
- apply sub_sound; auto.
- apply sub_sound; auto using shift_sound.
- apply mul_sound; auto.
- apply add_sound; auto. apply mul_sound; rewrite modarith_idem; auto.
- apply sub_sound; auto. apply mul_sound; rewrite modarith_idem; auto.
- apply and_sound; auto.
- apply and_sound; auto using shift_sound.
- apply andimm_sound; auto.
- apply or_sound; auto.
- apply or_sound; auto using shift_sound.
- apply orimm_sound; auto.
- apply xor_sound; auto.
- apply xor_sound; auto using shift_sound.
- apply xor_sound; auto with na.
- apply notint_sound; auto.
- apply notint_sound; auto using shift_sound.
- apply and_sound; auto. apply notint_sound; rewrite bitwise_idem; auto.
- apply and_sound; auto. apply notint_sound; rewrite bitwise_idem; auto using shift_sound.
- apply or_sound; auto. apply notint_sound; rewrite bitwise_idem; auto.
- apply or_sound; auto. apply notint_sound; rewrite bitwise_idem; auto using shift_sound.
- apply xor_sound; auto. apply notint_sound; rewrite bitwise_idem; auto.
- apply xor_sound; auto. apply notint_sound; rewrite bitwise_idem; auto using shift_sound.
- apply zero_ext'_sound; auto.
- apply sign_ext'_sound; auto.
- apply shlimm_sound; apply zero_ext'_sound; auto.
- apply shlimm_sound; apply sign_ext'_sound; auto.
- apply zero_ext'_sound; apply shruimm_sound; auto.
- apply sign_ext'_sound; apply shrimm_sound; auto.
- destruct (eval_condition cond args m) as [b|] eqn:EC.
erewrite needs_of_condition_sound by eauto.
apply select_sound; auto.
simpl; auto with na.
Qed.
Lemma operation_is_redundant_sound:
forall op nv arg1 args v arg1' args',
operation_is_redundant op nv = true ->
eval_operation ge (Vptr sp Ptrofs.zero) op (arg1 :: args) m = Some v ->
vagree_list (arg1 :: args) (arg1' :: args') (needs_of_operation op nv) ->
vagree v arg1' nv.
Proof.
intros. destruct op; simpl in *; try discriminate; inv H1; FuncInv; subst.
- apply andimm_redundant_sound; auto.
- apply orimm_redundant_sound; auto.
- InvBooleans. unfold zero_ext' in H5; rewrite zle_true in H5 by auto.
apply zero_ext_redundant_sound; auto.
- InvBooleans. unfold sign_ext' in H5; rewrite zlt_true in H5 by auto.
apply sign_ext_redundant_sound; auto.
Qed.
End SOUNDNESS.
|
