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
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
|
(*|
..
Vericert: Verified high-level synthesis.
Copyright (C) 2020-2022 Yann Herklotz <yann@yannherklotz.com>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
================
RTLBlockgenproof
================
.. coq:: none
|*)
Require compcert.backend.RTL.
Require Import compcert.common.AST.
Require Import compcert.common.Errors.
Require Import compcert.common.Globalenvs.
Require Import compcert.lib.Maps.
Require Import compcert.backend.Registers.
Require Import vericert.common.Vericertlib.
Require Import vericert.hls.RTLBlockInstr.
Require Import vericert.hls.RTLBlock.
Require Import vericert.hls.RTLBlockgen.
#[local] Open Scope positive.
(*|
Defining a find block specification
===================================
Basically, it should be able to find the location of the block without using the
``find_block`` function, so that this is more useful for the proofs. There are
various different types of options that could come up though:
1. The instruction is a standard instruction present inside of a basic block.
2. The instruction is a standard instruction which ends with a ``goto``.
3. The instruction is a control-flow instruction.
For case number 1, there should exist a value in the list of instructions, such
that the instructions match exactly, and the indices match as well. In the
original code, this instruction must have been going from the current node to
the node - 1.
For case number 2, there should be an instruction at the right index again,
however, this time there will also be a ``goto`` instruction in the control-flow
part of the basic block.
For case number 3, there should be a ``nop`` instruction in the basic block, and
then the equivalent control-flow instruction ending the basic block.
In the end though, it seems like two cases are actually enough, as the last two
cases are similar enough that they can be merged into one.
|*)
Definition all_max {A} (c: PTree.t A) p: Prop :=
Forall (fun x => x <= p) (map fst (PTree.elements c)).
Definition find_block_spec (c: code) (n1 n2: node): Prop :=
forall x, all_max c x -> find_block x (map fst (PTree.elements c)) n1 = n2.
Definition offset (pc pc': positive): nat := Pos.to_nat pc' - Pos.to_nat pc.
Definition find_instr_spec (c: code) (n: node) (i: RTL.instruction)
(n': node) (i': instr) :=
find_block_spec c n n' /\
exists il, c ! n' = Some il
/\ nth_error il.(bb_body) (offset n n') = Some i'.
(*|
.. index::
pair: semantics; transl_code_spec
Translation Specification
-------------------------
This specification should describe the translation that the unverified
transformation performs. This should be proven from the validation of the
transformation.
This also specifies ``x'`` relative to x given the code.
|*)
Variant transl_code_spec (c: RTL.code) (tc: code) (x x': node): Prop :=
| transl_code_standard_bb :
forall i i',
c ! x = Some i ->
find_instr_spec tc x i x' i' ->
Is_true (check_instr x i i') ->
transl_code_spec c tc x x'
| transl_code_standard_cf :
forall i i' il,
c ! x = Some i ->
tc ! x' = Some il ->
find_instr_spec tc x i x' i' ->
Is_true (check_cf_instr_body i i') ->
Is_true (check_cf_instr i il.(bb_exit)) ->
transl_code_spec c tc x x'
.
(*|
Matches the basic block that should be present in the state. This simulates the
small step execution of the basic block from the big step semantics that are
currently present.
Why does it not need to find the pc' value using ``find_block``?
It doesn't have to find the value because it's given as an input, and the
finding is actually done at that higher level already.
|*)
Variant match_bblock (tc: code) (pc pc': node): list instr -> Prop :=
| match_bblock_intro :
forall bb cf,
tc ! pc' = Some (mk_bblock bb cf) ->
match_bblock tc pc pc' (list_drop (offset pc pc') bb).
Variant match_stackframe : RTL.stackframe -> stackframe -> Prop :=
| match_stackframe_init :
forall res f tf sp pc pc' rs
(TF: transl_function f = OK tf)
(PC: transl_code_spec f.(RTL.fn_code) tf.(fn_code) pc pc'),
match_stackframe (RTL.Stackframe res f sp pc rs)
(Stackframe res tf sp pc' rs (PMap.init false)).
(*|
The ``match_states`` predicate defines how to find the correct ``bb`` that
should be executed, as well as the value of ``pc``.
|*)
Variant match_states : RTL.state -> RTLBlock.state -> Prop :=
| match_state :
forall stk stk' f tf sp pc rs m pc' bb
(TF: transl_function f = OK tf)
(PC: transl_code_spec f.(RTL.fn_code) tf.(fn_code) pc pc')
(STK: Forall2 match_stackframe stk stk')
(BB: match_bblock tf.(fn_code) pc pc' bb),
match_states (RTL.State stk f sp pc rs m)
(State stk' tf sp pc' bb rs (PMap.init false) m)
| match_callstate :
forall cs cs' f tf args m
(TF: transl_fundef f = OK tf)
(STK: Forall2 match_stackframe cs cs'),
match_states (RTL.Callstate cs f args m) (Callstate cs' tf args m)
| match_returnstate :
forall cs cs' v m
(STK: Forall2 match_stackframe cs cs'),
match_states (RTL.Returnstate cs v m) (Returnstate cs' v m).
Definition match_prog (p: RTL.program) (tp: RTLBlock.program) :=
Linking.match_program (fun cu f tf => transl_fundef f = Errors.OK tf) eq p tp.
Section CORRECTNESS.
Context (prog : RTL.program).
Context (tprog : RTLBlock.program).
Context (TRANSL : match_prog prog tprog).
Let ge : RTL.genv := Globalenvs.Genv.globalenv prog.
Let tge : genv := Globalenvs.Genv.globalenv tprog.
Lemma symbols_preserved:
forall (s: AST.ident), Genv.find_symbol tge s = Genv.find_symbol ge s.
Proof using TRANSL. intros. eapply (Genv.find_symbol_match TRANSL). Qed.
Lemma senv_preserved:
Senv.equiv (Genv.to_senv ge) (Genv.to_senv tge).
Proof using TRANSL. intros; eapply (Genv.senv_transf_partial TRANSL). Qed.
#[local] Hint Resolve senv_preserved : rtlgp.
Lemma function_ptr_translated:
forall b f,
Genv.find_funct_ptr ge b = Some f ->
exists tf, Genv.find_funct_ptr tge b = Some tf /\ transl_fundef f = OK tf.
Proof
(Genv.find_funct_ptr_transf_partial TRANSL).
Lemma sig_transl_function:
forall (f: RTL.fundef) (tf: RTLBlock.fundef),
transl_fundef f = OK tf ->
RTLBlock.funsig tf = RTL.funsig f.
Proof using.
unfold transl_fundef. unfold transf_partial_fundef.
intros. destruct_match. unfold bind in *. destruct_match; try discriminate.
inv H. unfold transl_function in Heqr.
repeat (destruct_match; try discriminate). inv Heqr. auto.
inv H; auto.
Qed.
Lemma transl_initial_states :
forall s1, RTL.initial_state prog s1 ->
exists s2, RTLBlock.initial_state tprog s2 /\ match_states s1 s2.
Proof using TRANSL.
induction 1.
exploit function_ptr_translated; eauto. intros [tf [A B]].
econstructor. simplify. econstructor.
apply (Genv.init_mem_transf_partial TRANSL); eauto.
replace (prog_main tprog) with (prog_main prog). rewrite symbols_preserved; eauto.
symmetry; eapply Linking.match_program_main; eauto. eauto.
erewrite sig_transl_function; eauto. constructor. auto. auto.
Qed.
Lemma transl_final_states :
forall s1 s2 r,
match_states s1 s2 ->
RTL.final_state s1 r ->
RTLBlock.final_state s2 r.
Proof using.
inversion 2; crush. subst. inv H. inv STK. econstructor.
Qed.
Lemma transl_Inop_correct:
forall s f sp pc rs m pc',
(RTL.fn_code f) ! pc = Some (RTL.Inop pc') ->
forall s2, match_states (RTL.State s f sp pc rs m) s2 ->
exists s2', Smallstep.plus step tge s2 Events.E0 s2'
/\ match_states (RTL.State s f sp pc' rs m) s2'.
Proof.
intros s f sp pc rs m pc' H.
Admitted.
Lemma transl_Iop_correct:
forall s f sp pc rs m op args res pc' v,
(RTL.fn_code f) ! pc = Some (RTL.Iop op args res pc') ->
Op.eval_operation ge sp op rs##args m = Some v ->
forall s2, match_states (RTL.State s f sp pc rs m) s2 ->
exists s2', Smallstep.plus step tge s2 Events.E0 s2'
/\ match_states (RTL.State s f sp pc' (Registers.Regmap.set res v rs) m) s2'.
Proof.
intros s f sp pc rs m op args res pc' v H H0.
Admitted.
Lemma transl_Iload_correct:
forall s f sp pc rs m chunk addr args dst pc' a v,
(RTL.fn_code f) ! pc = Some (RTL.Iload chunk addr args dst pc') ->
Op.eval_addressing ge sp addr rs##args = Some a ->
Memory.Mem.loadv chunk m a = Some v ->
forall s2, match_states (RTL.State s f sp pc rs m) s2 ->
exists s2', Smallstep.plus step tge s2 Events.E0 s2'
/\ match_states (RTL.State s f sp pc' (Registers.Regmap.set dst v rs) m) s2'.
Proof.
intros s f sp pc rs m chunk addr args dst pc' a v H H0 H1.
Admitted.
Lemma transl_correct:
forall s1 t s1',
RTL.step ge s1 t s1' ->
forall s2, match_states s1 s2 ->
exists s2', Smallstep.plus step tge s2 t s2' /\ match_states s1' s2'.
Proof.
induction 1.
- eauto using transl_Inop_correct.
- eauto using transl_Iop_correct.
- eauto using transl_Iload_correct.
Admitted.
Theorem transf_program_correct:
Smallstep.forward_simulation (RTL.semantics prog)
(RTLBlock.semantics tprog).
Proof using TRANSL.
eapply Smallstep.forward_simulation_plus.
apply senv_preserved.
eauto using transl_initial_states.
eapply transl_final_states.
eauto using transl_correct.
Qed.
End CORRECTNESS.
|
