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
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
|
(*** Interface for bounded arithmetic *)
Require Import Coq.ZArith.ZArith.
Require Import Crypto.Util.ZUtil.
Require Import Crypto.Util.Notations.
Local Open Scope Z_scope.
Local Open Scope type_scope.
Create HintDb push_decode discriminated.
Create HintDb pull_decode discriminated.
Hint Extern 1 => progress autorewrite with push_decode in * : push_decode.
Hint Extern 1 => progress autorewrite with pull_decode in * : pull_decode.
(* TODO(from jgross): Try dropping the record wrappers. See
https://github.com/mit-plv/fiat-crypto/pull/52#discussion_r74627992
and
https://github.com/mit-plv/fiat-crypto/pull/52#discussion_r74658417
and
https://github.com/mit-plv/fiat-crypto/pull/52#issuecomment-239536847.
The wrappers are here to make [autorewrite] databases feasable and
fast, based on design patterns learned from past experience. There
might be better ways. *)
Class decoder (n : Z) W :=
{ decode : W -> Z }.
Coercion decode : decoder >-> Funclass.
Global Arguments decode {n W _} _.
Class is_decode {n W} (decode : decoder n W) :=
decode_range : forall x, 0 <= decode x < 2^n.
Section InstructionGallery.
Context (n : Z) (* bit-width of width of [W] *)
{W : Type} (* bounded type, [W] for word *)
(Wdecoder : decoder n W).
Local Notation imm := Z (only parsing). (* immediate (compile-time) argument *)
Record load_immediate := { ldi :> imm -> W }.
Class is_load_immediate {ldi : load_immediate} :=
decode_load_immediate : forall x, 0 <= x < 2^n -> decode (ldi x) = x.
Record shift_right_doubleword_immediate := { shrd :> W -> W -> imm -> W }.
Class is_shift_right_doubleword_immediate (shrd : shift_right_doubleword_immediate) :=
decode_shift_right_doubleword :
forall high low count,
0 <= count < n
-> decode (shrd high low count) = (((decode high << n) + decode low) >> count) mod 2^n.
Record shift_left_immediate := { shl :> W -> imm -> W }.
Class is_shift_left_immediate (shl : shift_left_immediate) :=
decode_shift_left_immediate :
forall r count, 0 <= count < n -> decode (shl r count) = (decode r << count) mod 2^n.
Record shift_right_immediate := { shr :> W -> imm -> W }.
Class is_shift_right_immediate (shr : shift_right_immediate) :=
decode_shift_right_immediate :
forall r count, 0 <= count < n -> decode (shr r count) = (decode r >> count).
Record spread_left_immediate := { sprl :> W -> imm -> W * W (* [(low, high)] *) }.
Class is_spread_left_immediate (sprl : spread_left_immediate) :=
{
decode_fst_spread_left_immediate : forall r count,
0 <= count < n
-> decode (fst (sprl r count)) = (decode r << count) mod 2^n;
decode_snd_spread_left_immediate : forall r count,
0 <= count < n
-> decode (snd (sprl r count)) = (decode r << count) >> n;
}.
Definition Build_is_spread_left_immediate' (sprl : spread_left_immediate)
(pf : forall r count, 0 <= count < n
-> decode (fst (sprl r count)) = (decode r << count) mod 2^n
/\ decode (snd (sprl r count)) = (decode r << count) >> n)
:= {| decode_fst_spread_left_immediate r count H := proj1 (pf r count H);
decode_snd_spread_left_immediate r count H := proj2 (pf r count H) |}.
Record mask_keep_low := { mkl :> W -> imm -> W }.
Class is_mask_keep_low (mkl : mask_keep_low) :=
decode_mask_keep_low : forall r count,
0 <= count < n -> decode (mkl r count) = decode r mod 2^count.
Local Notation bit b := (if b then 1 else 0).
Record add_with_carry := { adc :> W -> W -> bool -> bool * W }.
Class is_add_with_carry (adc : add_with_carry) :=
{
bit_fst_add_with_carry : forall x y c, bit (fst (adc x y c)) = (decode x + decode y + bit c) >> n;
decode_snd_add_with_carry : forall x y c, decode (snd (adc x y c)) = (decode x + decode y + bit c) mod (2^n)
}.
Definition Build_is_add_with_carry' (adc : add_with_carry)
(pf : forall x y c, bit (fst (adc x y c)) = (decode x + decode y + bit c) >> n /\ decode (snd (adc x y c)) = (decode x + decode y + bit c) mod (2^n))
:= {| bit_fst_add_with_carry x y c := proj1 (pf x y c);
decode_snd_add_with_carry x y c := proj2 (pf x y c) |}.
Record sub_with_carry := { subc :> W -> W -> bool -> bool * W }.
Class is_sub_with_carry (subc:W->W->bool->bool*W) :=
{
fst_sub_with_carry : forall x y c, fst (subc x y c) = ((decode x - decode y - bit c) <? 0);
decode_snd_sub_with_carry : forall x y c, decode (snd (subc x y c)) = (decode x - decode y - bit c) mod 2^n
}.
Definition Build_is_sub_with_carry' (subc : sub_with_carry)
(pf : forall x y c, fst (subc x y c) = ((decode x - decode y - bit c) <? 0) /\ decode (snd (subc x y c)) = (decode x - decode y - bit c) mod 2^n)
:= {| fst_sub_with_carry x y c := proj1 (pf x y c);
decode_snd_sub_with_carry x y c := proj2 (pf x y c) |}.
Record multiply := { mul :> W -> W -> W }.
Class is_mul (mul : multiply) :=
decode_mul : forall x y, decode (mul x y) = (decode x * decode y) mod 2^n.
Record multiply_low_low := { mulhwll :> W -> W -> W }.
Record multiply_high_low := { mulhwhl :> W -> W -> W }.
Record multiply_high_high := { mulhwhh :> W -> W -> W }.
Class is_mul_low_low (w:Z) (mulhwll : multiply_low_low) :=
decode_mul_low_low :
forall x y, decode (mulhwll x y) = ((decode x mod 2^w) * (decode y mod 2^w)) mod 2^n.
Class is_mul_high_low (w:Z) (mulhwhl : multiply_high_low) :=
decode_mul_high_low :
forall x y, decode (mulhwhl x y) = ((decode x >> w) * (decode y mod 2^w)) mod 2^n.
Class is_mul_high_high (w:Z) (mulhwhh : multiply_high_high) :=
decode_mul_high_high :
forall x y, decode (mulhwhh x y) = ((decode x >> w) * (decode y >> w)) mod 2^n.
Record select_conditional := { selc :> bool -> W -> W -> W }.
Class is_select_conditional (selc : select_conditional) :=
decode_select_conditional : forall b x y,
decode (selc b x y) = if b then decode x else decode y.
Record add_modulo := { addm :> W -> W -> W (* modulus *) -> W }.
Class is_add_modulo (addm : add_modulo) :=
decode_add_modulo : forall x y modulus,
decode (addm x y modulus) = (if (decode x + decode y) <? decode modulus
then (decode x + decode y)
else (decode x + decode y) - decode modulus)%Z.
End InstructionGallery.
Global Arguments load_immediate : clear implicits.
Global Arguments shift_right_doubleword_immediate : clear implicits.
Global Arguments shift_left_immediate : clear implicits.
Global Arguments shift_right_immediate : clear implicits.
Global Arguments spread_left_immediate : clear implicits.
Global Arguments mask_keep_low : clear implicits.
Global Arguments add_with_carry : clear implicits.
Global Arguments sub_with_carry : clear implicits.
Global Arguments multiply : clear implicits.
Global Arguments multiply_low_low : clear implicits.
Global Arguments multiply_high_low : clear implicits.
Global Arguments multiply_high_high : clear implicits.
Global Arguments select_conditional : clear implicits.
Global Arguments add_modulo : clear implicits.
Global Arguments ldi {_ _} _.
Global Arguments shrd {_ _} _ _ _.
Global Arguments shl {_ _} _ _.
Global Arguments shr {_ _} _ _.
Global Arguments sprl {_ _} _ _.
Global Arguments mkl {_ _} _ _.
Global Arguments adc {_ _} _ _ _.
Global Arguments subc {_ _} _ _ _.
Global Arguments mul {_ _} _ _.
Global Arguments mulhwll {_ _} _ _.
Global Arguments mulhwhl {_ _} _ _.
Global Arguments mulhwhh {_ _} _ _.
Global Arguments selc {_ _} _ _ _.
Global Arguments addm {_ _} _ _ _.
Existing Class load_immediate.
Existing Class shift_right_doubleword_immediate.
Existing Class shift_left_immediate.
Existing Class shift_right_immediate.
Existing Class spread_left_immediate.
Existing Class mask_keep_low.
Existing Class add_with_carry.
Existing Class sub_with_carry.
Existing Class multiply.
Existing Class multiply_low_low.
Existing Class multiply_high_low.
Existing Class multiply_high_high.
Existing Class select_conditional.
Existing Class add_modulo.
Global Arguments is_decode {_ _} _.
Global Arguments is_load_immediate {_ _ _} _.
Global Arguments is_shift_right_doubleword_immediate {_ _ _} _.
Global Arguments is_shift_left_immediate {_ _ _} _.
Global Arguments is_shift_right_immediate {_ _ _} _.
Global Arguments is_spread_left_immediate {_ _ _} _.
Global Arguments is_mask_keep_low {_ _ _} _.
Global Arguments is_add_with_carry {_ _ _} _.
Global Arguments is_sub_with_carry {_ _ _} _.
Global Arguments is_mul {_ _ _} _.
Global Arguments is_mul_low_low {_ _ _} _ _.
Global Arguments is_mul_high_low {_ _ _} _ _.
Global Arguments is_mul_high_high {_ _ _} _ _.
Global Arguments is_select_conditional {_ _ _} _.
Global Arguments is_add_modulo {_ _ _} _.
Ltac bounded_solver_tac :=
solve [ eassumption | typeclasses eauto | omega ].
Lemma decode_proj n W (dec : W -> Z)
: @decode n W {| decode := dec |} = dec.
Proof. reflexivity. Qed.
Lemma decode_exponent_nonnegative {n W} (decode : decoder n W) {isdecode : is_decode decode}
(isinhabited : W)
: 0 <= n.
Proof.
pose proof (decode_range isinhabited).
assert (0 < 2^n) by omega.
destruct (Z_lt_ge_dec n 0) as [H'|]; [ | omega ].
assert (2^n = 0) by auto using Z.pow_neg_r.
omega.
Qed.
Hint Rewrite @decode_load_immediate @decode_shift_right_doubleword @decode_shift_left_immediate @decode_shift_right_immediate @decode_fst_spread_left_immediate @decode_snd_spread_left_immediate @decode_mask_keep_low @bit_fst_add_with_carry @decode_snd_add_with_carry @fst_sub_with_carry @decode_snd_sub_with_carry @decode_mul @decode_mul_low_low @decode_mul_high_low @decode_mul_high_high @decode_select_conditional @decode_add_modulo @decode_proj using bounded_solver_tac : push_decode.
Ltac push_decode_step :=
first [ rewrite !decode_proj
| erewrite !decode_load_immediate by bounded_solver_tac
| erewrite !decode_shift_right_doubleword by bounded_solver_tac
| erewrite !decode_shift_left_immediate by bounded_solver_tac
| erewrite !decode_shift_right_immediate by bounded_solver_tac
| erewrite !decode_fst_spread_left_immediate by bounded_solver_tac
| erewrite !decode_snd_spread_left_immediate by bounded_solver_tac
| erewrite !decode_mask_keep_low by bounded_solver_tac
| erewrite !bit_fst_add_with_carry by bounded_solver_tac
| erewrite !decode_snd_add_with_carry by bounded_solver_tac
| erewrite !fst_sub_with_carry by bounded_solver_tac
| erewrite !decode_snd_sub_with_carry by bounded_solver_tac
| erewrite !decode_mul by bounded_solver_tac
| erewrite !decode_mul_low_low by bounded_solver_tac
| erewrite !decode_mul_high_low by bounded_solver_tac
| erewrite !decode_mul_high_high by bounded_solver_tac
| erewrite !decode_select_conditional by bounded_solver_tac
| erewrite !decode_add_modulo by bounded_solver_tac ].
Ltac pull_decode_step :=
first [ erewrite <- !decode_load_immediate by bounded_solver_tac
| erewrite <- !decode_shift_right_doubleword by bounded_solver_tac
| erewrite <- !decode_shift_left_immediate by bounded_solver_tac
| erewrite <- !decode_shift_right_immediate by bounded_solver_tac
| erewrite <- !decode_fst_spread_left_immediate by bounded_solver_tac
| erewrite <- !decode_snd_spread_left_immediate by bounded_solver_tac
| erewrite <- !decode_mask_keep_low by bounded_solver_tac
| erewrite <- !bit_fst_add_with_carry by bounded_solver_tac
| erewrite <- !decode_snd_add_with_carry by bounded_solver_tac
| erewrite <- !fst_sub_with_carry by bounded_solver_tac
| erewrite <- !decode_snd_sub_with_carry by bounded_solver_tac
| erewrite <- !decode_mul by bounded_solver_tac
| erewrite <- !decode_mul_low_low by bounded_solver_tac
| erewrite <- !decode_mul_high_low by bounded_solver_tac
| erewrite <- !decode_mul_high_high by bounded_solver_tac
| erewrite <- !decode_select_conditional by bounded_solver_tac
| erewrite <- !decode_add_modulo by bounded_solver_tac ].
Ltac push_decode := repeat push_decode_step.
Ltac pull_decode := repeat pull_decode_step.
(* We take special care to handle the case where the decoder is
syntactically different but the decoded expression is judgmentally
the same; we don't want to split apart variables that should be the
same. *)
Ltac set_decode_step check :=
match goal with
| [ |- context G[@Interface.decode ?n ?W ?dr ?w] ]
=> check w;
first [ match goal with
| [ d := @Interface.decode _ _ _ w |- _ ]
=> change (@Interface.decode n W dr w) with d
end
| generalize (@decode_range n W dr _ w);
let d := fresh "d" in
set (d := @Interface.decode n W dr w);
intro ]
end.
Ltac set_decode check := repeat set_decode_step check.
Ltac clearbody_decode :=
repeat match goal with
| [ H := @Interface.decode _ _ _ _ |- _ ] => clearbody H
end.
Ltac generalize_decode_by check := set_decode check; clearbody_decode.
Ltac generalize_decode := generalize_decode_by ltac:(fun w => idtac).
Ltac generalize_decode_var := generalize_decode_by ltac:(fun w => is_var w).
Module fancy_machine.
Local Notation imm := Z (only parsing).
Class instructions (n : Z) :=
{
W : Type (* [n]-bit word *);
decode :> decoder n W;
ldi :> load_immediate W;
shrd :> shift_right_doubleword_immediate W;
shl :> shift_left_immediate W;
shr :> shift_right_immediate W;
mkl :> mask_keep_low W;
adc :> add_with_carry W;
subc :> sub_with_carry W;
mulhwll :> multiply_low_low W;
mulhwhl :> multiply_high_low W;
mulhwhh :> multiply_high_high W;
selc :> select_conditional W;
addm :> add_modulo W
}.
Class arithmetic {n_over_two} (ops:instructions (2 * n_over_two)) :=
{
decode_range :> is_decode decode;
load_immediate :> is_load_immediate ldi;
shift_right_doubleword_immediate :> is_shift_right_doubleword_immediate shrd;
shift_left_immediate :> is_shift_left_immediate shl;
shift_right_immediate :> is_shift_right_immediate shr;
mask_keep_low :> is_mask_keep_low mkl;
add_with_carry :> is_add_with_carry adc;
sub_with_carry :> is_sub_with_carry subc;
multiply_low_low :> is_mul_low_low n_over_two mulhwll;
multiply_high_low :> is_mul_high_low n_over_two mulhwhl;
multiply_high_high :> is_mul_high_high n_over_two mulhwhh;
select_conditional :> is_select_conditional selc;
add_modulo :> is_add_modulo addm
}.
End fancy_machine.
|
