author | haftmann |
Sat, 17 Dec 2016 15:22:14 +0100 | |
changeset 64593 | 50c715579715 |
parent 64242 | 93c6f0da5c70 |
child 66453 | cc19f7ca2ed6 |
permissions | -rw-r--r-- |
64015 | 1 |
(* Author: Florian Haftmann, TUM |
2 |
*) |
|
3 |
||
4 |
section \<open>Proof of concept for algebraically founded bit word types\<close> |
|
5 |
||
6 |
theory Word_Type |
|
7 |
imports |
|
8 |
Main |
|
9 |
"~~/src/HOL/Library/Type_Length" |
|
10 |
begin |
|
11 |
||
12 |
subsection \<open>Truncating bit representations of numeric types\<close> |
|
13 |
||
14 |
class semiring_bits = semiring_div_parity + |
|
15 |
assumes semiring_bits: "(1 + 2 * a) mod of_nat (2 * n) = 1 + 2 * (a mod of_nat n)" |
|
16 |
begin |
|
17 |
||
64114 | 18 |
definition bitrunc :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" |
19 |
where bitrunc_eq_mod: "bitrunc n a = a mod of_nat (2 ^ n)" |
|
64015 | 20 |
|
64114 | 21 |
lemma bitrunc_bitrunc [simp]: |
22 |
"bitrunc n (bitrunc n a) = bitrunc n a" |
|
23 |
by (simp add: bitrunc_eq_mod) |
|
64015 | 24 |
|
64114 | 25 |
lemma bitrunc_0 [simp]: |
26 |
"bitrunc 0 a = 0" |
|
27 |
by (simp add: bitrunc_eq_mod) |
|
64015 | 28 |
|
64114 | 29 |
lemma bitrunc_Suc [simp]: |
30 |
"bitrunc (Suc n) a = bitrunc n (a div 2) * 2 + a mod 2" |
|
64015 | 31 |
proof - |
32 |
define b and c |
|
33 |
where "b = a div 2" and "c = a mod 2" |
|
34 |
then have a: "a = b * 2 + c" |
|
35 |
and "c = 0 \<or> c = 1" |
|
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
64114
diff
changeset
|
36 |
by (simp_all add: div_mult_mod_eq parity) |
64015 | 37 |
from \<open>c = 0 \<or> c = 1\<close> |
64114 | 38 |
have "bitrunc (Suc n) (b * 2 + c) = bitrunc n b * 2 + c" |
64015 | 39 |
proof |
40 |
assume "c = 0" |
|
41 |
moreover have "(2 * b) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n)" |
|
42 |
by (simp add: mod_mult_mult1) |
|
43 |
ultimately show ?thesis |
|
64114 | 44 |
by (simp add: bitrunc_eq_mod ac_simps) |
64015 | 45 |
next |
46 |
assume "c = 1" |
|
47 |
with semiring_bits [of b "2 ^ n"] show ?thesis |
|
64114 | 48 |
by (simp add: bitrunc_eq_mod ac_simps) |
64015 | 49 |
qed |
50 |
with a show ?thesis |
|
51 |
by (simp add: b_def c_def) |
|
52 |
qed |
|
53 |
||
64114 | 54 |
lemma bitrunc_of_0 [simp]: |
55 |
"bitrunc n 0 = 0" |
|
56 |
by (simp add: bitrunc_eq_mod) |
|
64015 | 57 |
|
64114 | 58 |
lemma bitrunc_plus: |
59 |
"bitrunc n (bitrunc n a + bitrunc n b) = bitrunc n (a + b)" |
|
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64242
diff
changeset
|
60 |
by (simp add: bitrunc_eq_mod mod_add_eq) |
64015 | 61 |
|
64114 | 62 |
lemma bitrunc_of_1_eq_0_iff [simp]: |
63 |
"bitrunc n 1 = 0 \<longleftrightarrow> n = 0" |
|
64015 | 64 |
by (induct n) simp_all |
65 |
||
66 |
end |
|
67 |
||
68 |
instance nat :: semiring_bits |
|
69 |
by standard (simp add: mod_Suc Suc_double_not_eq_double) |
|
70 |
||
71 |
instance int :: semiring_bits |
|
72 |
by standard (simp add: pos_zmod_mult_2) |
|
73 |
||
64114 | 74 |
lemma bitrunc_uminus: |
64015 | 75 |
fixes k :: int |
64114 | 76 |
shows "bitrunc n (- (bitrunc n k)) = bitrunc n (- k)" |
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64242
diff
changeset
|
77 |
by (simp add: bitrunc_eq_mod mod_minus_eq) |
64015 | 78 |
|
64114 | 79 |
lemma bitrunc_minus: |
64015 | 80 |
fixes k l :: int |
64114 | 81 |
shows "bitrunc n (bitrunc n k - bitrunc n l) = bitrunc n (k - l)" |
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64242
diff
changeset
|
82 |
by (simp add: bitrunc_eq_mod mod_diff_eq) |
64015 | 83 |
|
64114 | 84 |
lemma bitrunc_nonnegative [simp]: |
64015 | 85 |
fixes k :: int |
64114 | 86 |
shows "bitrunc n k \<ge> 0" |
87 |
by (simp add: bitrunc_eq_mod) |
|
64015 | 88 |
|
64114 | 89 |
definition signed_bitrunc :: "nat \<Rightarrow> int \<Rightarrow> int" |
90 |
where signed_bitrunc_eq_bitrunc: |
|
91 |
"signed_bitrunc n k = bitrunc (Suc n) (k + 2 ^ n) - 2 ^ n" |
|
64015 | 92 |
|
64114 | 93 |
lemma signed_bitrunc_eq_bitrunc': |
64015 | 94 |
assumes "n > 0" |
64114 | 95 |
shows "signed_bitrunc (n - Suc 0) k = bitrunc n (k + 2 ^ (n - 1)) - 2 ^ (n - 1)" |
96 |
using assms by (simp add: signed_bitrunc_eq_bitrunc) |
|
64015 | 97 |
|
64114 | 98 |
lemma signed_bitrunc_0 [simp]: |
99 |
"signed_bitrunc 0 k = - (k mod 2)" |
|
64015 | 100 |
proof (cases "even k") |
101 |
case True |
|
102 |
then have "odd (k + 1)" |
|
103 |
by simp |
|
104 |
then have "(k + 1) mod 2 = 1" |
|
105 |
by (simp add: even_iff_mod_2_eq_zero) |
|
106 |
with True show ?thesis |
|
64114 | 107 |
by (simp add: signed_bitrunc_eq_bitrunc) |
64015 | 108 |
next |
109 |
case False |
|
110 |
then show ?thesis |
|
64114 | 111 |
by (simp add: signed_bitrunc_eq_bitrunc odd_iff_mod_2_eq_one) |
64015 | 112 |
qed |
113 |
||
64114 | 114 |
lemma signed_bitrunc_Suc [simp]: |
115 |
"signed_bitrunc (Suc n) k = signed_bitrunc n (k div 2) * 2 + k mod 2" |
|
116 |
using zero_not_eq_two by (simp add: signed_bitrunc_eq_bitrunc algebra_simps) |
|
64015 | 117 |
|
64114 | 118 |
lemma signed_bitrunc_of_0 [simp]: |
119 |
"signed_bitrunc n 0 = 0" |
|
120 |
by (simp add: signed_bitrunc_eq_bitrunc bitrunc_eq_mod) |
|
64015 | 121 |
|
64114 | 122 |
lemma signed_bitrunc_of_minus_1 [simp]: |
123 |
"signed_bitrunc n (- 1) = - 1" |
|
64015 | 124 |
by (induct n) simp_all |
125 |
||
64114 | 126 |
lemma signed_bitrunc_eq_iff_bitrunc_eq: |
64015 | 127 |
assumes "n > 0" |
64114 | 128 |
shows "signed_bitrunc (n - Suc 0) k = signed_bitrunc (n - Suc 0) l \<longleftrightarrow> bitrunc n k = bitrunc n l" (is "?P \<longleftrightarrow> ?Q") |
64015 | 129 |
proof - |
130 |
from assms obtain m where m: "n = Suc m" |
|
131 |
by (cases n) auto |
|
132 |
show ?thesis |
|
133 |
proof |
|
134 |
assume ?Q |
|
64114 | 135 |
have "bitrunc (Suc m) (k + 2 ^ m) = |
136 |
bitrunc (Suc m) (bitrunc (Suc m) k + bitrunc (Suc m) (2 ^ m))" |
|
137 |
by (simp only: bitrunc_plus) |
|
64015 | 138 |
also have "\<dots> = |
64114 | 139 |
bitrunc (Suc m) (bitrunc (Suc m) l + bitrunc (Suc m) (2 ^ m))" |
64015 | 140 |
by (simp only: \<open>?Q\<close> m [symmetric]) |
64114 | 141 |
also have "\<dots> = bitrunc (Suc m) (l + 2 ^ m)" |
142 |
by (simp only: bitrunc_plus) |
|
64015 | 143 |
finally show ?P |
64114 | 144 |
by (simp only: signed_bitrunc_eq_bitrunc m) simp |
64015 | 145 |
next |
146 |
assume ?P |
|
147 |
with assms have "(k + 2 ^ (n - Suc 0)) mod 2 ^ n = (l + 2 ^ (n - Suc 0)) mod 2 ^ n" |
|
64114 | 148 |
by (simp add: signed_bitrunc_eq_bitrunc' bitrunc_eq_mod) |
64015 | 149 |
then have "(i + (k + 2 ^ (n - Suc 0))) mod 2 ^ n = (i + (l + 2 ^ (n - Suc 0))) mod 2 ^ n" for i |
150 |
by (metis mod_add_eq) |
|
151 |
then have "k mod 2 ^ n = l mod 2 ^ n" |
|
152 |
by (metis add_diff_cancel_right' uminus_add_conv_diff) |
|
153 |
then show ?Q |
|
64114 | 154 |
by (simp add: bitrunc_eq_mod) |
64015 | 155 |
qed |
156 |
qed |
|
157 |
||
158 |
||
159 |
subsection \<open>Bit strings as quotient type\<close> |
|
160 |
||
161 |
subsubsection \<open>Basic properties\<close> |
|
162 |
||
64114 | 163 |
quotient_type (overloaded) 'a word = int / "\<lambda>k l. bitrunc LENGTH('a) k = bitrunc LENGTH('a::len0) l" |
64015 | 164 |
by (auto intro!: equivpI reflpI sympI transpI) |
165 |
||
166 |
instantiation word :: (len0) "{semiring_numeral, comm_semiring_0, comm_ring}" |
|
167 |
begin |
|
168 |
||
169 |
lift_definition zero_word :: "'a word" |
|
170 |
is 0 |
|
171 |
. |
|
172 |
||
173 |
lift_definition one_word :: "'a word" |
|
174 |
is 1 |
|
175 |
. |
|
176 |
||
177 |
lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" |
|
178 |
is plus |
|
64114 | 179 |
by (subst bitrunc_plus [symmetric]) (simp add: bitrunc_plus) |
64015 | 180 |
|
181 |
lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" |
|
182 |
is uminus |
|
64114 | 183 |
by (subst bitrunc_uminus [symmetric]) (simp add: bitrunc_uminus) |
64015 | 184 |
|
185 |
lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" |
|
186 |
is minus |
|
64114 | 187 |
by (subst bitrunc_minus [symmetric]) (simp add: bitrunc_minus) |
64015 | 188 |
|
189 |
lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" |
|
190 |
is times |
|
64114 | 191 |
by (auto simp add: bitrunc_eq_mod intro: mod_mult_cong) |
64015 | 192 |
|
193 |
instance |
|
194 |
by standard (transfer; simp add: algebra_simps)+ |
|
195 |
||
196 |
end |
|
197 |
||
198 |
instance word :: (len) comm_ring_1 |
|
199 |
by standard (transfer; simp)+ |
|
200 |
||
201 |
||
202 |
subsubsection \<open>Conversions\<close> |
|
203 |
||
204 |
lemma [transfer_rule]: |
|
205 |
"rel_fun HOL.eq pcr_word int of_nat" |
|
206 |
proof - |
|
207 |
note transfer_rule_of_nat [transfer_rule] |
|
208 |
show ?thesis by transfer_prover |
|
209 |
qed |
|
210 |
||
211 |
lemma [transfer_rule]: |
|
212 |
"rel_fun HOL.eq pcr_word (\<lambda>k. k) of_int" |
|
213 |
proof - |
|
214 |
note transfer_rule_of_int [transfer_rule] |
|
215 |
have "rel_fun HOL.eq pcr_word (of_int :: int \<Rightarrow> int) (of_int :: int \<Rightarrow> 'a word)" |
|
216 |
by transfer_prover |
|
217 |
then show ?thesis by (simp add: id_def) |
|
218 |
qed |
|
219 |
||
220 |
context semiring_1 |
|
221 |
begin |
|
222 |
||
223 |
lift_definition unsigned :: "'b::len0 word \<Rightarrow> 'a" |
|
64114 | 224 |
is "of_nat \<circ> nat \<circ> bitrunc LENGTH('b)" |
64015 | 225 |
by simp |
226 |
||
227 |
lemma unsigned_0 [simp]: |
|
228 |
"unsigned 0 = 0" |
|
229 |
by transfer simp |
|
230 |
||
231 |
end |
|
232 |
||
233 |
context semiring_char_0 |
|
234 |
begin |
|
235 |
||
236 |
lemma word_eq_iff_unsigned: |
|
237 |
"a = b \<longleftrightarrow> unsigned a = unsigned b" |
|
238 |
by safe (transfer; simp add: eq_nat_nat_iff) |
|
239 |
||
240 |
end |
|
241 |
||
242 |
context ring_1 |
|
243 |
begin |
|
244 |
||
245 |
lift_definition signed :: "'b::len word \<Rightarrow> 'a" |
|
64114 | 246 |
is "of_int \<circ> signed_bitrunc (LENGTH('b) - 1)" |
247 |
by (simp add: signed_bitrunc_eq_iff_bitrunc_eq [symmetric]) |
|
64015 | 248 |
|
249 |
lemma signed_0 [simp]: |
|
250 |
"signed 0 = 0" |
|
251 |
by transfer simp |
|
252 |
||
253 |
end |
|
254 |
||
255 |
lemma unsigned_of_nat [simp]: |
|
64114 | 256 |
"unsigned (of_nat n :: 'a word) = bitrunc LENGTH('a::len) n" |
257 |
by transfer (simp add: nat_eq_iff bitrunc_eq_mod zmod_int) |
|
64015 | 258 |
|
259 |
lemma of_nat_unsigned [simp]: |
|
260 |
"of_nat (unsigned a) = a" |
|
261 |
by transfer simp |
|
262 |
||
263 |
lemma of_int_unsigned [simp]: |
|
264 |
"of_int (unsigned a) = a" |
|
265 |
by transfer simp |
|
266 |
||
267 |
context ring_char_0 |
|
268 |
begin |
|
269 |
||
270 |
lemma word_eq_iff_signed: |
|
271 |
"a = b \<longleftrightarrow> signed a = signed b" |
|
64114 | 272 |
by safe (transfer; auto simp add: signed_bitrunc_eq_iff_bitrunc_eq) |
64015 | 273 |
|
274 |
end |
|
275 |
||
276 |
lemma signed_of_int [simp]: |
|
64114 | 277 |
"signed (of_int k :: 'a word) = signed_bitrunc (LENGTH('a::len) - 1) k" |
64015 | 278 |
by transfer simp |
279 |
||
280 |
lemma of_int_signed [simp]: |
|
281 |
"of_int (signed a) = a" |
|
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64242
diff
changeset
|
282 |
by transfer (simp add: signed_bitrunc_eq_bitrunc bitrunc_eq_mod mod_simps) |
64015 | 283 |
|
284 |
||
285 |
subsubsection \<open>Properties\<close> |
|
286 |
||
287 |
||
288 |
subsubsection \<open>Division\<close> |
|
289 |
||
290 |
instantiation word :: (len0) modulo |
|
291 |
begin |
|
292 |
||
293 |
lift_definition divide_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" |
|
64114 | 294 |
is "\<lambda>a b. bitrunc LENGTH('a) a div bitrunc LENGTH('a) b" |
64015 | 295 |
by simp |
296 |
||
297 |
lift_definition modulo_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" |
|
64114 | 298 |
is "\<lambda>a b. bitrunc LENGTH('a) a mod bitrunc LENGTH('a) b" |
64015 | 299 |
by simp |
300 |
||
301 |
instance .. |
|
302 |
||
303 |
end |
|
304 |
||
305 |
||
306 |
subsubsection \<open>Orderings\<close> |
|
307 |
||
308 |
instantiation word :: (len0) linorder |
|
309 |
begin |
|
310 |
||
311 |
lift_definition less_eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" |
|
64114 | 312 |
is "\<lambda>a b. bitrunc LENGTH('a) a \<le> bitrunc LENGTH('a) b" |
64015 | 313 |
by simp |
314 |
||
315 |
lift_definition less_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" |
|
64114 | 316 |
is "\<lambda>a b. bitrunc LENGTH('a) a < bitrunc LENGTH('a) b" |
64015 | 317 |
by simp |
318 |
||
319 |
instance |
|
320 |
by standard (transfer; auto)+ |
|
321 |
||
322 |
end |
|
323 |
||
324 |
context linordered_semidom |
|
325 |
begin |
|
326 |
||
327 |
lemma word_less_eq_iff_unsigned: |
|
328 |
"a \<le> b \<longleftrightarrow> unsigned a \<le> unsigned b" |
|
329 |
by (transfer fixing: less_eq) (simp add: nat_le_eq_zle) |
|
330 |
||
331 |
lemma word_less_iff_unsigned: |
|
332 |
"a < b \<longleftrightarrow> unsigned a < unsigned b" |
|
64114 | 333 |
by (transfer fixing: less) (auto dest: preorder_class.le_less_trans [OF bitrunc_nonnegative]) |
64015 | 334 |
|
335 |
end |
|
336 |
||
337 |
end |