1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/assembly/byte".
17 include "assembly/exadecimal.ma".
19 record byte : Type ≝ {
24 notation "〈 x, y 〉" non associative with precedence 80 for @{ 'mk_byte $x $y }.
25 interpretation "mk_byte" 'mk_byte x y =
26 (cic:/matita/assembly/byte/byte.ind#xpointer(1/1/1) x y).
29 λb,b'. eqex (bh b) (bh b') ∧ eqex (bl b) (bl b').
33 match plusex (bl b1) (bl b2) c with
35 match plusex (bh b1) (bh b2) c' with
36 [ couple h c'' ⇒ couple ? ? (mk_byte h l) c'' ]].
38 definition nat_of_byte ≝ λb:byte. 16*(bh b) + (bl b).
40 coercion cic:/matita/assembly/byte/nat_of_byte.con.
42 definition byte_of_nat ≝
43 λn. mk_byte (exadecimal_of_nat (n / 16)) (exadecimal_of_nat n).
45 interpretation "byte_of_nat" 'byte_of_opcode a =
46 (cic:/matita/assembly/byte/byte_of_nat.con a).
48 lemma byte_of_nat_nat_of_byte: ∀b. byte_of_nat (nat_of_byte b) = b.
56 lemma lt_nat_of_byte_256: ∀b. nat_of_byte b < 256.
59 letin H ≝ (lt_nat_of_exadecimal_16 (bh b)); clearbody H;
60 letin K ≝ (lt_nat_of_exadecimal_16 (bl b)); clearbody K;
62 letin H' ≝ (le_S_S_to_le ? ? H); clearbody H'; clear H;
63 letin K' ≝ (le_S_S_to_le ? ? K); clearbody K'; clear K;
65 cut (16*bh b ≤ 16*15);
66 [ letin Hf ≝ (le_plus ? ? ? ? Hcut K'); clearbody Hf;
67 simplify in Hf:(? ? %);
73 lemma nat_of_byte_byte_of_nat: ∀n. nat_of_byte (byte_of_nat n) = n \mod 256.
75 letin H ≝ (lt_nat_of_byte_256 (byte_of_nat n)); clearbody H;
76 rewrite < (lt_to_eq_mod ? ? H); clear H;
79 change with ((16*(exadecimal_of_nat (n/16)) + exadecimal_of_nat n) \mod 256 = n \mod 256);
80 letin H ≝ (div_mod n 16 ?); clearbody H; [ autobatch | ];
81 rewrite > symmetric_times in H;
82 rewrite > nat_of_exadecimal_exadecimal_of_nat in ⊢ (? ? (? (? % ?) ?) ?);
83 rewrite > nat_of_exadecimal_exadecimal_of_nat in ⊢ (? ? (? (? ? %) ?) ?);
84 rewrite > H in ⊢ (? ? ? (? % ?)); clear H;
85 rewrite > mod_plus in ⊢ (? ? % ?);
86 rewrite > mod_plus in ⊢ (? ? ? %);
87 apply eq_mod_to_eq_plus_mod;
88 rewrite < mod_mod in ⊢ (? ? ? %); [ | autobatch];
89 rewrite < mod_mod in ⊢ (? ? % ?); [ | autobatch];
90 rewrite < (eq_mod_times_times_mod ? ? 16 256) in ⊢ (? ? (? % ?) ?); [2: reflexivity | ];
91 rewrite < mod_mod in ⊢ (? ? % ?);
97 lemma eq_nat_of_byte_n_nat_of_byte_mod_n_256:
98 ∀n. byte_of_nat n = byte_of_nat (n \mod 256).
102 [ rewrite > exadecimal_of_nat_mod in ⊢ (? ? % ?);
103 rewrite > exadecimal_of_nat_mod in ⊢ (? ? ? %);
106 | rewrite > exadecimal_of_nat_mod;
107 rewrite > exadecimal_of_nat_mod in ⊢ (? ? ? %);
108 rewrite > divides_to_eq_mod_mod_mod;
117 match plusbyte b1 b2 c with
118 [ couple r c' ⇒ b1 + b2 + nat_of_bool c = nat_of_byte r + nat_of_bool c' * 256
122 generalize in match (plusex_ok (bl b1) (bl b2) c);
123 elim (plusex (bl b1) (bl b2) c);
125 generalize in match (plusex_ok (bh b1) (bh b2) t1);
126 elim (plusex (bh b1) (bh b2) t1);
128 change in ⊢ (? ? ? (? (? % ?) ?)) with (16 * t2);
130 letin K ≝ (eq_f ? ? (λy.16*y) ? ? H1); clearbody K; clear H1;
131 rewrite > distr_times_plus in K:(? ? ? %);
132 rewrite > symmetric_times in K:(? ? ? (? ? (? ? %)));
133 rewrite < associative_times in K:(? ? ? (? ? %));
134 normalize in K:(? ? ? (? ? (? % ?)));
135 rewrite > symmetric_times in K:(? ? ? (? ? %));
136 rewrite > sym_plus in ⊢ (? ? ? (? % ?));
137 rewrite > associative_plus in ⊢ (? ? ? %);
138 letin K' ≝ (eq_f ? ? (plus t) ? ? K); clearbody K'; clear K;
139 apply transitive_eq; [3: apply K' | skip | ];
141 rewrite > sym_plus in ⊢ (? ? (? (? ? %) ?) ?);
142 rewrite > associative_plus in ⊢ (? ? (? % ?) ?);
143 rewrite > associative_plus in ⊢ (? ? % ?);
144 rewrite > associative_plus in ⊢ (? ? (? ? %) ?);
145 rewrite > associative_plus in ⊢ (? ? (? ? (? ? %)) ?);
146 rewrite > sym_plus in ⊢ (? ? (? ? (? ? (? ? %))) ?);
147 rewrite < associative_plus in ⊢ (? ? (? ? (? ? %)) ?);
148 rewrite < associative_plus in ⊢ (? ? (? ? %) ?);
149 rewrite < associative_plus in ⊢ (? ? (? ? (? % ?)) ?);
150 rewrite > H; clear H;
151 autobatch paramodulation.
156 match eqex (bl b) x0 with
157 [ true ⇒ mk_byte (xpred (bh b)) (xpred (bl b))
158 | false ⇒ mk_byte (bh b) (xpred (bl b))
162 ∀b. plusbyte (mk_byte x0 x0) b false = couple ? ? b false.
170 definition plusbytenc ≝
172 match plusbyte x y false with
173 [couple res _ ⇒ res].
175 definition plusbytec ≝
177 match plusbyte x y false with
180 lemma plusbytenc_O_x:
181 ∀x. plusbytenc (mk_byte x0 x0) x = x.
184 rewrite > plusbyte_O_x;
188 lemma eq_nat_of_byte_mod: ∀b. nat_of_byte b = nat_of_byte b \mod 256.
190 lapply (lt_nat_of_byte_256 b);
191 rewrite > (lt_to_eq_mod ? ? Hletin) in ⊢ (? ? ? %);
195 theorem plusbytenc_ok:
196 ∀b1,b2:byte. nat_of_byte (plusbytenc b1 b2) = (b1 + b2) \mod 256.
199 generalize in match (plusbyte_ok b1 b2 false);
200 elim (plusbyte b1 b2 false);
202 change with (nat_of_byte t = (b1 + b2) \mod 256);
203 rewrite < plus_n_O in H;
204 rewrite > H; clear H;
206 letin K ≝ (eq_nat_of_byte_mod t); clearbody K;
207 letin K' ≝ (eq_mod_times_n_m_m_O (nat_of_bool t1) 256 ?); clearbody K';
209 autobatch paramodulation.
214 lemma eq_eqbyte_x0_x0_byte_of_nat_S_false:
215 ∀b. b < 255 → eqbyte (mk_byte x0 x0) (byte_of_nat (S b)) = false.
218 cut (b < 15 ∨ b ≥ 15);
221 change in ⊢ (? ? (? ? %) ?) with (eqex x0 (exadecimal_of_nat (S b)));
222 rewrite > eq_eqex_S_x0_false;
223 [ elim (eqex (bh (mk_byte x0 x0))
224 (bh (mk_byte (exadecimal_of_nat (S b/16)) (exadecimal_of_nat (S b)))));
230 change in ⊢ (? ? (? % ?) ?) with (eqex x0 (exadecimal_of_nat (S b/16)));
231 letin K ≝ (leq_m_n_to_eq_div_n_m_S (S b) 16 ? ?);
239 rewrite > eq_eqex_S_x0_false;
244 clear H2; clear a; clear H1; clear Hcut;
245 apply (le_times_to_le 16) [ autobatch | ] ;
246 rewrite > (div_mod (S b) 16) in H;[2:autobatch|]
247 rewrite > (div_mod 255 16) in H:(? ? %);[2:autobatch|]
248 lapply (le_to_le_plus_to_le ? ? ? ? ? H);
250 apply lt_mod_m_m;autobatch
251 |rewrite > sym_times;
252 rewrite > sym_times in ⊢ (? ? %); (* just to speed up qed *)
253 normalize in ⊢ (? ? %);apply Hletin;
258 | elim (or_lt_le b 15);unfold ge;autobatch
262 axiom eq_mod_O_to_exists: ∀n,m. n \mod m = 0 → ∃z. n = z*m.
264 lemma eq_bpred_S_a_a:
265 ∀a. a < 255 → bpred (byte_of_nat (S a)) = byte_of_nat a.
268 apply (bool_elim ? (eqex (bl (byte_of_nat (S a))) x0)); intros;
269 [ change with (mk_byte (xpred (bh (byte_of_nat (S a)))) (xpred (bl (byte_of_nat (S a))))
271 rewrite > (eqex_true_to_eq ? ? H1);
272 normalize in ⊢ (? ? (? ? %) ?);
274 change with (mk_byte (xpred (exadecimal_of_nat (S a/16))) xF =
275 mk_byte (exadecimal_of_nat (a/16)) (exadecimal_of_nat a));
276 lapply (eqex_true_to_eq ? ? H1); clear H1;
277 unfold byte_of_nat in Hletin;
278 change in Hletin with (exadecimal_of_nat (S a) = x0);
279 lapply (eq_f ? ? nat_of_exadecimal ? ? Hletin); clear Hletin;
280 normalize in Hletin1:(? ? ? %);
281 rewrite > nat_of_exadecimal_exadecimal_of_nat in Hletin1;
282 elim (eq_mod_O_to_exists ? ? Hletin1); clear Hletin1;
284 rewrite > div_times_ltO; [2: autobatch | ]
285 lapply (eq_f ? ? (λx.x/16) ? ? H1);
286 rewrite > div_times_ltO in Hletin; [2: autobatch | ]
287 lapply (eq_f ? ? (λx.x \mod 16) ? ? H1);
288 rewrite > eq_mod_times_n_m_m_O in Hletin1;
290 | change with (mk_byte (bh (byte_of_nat (S a))) (xpred (bl (byte_of_nat (S a))))
293 change with (mk_byte (exadecimal_of_nat (S a/16)) (xpred (exadecimal_of_nat (S a)))
294 = mk_byte (exadecimal_of_nat (a/16)) (exadecimal_of_nat a));
295 lapply (eqex_false_to_not_eq ? ? H1);
296 unfold byte_of_nat in Hletin;
297 change in Hletin with (exadecimal_of_nat (S a) ≠ x0);
298 cut (nat_of_exadecimal (exadecimal_of_nat (S a)) ≠ 0);
301 lapply (eq_f ? ? exadecimal_of_nat ? ? H2);
302 rewrite > exadecimal_of_nat_nat_of_exadecimal in Hletin1;
311 ∀x:byte.∀n.plusbytenc (byte_of_nat (x*n)) x = byte_of_nat (x * S n).
313 rewrite < byte_of_nat_nat_of_byte;
314 rewrite > (plusbytenc_ok (byte_of_nat (x*n)) x);
315 rewrite < times_n_Sm;
316 rewrite > nat_of_byte_byte_of_nat in ⊢ (? ? (? (? (? % ?) ?)) ?);
317 rewrite > eq_nat_of_byte_n_nat_of_byte_mod_n_256 in ⊢ (? ? ? %);
318 rewrite > mod_plus in ⊢ (? ? (? %) ?);
319 rewrite > mod_plus in ⊢ (? ? ? (? %));
320 rewrite < mod_mod in ⊢ (? ? (? (? (? % ?) ?)) ?); [2: autobatch | ];
321 rewrite > sym_plus in ⊢ (? ? (? (? % ?)) ?);
325 lemma eq_plusbytec_x0_x0_x_false:
326 ∀x.plusbytec (mk_byte x0 x0) x = false.