1 | include "ASM/ASM.ma". |
---|
2 | include "ASM/Arithmetic.ma". |
---|
3 | include "ASM/Fetch.ma". |
---|
4 | include "ASM/Status.ma". |
---|
5 | include alias "basics/logic.ma". |
---|
6 | include alias "arithmetics/nat.ma". |
---|
7 | include "utilities/extralib.ma". |
---|
8 | |
---|
9 | (**************************************** START OF POLICY ABSTRACTION ********************) |
---|
10 | |
---|
11 | (* definition of & operations on jump length *) |
---|
12 | inductive jump_length: Type[0] ≝ |
---|
13 | | short_jump: jump_length |
---|
14 | | medium_jump: jump_length |
---|
15 | | long_jump: jump_length. |
---|
16 | |
---|
17 | definition assembly_preinstruction ≝ |
---|
18 | λA: Type[0]. |
---|
19 | λaddr_of: A → Byte. (* relative *) |
---|
20 | λpre: preinstruction A. |
---|
21 | match pre with |
---|
22 | [ ADD addr1 addr2 ⇒ |
---|
23 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
---|
24 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;false;true]]) @@ r ] |
---|
25 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;false;false;true;false;true]]); b1 ] |
---|
26 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;false;false;true;true;i1]]) ] |
---|
27 | | DATA b1 ⇒ λ_. [ ([[false;false;true;false;false;true;false;false]]) ; b1 ] |
---|
28 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
29 | | ADDC addr1 addr2 ⇒ |
---|
30 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
---|
31 | [ REGISTER r ⇒ λ_.[ ([[false;false;true;true;true]]) @@ r ] |
---|
32 | | DIRECT b1 ⇒ λ_.[ ([[false;false;true;true;false;true;false;true]]); b1 ] |
---|
33 | | INDIRECT i1 ⇒ λ_. [ ([[false;false;true;true;false;true;true;i1]]) ] |
---|
34 | | DATA b1 ⇒ λ_. [ ([[false;false;true;true;false;true;false;false]]) ; b1 ] |
---|
35 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
36 | | ANL addrs ⇒ |
---|
37 | match addrs with |
---|
38 | [ inl addrs ⇒ match addrs with |
---|
39 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
40 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
---|
41 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;true;true]]) @@ r ] |
---|
42 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;true;false;true;false;true]]); b1 ] |
---|
43 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;true;false;true;true;i1]]) ] |
---|
44 | | DATA b1 ⇒ λ_. [ ([[false;true;false;true;false;true;false;false]]) ; b1 ] |
---|
45 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
46 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
47 | let b1 ≝ |
---|
48 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
---|
49 | [ DIRECT b1 ⇒ λ_.b1 |
---|
50 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
---|
51 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
---|
52 | [ ACC_A ⇒ λ_.[ ([[false;true;false;true;false;false;true;false]]) ; b1 ] |
---|
53 | | DATA b2 ⇒ λ_. [ ([[false;true;false;true;false;false;true;true]]) ; b1 ; b2 ] |
---|
54 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
55 | ] |
---|
56 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
57 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
---|
58 | [ BIT_ADDR b1 ⇒ λ_.[ ([[true;false;false;false;false;false;true;false]]) ; b1 ] |
---|
59 | | N_BIT_ADDR b1 ⇒ λ_. [ ([[true;false;true;true;false;false;false;false]]) ; b1 ] |
---|
60 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
---|
61 | | CLR addr ⇒ |
---|
62 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
---|
63 | [ ACC_A ⇒ λ_. |
---|
64 | [ ([[true; true; true; false; false; true; false; false]]) ] |
---|
65 | | CARRY ⇒ λ_. |
---|
66 | [ ([[true; true; false; false; false; false; true; true]]) ] |
---|
67 | | BIT_ADDR b1 ⇒ λ_. |
---|
68 | [ ([[true; true; false; false; false; false; true; false]]) ; b1 ] |
---|
69 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
70 | | CPL addr ⇒ |
---|
71 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;carry;bit_addr]] x) → ? with |
---|
72 | [ ACC_A ⇒ λ_. |
---|
73 | [ ([[true; true; true; true; false; true; false; false]]) ] |
---|
74 | | CARRY ⇒ λ_. |
---|
75 | [ ([[true; false; true; true; false; false; true; true]]) ] |
---|
76 | | BIT_ADDR b1 ⇒ λ_. |
---|
77 | [ ([[true; false; true; true; false; false; true; false]]) ; b1 ] |
---|
78 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
79 | | DA addr ⇒ |
---|
80 | [ ([[true; true; false; true; false; true; false; false]]) ] |
---|
81 | | DEC addr ⇒ |
---|
82 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect]] x) → ? with |
---|
83 | [ ACC_A ⇒ λ_. |
---|
84 | [ ([[false; false; false; true; false; true; false; false]]) ] |
---|
85 | | REGISTER r ⇒ λ_. |
---|
86 | [ ([[false; false; false; true; true]]) @@ r ] |
---|
87 | | DIRECT b1 ⇒ λ_. |
---|
88 | [ ([[false; false; false; true; false; true; false; true]]); b1 ] |
---|
89 | | INDIRECT i1 ⇒ λ_. |
---|
90 | [ ([[false; false; false; true; false; true; true; i1]]) ] |
---|
91 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
92 | | DJNZ addr1 addr2 ⇒ |
---|
93 | let b2 ≝ addr_of addr2 in |
---|
94 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;direct]] x) → ? with |
---|
95 | [ REGISTER r ⇒ λ_. |
---|
96 | [ ([[true; true; false; true; true]]) @@ r ; b2 ] |
---|
97 | | DIRECT b1 ⇒ λ_. |
---|
98 | [ ([[true; true; false; true; false; true; false; true]]); b1; b2 ] |
---|
99 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
---|
100 | | JC addr ⇒ |
---|
101 | let b1 ≝ addr_of addr in |
---|
102 | [ ([[false; true; false; false; false; false; false; false]]); b1 ] |
---|
103 | | JNC addr ⇒ |
---|
104 | let b1 ≝ addr_of addr in |
---|
105 | [ ([[false; true; false; true; false; false; false; false]]); b1 ] |
---|
106 | | JZ addr ⇒ |
---|
107 | let b1 ≝ addr_of addr in |
---|
108 | [ ([[false; true; true; false; false; false; false; false]]); b1 ] |
---|
109 | | JNZ addr ⇒ |
---|
110 | let b1 ≝ addr_of addr in |
---|
111 | [ ([[false; true; true; true; false; false; false; false]]); b1 ] |
---|
112 | | JB addr1 addr2 ⇒ |
---|
113 | let b2 ≝ addr_of addr2 in |
---|
114 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
---|
115 | [ BIT_ADDR b1 ⇒ λ_. |
---|
116 | [ ([[false; false; true; false; false; false; false; false]]); b1; b2 ] |
---|
117 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
---|
118 | | JNB addr1 addr2 ⇒ |
---|
119 | let b2 ≝ addr_of addr2 in |
---|
120 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
---|
121 | [ BIT_ADDR b1 ⇒ λ_. |
---|
122 | [ ([[false; false; true; true; false; false; false; false]]); b1; b2 ] |
---|
123 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
---|
124 | | JBC addr1 addr2 ⇒ |
---|
125 | let b2 ≝ addr_of addr2 in |
---|
126 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
---|
127 | [ BIT_ADDR b1 ⇒ λ_. |
---|
128 | [ ([[false; false; false; true; false; false; false; false]]); b1; b2 ] |
---|
129 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
---|
130 | | CJNE addrs addr3 ⇒ |
---|
131 | let b3 ≝ addr_of addr3 in |
---|
132 | match addrs with |
---|
133 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
134 | match addr2 return λx. bool_to_Prop (is_in ? [[direct;data]] x) → ? with |
---|
135 | [ DIRECT b1 ⇒ λ_. |
---|
136 | [ ([[true; false; true; true; false; true; false; true]]); b1; b3 ] |
---|
137 | | DATA b1 ⇒ λ_. |
---|
138 | [ ([[true; false; true; true; false; true; false; false]]); b1; b3 ] |
---|
139 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
140 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
141 | let b2 ≝ |
---|
142 | match addr2 return λx. bool_to_Prop (is_in ? [[data]] x) → ? with |
---|
143 | [ DATA b2 ⇒ λ_. b2 |
---|
144 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) in |
---|
145 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → list Byte with |
---|
146 | [ REGISTER r ⇒ λ_. |
---|
147 | [ ([[true; false; true; true; true]]) @@ r; b2; b3 ] |
---|
148 | | INDIRECT i1 ⇒ λ_. |
---|
149 | [ ([[true; false; true; true; false; true; true; i1]]); b2; b3 ] |
---|
150 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) |
---|
151 | ] |
---|
152 | | DIV addr1 addr2 ⇒ |
---|
153 | [ ([[true;false;false;false;false;true;false;false]]) ] |
---|
154 | | INC addr ⇒ |
---|
155 | match addr return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;dptr]] x) → ? with |
---|
156 | [ ACC_A ⇒ λ_. |
---|
157 | [ ([[false;false;false;false;false;true;false;false]]) ] |
---|
158 | | REGISTER r ⇒ λ_. |
---|
159 | [ ([[false;false;false;false;true]]) @@ r ] |
---|
160 | | DIRECT b1 ⇒ λ_. |
---|
161 | [ ([[false; false; false; false; false; true; false; true]]); b1 ] |
---|
162 | | INDIRECT i1 ⇒ λ_. |
---|
163 | [ ([[false; false; false; false; false; true; true; i1]]) ] |
---|
164 | | DPTR ⇒ λ_. |
---|
165 | [ ([[true;false;true;false;false;false;true;true]]) ] |
---|
166 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
167 | | MOV addrs ⇒ |
---|
168 | match addrs with |
---|
169 | [ inl addrs ⇒ |
---|
170 | match addrs with |
---|
171 | [ inl addrs ⇒ |
---|
172 | match addrs with |
---|
173 | [ inl addrs ⇒ |
---|
174 | match addrs with |
---|
175 | [ inl addrs ⇒ |
---|
176 | match addrs with |
---|
177 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
178 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
---|
179 | [ REGISTER r ⇒ λ_.[ ([[true;true;true;false;true]]) @@ r ] |
---|
180 | | DIRECT b1 ⇒ λ_.[ ([[true;true;true;false;false;true;false;true]]); b1 ] |
---|
181 | | INDIRECT i1 ⇒ λ_. [ ([[true;true;true;false;false;true;true;i1]]) ] |
---|
182 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;false;false]]) ; b1 ] |
---|
183 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
184 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
185 | match addr1 return λx. bool_to_Prop (is_in ? [[registr;indirect]] x) → ? with |
---|
186 | [ REGISTER r ⇒ λ_. |
---|
187 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
---|
188 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;true]]) @@ r ] |
---|
189 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;true]]) @@ r; b1 ] |
---|
190 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;true]]) @@ r; b1 ] |
---|
191 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
192 | | INDIRECT i1 ⇒ λ_. |
---|
193 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;direct;data]] x) → ? with |
---|
194 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;true;i1]]) ] |
---|
195 | | DIRECT b1 ⇒ λ_.[ ([[true;false;true;false;false;true;true;i1]]); b1 ] |
---|
196 | | DATA b1 ⇒ λ_. [ ([[false;true;true;true;false;true;true;i1]]) ; b1 ] |
---|
197 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
198 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
---|
199 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
200 | let b1 ≝ |
---|
201 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
---|
202 | [ DIRECT b1 ⇒ λ_. b1 |
---|
203 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
---|
204 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;registr;direct;indirect;data]] x) → ? with |
---|
205 | [ ACC_A ⇒ λ_.[ ([[true;true;true;true;false;true;false;true]]); b1] |
---|
206 | | REGISTER r ⇒ λ_.[ ([[true;false;false;false;true]]) @@ r; b1 ] |
---|
207 | | DIRECT b2 ⇒ λ_.[ ([[true;false;false;false;false;true;false;true]]); b1; b2 ] |
---|
208 | | INDIRECT i1 ⇒ λ_. [ ([[true;false;false;false;false;true;true;i1]]); b1 ] |
---|
209 | | DATA b2 ⇒ λ_. [ ([[false;true;true;true;false;true;false;true]]); b1; b2 ] |
---|
210 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
---|
211 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
212 | match addr2 return λx. bool_to_Prop (is_in ? [[data16]] x) → ? with |
---|
213 | [ DATA16 w ⇒ λ_. |
---|
214 | let b1_b2 ≝ split ? 8 8 w in |
---|
215 | let b1 ≝ \fst b1_b2 in |
---|
216 | let b2 ≝ \snd b1_b2 in |
---|
217 | [ ([[true;false;false;true;false;false;false;false]]); b1; b2] |
---|
218 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
---|
219 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
220 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
---|
221 | [ BIT_ADDR b1 ⇒ λ_. |
---|
222 | [ ([[true;false;true;false;false;false;true;false]]); b1 ] |
---|
223 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
---|
224 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
225 | match addr1 return λx. bool_to_Prop (is_in ? [[bit_addr]] x) → ? with |
---|
226 | [ BIT_ADDR b1 ⇒ λ_. |
---|
227 | [ ([[true;false;false;true;false;false;true;false]]); b1 ] |
---|
228 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
---|
229 | | MOVX addrs ⇒ |
---|
230 | match addrs with |
---|
231 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
232 | match addr2 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
---|
233 | [ EXT_INDIRECT i1 ⇒ λ_. |
---|
234 | [ ([[true;true;true;false;false;false;true;i1]]) ] |
---|
235 | | EXT_INDIRECT_DPTR ⇒ λ_. |
---|
236 | [ ([[true;true;true;false;false;false;false;false]]) ] |
---|
237 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
238 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
239 | match addr1 return λx. bool_to_Prop (is_in ? [[ext_indirect;ext_indirect_dptr]] x) → ? with |
---|
240 | [ EXT_INDIRECT i1 ⇒ λ_. |
---|
241 | [ ([[true;true;true;true;false;false;true;i1]]) ] |
---|
242 | | EXT_INDIRECT_DPTR ⇒ λ_. |
---|
243 | [ ([[true;true;true;true;false;false;false;false]]) ] |
---|
244 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1)] |
---|
245 | | MUL addr1 addr2 ⇒ |
---|
246 | [ ([[true;false;true;false;false;true;false;false]]) ] |
---|
247 | | NOP ⇒ |
---|
248 | [ ([[false;false;false;false;false;false;false;false]]) ] |
---|
249 | | ORL addrs ⇒ |
---|
250 | match addrs with |
---|
251 | [ inl addrs ⇒ |
---|
252 | match addrs with |
---|
253 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
254 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;data;direct;indirect]] x) → ? with |
---|
255 | [ REGISTER r ⇒ λ_.[ ([[false;true;false;false;true]]) @@ r ] |
---|
256 | | DIRECT b1 ⇒ λ_.[ ([[false;true;false;false;false;true;false;true]]); b1 ] |
---|
257 | | INDIRECT i1 ⇒ λ_. [ ([[false;true;false;false;false;true;true;i1]]) ] |
---|
258 | | DATA b1 ⇒ λ_. [ ([[false;true;false;false;false;true;false;false]]) ; b1 ] |
---|
259 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
260 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
261 | let b1 ≝ |
---|
262 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
---|
263 | [ DIRECT b1 ⇒ λ_. b1 |
---|
264 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
---|
265 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
---|
266 | [ ACC_A ⇒ λ_. |
---|
267 | [ ([[false;true;false;false;false;false;true;false]]); b1 ] |
---|
268 | | DATA b2 ⇒ λ_. |
---|
269 | [ ([[false;true;false;false;false;false;true;true]]); b1; b2 ] |
---|
270 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
---|
271 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
272 | match addr2 return λx. bool_to_Prop (is_in ? [[bit_addr;n_bit_addr]] x) → ? with |
---|
273 | [ BIT_ADDR b1 ⇒ λ_. |
---|
274 | [ ([[false;true;true;true;false;false;true;false]]); b1 ] |
---|
275 | | N_BIT_ADDR b1 ⇒ λ_. |
---|
276 | [ ([[true;false;true;false;false;false;false;false]]); b1 ] |
---|
277 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
---|
278 | | POP addr ⇒ |
---|
279 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
---|
280 | [ DIRECT b1 ⇒ λ_. |
---|
281 | [ ([[true;true;false;true;false;false;false;false]]) ; b1 ] |
---|
282 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
283 | | PUSH addr ⇒ |
---|
284 | match addr return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
---|
285 | [ DIRECT b1 ⇒ λ_. |
---|
286 | [ ([[true;true;false;false;false;false;false;false]]) ; b1 ] |
---|
287 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
288 | | RET ⇒ |
---|
289 | [ ([[false;false;true;false;false;false;true;false]]) ] |
---|
290 | | RETI ⇒ |
---|
291 | [ ([[false;false;true;true;false;false;true;false]]) ] |
---|
292 | | RL addr ⇒ |
---|
293 | [ ([[false;false;true;false;false;false;true;true]]) ] |
---|
294 | | RLC addr ⇒ |
---|
295 | [ ([[false;false;true;true;false;false;true;true]]) ] |
---|
296 | | RR addr ⇒ |
---|
297 | [ ([[false;false;false;false;false;false;true;true]]) ] |
---|
298 | | RRC addr ⇒ |
---|
299 | [ ([[false;false;false;true;false;false;true;true]]) ] |
---|
300 | | SETB addr ⇒ |
---|
301 | match addr return λx. bool_to_Prop (is_in ? [[carry;bit_addr]] x) → ? with |
---|
302 | [ CARRY ⇒ λ_. |
---|
303 | [ ([[true;true;false;true;false;false;true;true]]) ] |
---|
304 | | BIT_ADDR b1 ⇒ λ_. |
---|
305 | [ ([[true;true;false;true;false;false;true;false]]); b1 ] |
---|
306 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
307 | | SUBB addr1 addr2 ⇒ |
---|
308 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect;data]] x) → ? with |
---|
309 | [ REGISTER r ⇒ λ_. |
---|
310 | [ ([[true;false;false;true;true]]) @@ r ] |
---|
311 | | DIRECT b1 ⇒ λ_. |
---|
312 | [ ([[true;false;false;true;false;true;false;true]]); b1] |
---|
313 | | INDIRECT i1 ⇒ λ_. |
---|
314 | [ ([[true;false;false;true;false;true;true;i1]]) ] |
---|
315 | | DATA b1 ⇒ λ_. |
---|
316 | [ ([[true;false;false;true;false;true;false;false]]); b1] |
---|
317 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
318 | | SWAP addr ⇒ |
---|
319 | [ ([[true;true;false;false;false;true;false;false]]) ] |
---|
320 | | XCH addr1 addr2 ⇒ |
---|
321 | match addr2 return λx. bool_to_Prop (is_in ? [[registr;direct;indirect]] x) → ? with |
---|
322 | [ REGISTER r ⇒ λ_. |
---|
323 | [ ([[true;true;false;false;true]]) @@ r ] |
---|
324 | | DIRECT b1 ⇒ λ_. |
---|
325 | [ ([[true;true;false;false;false;true;false;true]]); b1] |
---|
326 | | INDIRECT i1 ⇒ λ_. |
---|
327 | [ ([[true;true;false;false;false;true;true;i1]]) ] |
---|
328 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
329 | | XCHD addr1 addr2 ⇒ |
---|
330 | match addr2 return λx. bool_to_Prop (is_in ? [[indirect]] x) → ? with |
---|
331 | [ INDIRECT i1 ⇒ λ_. |
---|
332 | [ ([[true;true;false;true;false;true;true;i1]]) ] |
---|
333 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
334 | | XRL addrs ⇒ |
---|
335 | match addrs with |
---|
336 | [ inl addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
337 | match addr2 return λx. bool_to_Prop (is_in ? [[data;registr;direct;indirect]] x) → ? with |
---|
338 | [ REGISTER r ⇒ λ_. |
---|
339 | [ ([[false;true;true;false;true]]) @@ r ] |
---|
340 | | DIRECT b1 ⇒ λ_. |
---|
341 | [ ([[false;true;true;false;false;true;false;true]]); b1] |
---|
342 | | INDIRECT i1 ⇒ λ_. |
---|
343 | [ ([[false;true;true;false;false;true;true;i1]]) ] |
---|
344 | | DATA b1 ⇒ λ_. |
---|
345 | [ ([[false;true;true;false;false;true;false;false]]); b1] |
---|
346 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
347 | | inr addrs ⇒ let 〈addr1,addr2〉 ≝ addrs in |
---|
348 | let b1 ≝ |
---|
349 | match addr1 return λx. bool_to_Prop (is_in ? [[direct]] x) → ? with |
---|
350 | [ DIRECT b1 ⇒ λ_. b1 |
---|
351 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr1) in |
---|
352 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_a;data]] x) → ? with |
---|
353 | [ ACC_A ⇒ λ_. |
---|
354 | [ ([[false;true;true;false;false;false;true;false]]); b1 ] |
---|
355 | | DATA b2 ⇒ λ_. |
---|
356 | [ ([[false;true;true;false;false;false;true;true]]); b1; b2 ] |
---|
357 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2)] |
---|
358 | ]. |
---|
359 | |
---|
360 | definition assembly1 ≝ |
---|
361 | λi: instruction. |
---|
362 | match i with |
---|
363 | [ ACALL addr ⇒ |
---|
364 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
---|
365 | [ ADDR11 w ⇒ λ_. |
---|
366 | let v1_v2 ≝ split ? 3 8 w in |
---|
367 | let v1 ≝ \fst v1_v2 in |
---|
368 | let v2 ≝ \snd v1_v2 in |
---|
369 | [ (v1 @@ [[true; false; false; false; true]]) ; v2 ] |
---|
370 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
371 | | AJMP addr ⇒ |
---|
372 | match addr return λx. bool_to_Prop (is_in ? [[addr11]] x) → ? with |
---|
373 | [ ADDR11 w ⇒ λ_. |
---|
374 | let v1_v2 ≝ split ? 3 8 w in |
---|
375 | let v1 ≝ \fst v1_v2 in |
---|
376 | let v2 ≝ \snd v1_v2 in |
---|
377 | [ (v1 @@ [[false; false; false; false; true]]) ; v2 ] |
---|
378 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
379 | | JMP adptr ⇒ |
---|
380 | [ ([[false;true;true;true;false;false;true;true]]) ] |
---|
381 | | LCALL addr ⇒ |
---|
382 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
---|
383 | [ ADDR16 w ⇒ λ_. |
---|
384 | let b1_b2 ≝ split ? 8 8 w in |
---|
385 | let b1 ≝ \fst b1_b2 in |
---|
386 | let b2 ≝ \snd b1_b2 in |
---|
387 | [ ([[false;false;false;true;false;false;true;false]]); b1; b2 ] |
---|
388 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
389 | | LJMP addr ⇒ |
---|
390 | match addr return λx. bool_to_Prop (is_in ? [[addr16]] x) → ? with |
---|
391 | [ ADDR16 w ⇒ λ_. |
---|
392 | let b1_b2 ≝ split ? 8 8 w in |
---|
393 | let b1 ≝ \fst b1_b2 in |
---|
394 | let b2 ≝ \snd b1_b2 in |
---|
395 | [ ([[false;false;false;false;false;false;true;false]]); b1; b2 ] |
---|
396 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
397 | | MOVC addr1 addr2 ⇒ |
---|
398 | match addr2 return λx. bool_to_Prop (is_in ? [[acc_dptr;acc_pc]] x) → ? with |
---|
399 | [ ACC_DPTR ⇒ λ_. |
---|
400 | [ ([[true;false;false;true;false;false;true;true]]) ] |
---|
401 | | ACC_PC ⇒ λ_. |
---|
402 | [ ([[true;false;false;false;false;false;true;true]]) ] |
---|
403 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr2) |
---|
404 | | SJMP addr ⇒ |
---|
405 | match addr return λx. bool_to_Prop (is_in ? [[relative]] x) → ? with |
---|
406 | [ RELATIVE b1 ⇒ λ_. |
---|
407 | [ ([[true;false;false;false;false;false;false;false]]); b1 ] |
---|
408 | | _ ⇒ λK.match K in False with [ ] ] (subaddressing_modein … addr) |
---|
409 | | RealInstruction instr ⇒ |
---|
410 | assembly_preinstruction [[ relative ]] |
---|
411 | (λx. |
---|
412 | match x return λs. bool_to_Prop (is_in ? [[ relative ]] s) → ? with |
---|
413 | [ RELATIVE r ⇒ λ_. r |
---|
414 | | _ ⇒ λabsd. ⊥ |
---|
415 | ] (subaddressing_modein … x)) instr |
---|
416 | ]. |
---|
417 | cases absd |
---|
418 | qed. |
---|
419 | |
---|
420 | definition expand_relative_jump_internal: |
---|
421 | ∀lookup_labels:Identifier → Word.∀sigma:Word → Word × bool. |
---|
422 | Identifier → Word → ([[relative]] → preinstruction [[relative]]) → |
---|
423 | list instruction |
---|
424 | ≝ |
---|
425 | λlookup_labels.λsigma.λlbl.λppc,i. |
---|
426 | let lookup_address ≝ \fst (sigma (lookup_labels lbl)) in |
---|
427 | let pc ≝ \fst (sigma ppc) in |
---|
428 | let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in |
---|
429 | let 〈upper, lower〉 ≝ split ? 8 8 result in |
---|
430 | if eq_bv ? upper (zero 8) then |
---|
431 | let address ≝ RELATIVE lower in |
---|
432 | [ RealInstruction (i address) ] |
---|
433 | else |
---|
434 | [ RealInstruction (i (RELATIVE (bitvector_of_nat ? 2))); |
---|
435 | SJMP (RELATIVE (bitvector_of_nat ? 3)); (* LJMP size? *) |
---|
436 | LJMP (ADDR16 lookup_address) |
---|
437 | ]. |
---|
438 | @ I |
---|
439 | qed. |
---|
440 | |
---|
441 | (*definition rel_jump_length_ok ≝ |
---|
442 | λlookup_address:Word. |
---|
443 | λpc:Word. |
---|
444 | Σjump_len:jump_length. |
---|
445 | (* CSC,JPB: Cheating here, use Jaap's better definition select_reljump_length *) |
---|
446 | ∀(*x,*)y. expand_relative_jump_internal_safe lookup_address jump_len (*x*) pc y ≠ None ?. |
---|
447 | |
---|
448 | lemma eject_rel_jump_length: ∀x,y. rel_jump_length_ok x y → jump_length. |
---|
449 | #x #y #p @(pi1 … p) |
---|
450 | qed. |
---|
451 | |
---|
452 | coercion eject_rel_jump_length nocomposites: |
---|
453 | ∀x,y.∀pol:rel_jump_length_ok x y. jump_length ≝ |
---|
454 | eject_rel_jump_length on _pol:(rel_jump_length_ok ??) to jump_length.*) |
---|
455 | |
---|
456 | (*definition expand_relative_jump_internal: |
---|
457 | ∀lookup_address:Word. ∀pc:Word. ([[relative]] → preinstruction [[relative]]) → |
---|
458 | list instruction |
---|
459 | ≝ λlookup_address,pc,i. |
---|
460 | match expand_relative_jump_internal_safe lookup_address pc i |
---|
461 | return λres. res ≠ None ? → ? |
---|
462 | with |
---|
463 | [ None ⇒ λabs.⊥ |
---|
464 | | Some res ⇒ λ_.res ] (pi2 … jump_len i). |
---|
465 | cases abs /2/ |
---|
466 | qed.*) |
---|
467 | |
---|
468 | definition expand_relative_jump: |
---|
469 | ∀lookup_labels.∀sigma. |
---|
470 | Word → (*jump_length →*) |
---|
471 | preinstruction Identifier → list instruction ≝ |
---|
472 | λlookup_labels: Identifier → Word. |
---|
473 | λsigma:Word → Word × bool. |
---|
474 | λppc: Word. |
---|
475 | (*λjmp_len: jump_length.*) |
---|
476 | λi: preinstruction Identifier. |
---|
477 | (*let rel_jmp ≝ RELATIVE (bitvector_of_nat ? 2) in*) |
---|
478 | match i with |
---|
479 | [ JC jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JC ?) |
---|
480 | | JNC jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JNC ?) |
---|
481 | | JB baddr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JB ? baddr) |
---|
482 | | JZ jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JZ ?) |
---|
483 | | JNZ jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JNZ ?) |
---|
484 | | JBC baddr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JBC ? baddr) |
---|
485 | | JNB baddr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JNB ? baddr) |
---|
486 | | CJNE addr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (CJNE ? addr) |
---|
487 | | DJNZ addr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (DJNZ ? addr) |
---|
488 | | ADD arg1 arg2 ⇒ [ ADD ? arg1 arg2 ] |
---|
489 | | ADDC arg1 arg2 ⇒ [ ADDC ? arg1 arg2 ] |
---|
490 | | SUBB arg1 arg2 ⇒ [ SUBB ? arg1 arg2 ] |
---|
491 | | INC arg ⇒ [ INC ? arg ] |
---|
492 | | DEC arg ⇒ [ DEC ? arg ] |
---|
493 | | MUL arg1 arg2 ⇒ [ MUL ? arg1 arg2 ] |
---|
494 | | DIV arg1 arg2 ⇒ [ DIV ? arg1 arg2 ] |
---|
495 | | DA arg ⇒ [ DA ? arg ] |
---|
496 | | ANL arg ⇒ [ ANL ? arg ] |
---|
497 | | ORL arg ⇒ [ ORL ? arg ] |
---|
498 | | XRL arg ⇒ [ XRL ? arg ] |
---|
499 | | CLR arg ⇒ [ CLR ? arg ] |
---|
500 | | CPL arg ⇒ [ CPL ? arg ] |
---|
501 | | RL arg ⇒ [ RL ? arg ] |
---|
502 | | RR arg ⇒ [ RR ? arg ] |
---|
503 | | RLC arg ⇒ [ RLC ? arg ] |
---|
504 | | RRC arg ⇒ [ RRC ? arg ] |
---|
505 | | SWAP arg ⇒ [ SWAP ? arg ] |
---|
506 | | MOV arg ⇒ [ MOV ? arg ] |
---|
507 | | MOVX arg ⇒ [ MOVX ? arg ] |
---|
508 | | SETB arg ⇒ [ SETB ? arg ] |
---|
509 | | PUSH arg ⇒ [ PUSH ? arg ] |
---|
510 | | POP arg ⇒ [ POP ? arg ] |
---|
511 | | XCH arg1 arg2 ⇒ [ XCH ? arg1 arg2 ] |
---|
512 | | XCHD arg1 arg2 ⇒ [ XCHD ? arg1 arg2 ] |
---|
513 | | RET ⇒ [ RET ? ] |
---|
514 | | RETI ⇒ [ RETI ? ] |
---|
515 | | NOP ⇒ [ RealInstruction (NOP ?) ] |
---|
516 | ]. |
---|
517 | |
---|
518 | definition expand_pseudo_instruction: |
---|
519 | ∀lookup_labels.∀sigma.Word → ? → pseudo_instruction → list instruction ≝ |
---|
520 | λlookup_labels:Identifier → Word. |
---|
521 | λsigma:Word → Word × bool. |
---|
522 | λppc. |
---|
523 | λlookup_datalabels:Identifier → Word. |
---|
524 | λi. |
---|
525 | match i with |
---|
526 | [ Cost cost ⇒ [ ] |
---|
527 | | Comment comment ⇒ [ ] |
---|
528 | | Call call ⇒ |
---|
529 | let 〈addr_5, resta〉 ≝ split ? 5 11 (\fst (sigma (lookup_labels call))) in |
---|
530 | let 〈pc, do_a_long〉 ≝ sigma ppc in |
---|
531 | let 〈pc_5, restp〉 ≝ split ? 5 11 pc in |
---|
532 | if eq_bv ? addr_5 pc_5 ∧ ¬ do_a_long then |
---|
533 | let address ≝ ADDR11 resta in |
---|
534 | [ ACALL address ] |
---|
535 | else |
---|
536 | let address ≝ ADDR16 (\fst (sigma (lookup_labels call))) in |
---|
537 | [ LCALL address ] |
---|
538 | | Mov d trgt ⇒ |
---|
539 | let address ≝ DATA16 (lookup_datalabels trgt) in |
---|
540 | [ RealInstruction (MOV ? (inl ? ? (inl ? ? (inr ? ? 〈DPTR, address〉))))] |
---|
541 | | Instruction instr ⇒ expand_relative_jump lookup_labels sigma ppc instr |
---|
542 | | Jmp jmp ⇒ |
---|
543 | let 〈pc, do_a_long〉 ≝ sigma ppc in |
---|
544 | let lookup_address ≝ \fst (sigma (lookup_labels jmp)) in |
---|
545 | let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in |
---|
546 | let 〈upper, lower〉 ≝ split ? 8 8 result in |
---|
547 | if eq_bv ? upper (zero 8) ∧ ¬ do_a_long then |
---|
548 | let address ≝ RELATIVE lower in |
---|
549 | [ SJMP address ] |
---|
550 | else |
---|
551 | let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 (lookup_labels jmp) in |
---|
552 | let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 pc in |
---|
553 | if eq_bv ? fst_5_addr fst_5_pc ∧ ¬ do_a_long then |
---|
554 | let address ≝ ADDR11 rest_addr in |
---|
555 | [ AJMP address ] |
---|
556 | else |
---|
557 | let address ≝ ADDR16 lookup_address in |
---|
558 | [ LJMP address ] |
---|
559 | ]. |
---|
560 | @ I |
---|
561 | qed. |
---|
562 | |
---|
563 | (* |
---|
564 | (*X? |
---|
565 | definition jump_length_ok ≝ |
---|
566 | λlookup_labels:Identifier → Word. |
---|
567 | λpc:Word. |
---|
568 | Σjump_len:jump_length. |
---|
569 | (* CSC,JPB: Cheating here, use Jaap's better definition select_reljump_length *) |
---|
570 | ∀x,y.expand_pseudo_instruction_safe lookup_labels pc jump_len x y ≠ None ?. |
---|
571 | *) |
---|
572 | |
---|
573 | lemma eject_jump_length: ∀x,y. jump_length_ok x y → jump_length. |
---|
574 | #x #y #p @(pi1 … p) |
---|
575 | qed. |
---|
576 | |
---|
577 | coercion eject_jump_length nocomposites: |
---|
578 | ∀x,y.∀pol:jump_length_ok x y. jump_length ≝ |
---|
579 | eject_jump_length on _pol:(jump_length_ok ??) to jump_length. |
---|
580 | |
---|
581 | definition expand_pseudo_instruction: |
---|
582 | ∀lookup_labels:Identifier → Word. ∀pc:Word. jump_length_ok lookup_labels pc → |
---|
583 | ? → pseudo_instruction → list instruction ≝ |
---|
584 | λlookup_labels,pc,jump_len,lookup_datalabels,i. |
---|
585 | match expand_pseudo_instruction_safe lookup_labels pc jump_len lookup_datalabels i |
---|
586 | return λres. res ≠ None ? → ? |
---|
587 | with |
---|
588 | [ None ⇒ λabs.⊥ |
---|
589 | | Some res ⇒ λ_.res ] (pi2 … jump_len lookup_datalabels i). |
---|
590 | cases abs /2/ |
---|
591 | qed. |
---|
592 | *) |
---|
593 | (*X? |
---|
594 | definition policy_type ≝ |
---|
595 | λlookup_labels:Identifier → Word. |
---|
596 | ∀pc:Word. jump_length_ok lookup_labels pc. |
---|
597 | *) |
---|
598 | |
---|
599 | (*definition policy_type2 ≝ |
---|
600 | λprogram. |
---|
601 | Σpol:Word → jump_length. |
---|
602 | let lookup_labels ≝ |
---|
603 | (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) in |
---|
604 | ∀pc:Word. let jump_len ≝ pol pc in |
---|
605 | ∀x,y.expand_pseudo_instruction_safe lookup_labels pc jump_len x y ≠ None ?.*) |
---|
606 | |
---|
607 | definition assembly_1_pseudoinstruction ≝ |
---|
608 | λlookup_labels. |
---|
609 | λsigma. |
---|
610 | λppc: Word. |
---|
611 | λlookup_datalabels. |
---|
612 | λi. |
---|
613 | let pseudos ≝ expand_pseudo_instruction lookup_labels sigma ppc lookup_datalabels i in |
---|
614 | let mapped ≝ map ? ? assembly1 pseudos in |
---|
615 | let flattened ≝ flatten ? mapped in |
---|
616 | let pc_len ≝ length ? flattened in |
---|
617 | 〈pc_len, flattened〉. |
---|
618 | |
---|
619 | definition instruction_size ≝ |
---|
620 | λlookup_labels.λsigma.λppc.λi. |
---|
621 | \fst (assembly_1_pseudoinstruction lookup_labels sigma ppc (λx.zero …) i). |
---|
622 | |
---|
623 | (* Jaap: never used |
---|
624 | lemma fetch_pseudo_instruction_prefix: |
---|
625 | ∀prefix.∀x.∀ppc.ppc < (|prefix|) → |
---|
626 | fetch_pseudo_instruction prefix (bitvector_of_nat ? ppc) = |
---|
627 | fetch_pseudo_instruction (prefix@x) (bitvector_of_nat ? ppc). |
---|
628 | #prefix #x #ppc elim prefix |
---|
629 | [ #Hppc @⊥ @(absurd … Hppc) @le_to_not_lt @le_O_n |
---|
630 | | #h #t #Hind #Hppc whd in match (fetch_pseudo_instruction ??); |
---|
631 | whd in match (fetch_pseudo_instruction ((h::t)@x) ?); |
---|
632 | >nth_append_first |
---|
633 | [ // |
---|
634 | | >nat_of_bitvector_bitvector_of_nat |
---|
635 | [ @Hppc |
---|
636 | | cases daemon (* XXX invariant *) |
---|
637 | ] |
---|
638 | ] |
---|
639 | ] |
---|
640 | qed. |
---|
641 | *) |
---|
642 | |
---|
643 | (* |
---|
644 | (* This establishes the correspondence between pseudo program counters and |
---|
645 | program counters. It is at the heart of the proof. *) |
---|
646 | (*CSC: code taken from build_maps *) |
---|
647 | definition sigma00: |
---|
648 | ∀jump_expansion:policy_type2.∀l:list labelled_instruction.? → |
---|
649 | (Σppc_pc_map:ℕ×(ℕ×(BitVectorTrie Word 16)). |
---|
650 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
651 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
652 | ppc = |l| ∧ |
---|
653 | (ppc = |l| → |
---|
654 | (bvt_lookup ?? (bitvector_of_nat ? ppc) sigma_map (zero ?) = (bitvector_of_nat ? program_counter)) ∧ |
---|
655 | (∀x.x < |l| → |
---|
656 | ∀pi.\fst (fetch_pseudo_instruction l (bitvector_of_nat ? x)) = pi → |
---|
657 | let pc_x ≝ bvt_lookup ?? (bitvector_of_nat 16 x) sigma_map (zero ?) in |
---|
658 | bvt_lookup ?? (bitvector_of_nat 16 (S x)) sigma_map (zero ?) = |
---|
659 | bitvector_of_nat 16 ((nat_of_bitvector ? pc_x) + |
---|
660 | (\fst (assembly_1_pseudoinstruction lookup_labels(*X?(λx.pc_x)*) (jump_expansion (*?(λx.pc_x)*)) pc_x |
---|
661 | (λx.zero ?) pi))))) |
---|
662 | ) ≝ |
---|
663 | (*?*)λlookup_labels. |
---|
664 | λjump_expansion(*X?: policy_type2*). |
---|
665 | λl:list labelled_instruction. |
---|
666 | λacc. |
---|
667 | foldl_strong ? |
---|
668 | (λprefix.(Σppc_pc_map:ℕ×(ℕ×(BitVectorTrie Word 16)). |
---|
669 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
670 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
671 | (ppc = |prefix|) ∧ |
---|
672 | (ppc = |prefix| → |
---|
673 | (bvt_lookup ?? (bitvector_of_nat ? ppc) sigma_map (zero ?) = (bitvector_of_nat ? program_counter)) ∧ |
---|
674 | (∀x.x < |prefix| → |
---|
675 | ∀pi.\fst (fetch_pseudo_instruction l (bitvector_of_nat ? x)) = pi → |
---|
676 | let pc_x ≝ bvt_lookup ?? (bitvector_of_nat 16 x) sigma_map (zero ?) in |
---|
677 | bvt_lookup ?? (bitvector_of_nat 16 (S x)) sigma_map (zero ?) = |
---|
678 | bitvector_of_nat 16 ((nat_of_bitvector ? pc_x) + |
---|
679 | (\fst (assembly_1_pseudoinstruction (*X?(λx.pc_x)*)lookup_labels (jump_expansion (*X?(λx.pc_x)*)) pc_x |
---|
680 | (λx.zero ?) pi)))))) |
---|
681 | ) |
---|
682 | l |
---|
683 | (λhd.λi.λtl.λp.λppc_pc_map. |
---|
684 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
685 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
686 | let 〈label, i〉 ≝ i in |
---|
687 | let 〈pc,ignore〉 ≝ construct_costs lookup_labels program_counter (jump_expansion (*X?(λx.bitvector_of_nat ? program_counter)*)) ppc (Stub …) i in |
---|
688 | 〈S ppc, 〈pc, insert ?? (bitvector_of_nat 16 (S ppc)) (bitvector_of_nat 16 pc) sigma_map〉〉 |
---|
689 | ) acc. |
---|
690 | cases i in p; #label #ins #p @pair_elim #new_ppc #x normalize nodelta cases x -x #old_pc #old_map |
---|
691 | @pair_elim #new_pc #ignore #Hc #Heq normalize nodelta @conj |
---|
692 | [ lapply (pi2 ?? ppc_pc_map) >p1 >p2 normalize nodelta #Hind |
---|
693 | <(pair_eq1 ?????? Heq) >(proj1 ?? Hind) >append_length <commutative_plus normalize @refl |
---|
694 | | #Hnew <(pair_eq2 ?????? (pair_eq2 ?????? Heq)) <(pair_eq1 ?????? Heq) @conj |
---|
695 | [ >lookup_insert_hit >(pair_eq1 ?????? (pair_eq2 ?????? Heq)) @refl |
---|
696 | | #x <(pair_eq1 ?????? Heq) >append_length <commutative_plus #Hx normalize in Hx; |
---|
697 | #pi #Hpi <(pair_eq2 ?????? (pair_eq2 ?????? Heq)) <(pair_eq1 ?????? Heq) in Hnew; |
---|
698 | >append_length <commutative_plus #Hnew normalize in Hnew; >(injective_S … Hnew) |
---|
699 | elim (le_to_or_lt_eq … Hx) -Hx #Hx |
---|
700 | [ lapply (pi2 ?? ppc_pc_map) >p1 >p2 normalize nodelta #Hind |
---|
701 | lapply (proj2 ?? ((proj2 ?? Hind) (proj1 ?? Hind)) x (le_S_S_to_le … Hx) pi Hpi) |
---|
702 | -Hind #Hind >lookup_insert_miss |
---|
703 | [2: @bitvector_of_nat_abs |
---|
704 | [3: @lt_to_not_eq @Hx |
---|
705 | |1: @(transitive_le … Hx) |
---|
706 | ] |
---|
707 | cases daemon (* XXX invariant *) |
---|
708 | ] |
---|
709 | >lookup_insert_miss |
---|
710 | [2: @bitvector_of_nat_abs |
---|
711 | [3: @lt_to_not_eq @(transitive_le … (le_S_S_to_le … Hx)) @le_S @le_n |
---|
712 | |1: @(transitive_le … (le_S_S_to_le … Hx)) |
---|
713 | ] |
---|
714 | cases daemon (* XXX invariant *) |
---|
715 | ] |
---|
716 | @Hind |
---|
717 | | lapply (pi2 ?? ppc_pc_map) >p1 >p2 normalize nodelta |
---|
718 | #Hind lapply (proj1 ?? ((proj2 ?? Hind) (proj1 ?? Hind))) -Hind |
---|
719 | >(injective_S … Hnew) #Hind <(injective_S … Hx) >lookup_insert_hit >lookup_insert_miss |
---|
720 | [2: @bitvector_of_nat_abs |
---|
721 | [3: @lt_to_not_eq @le_n |
---|
722 | |1: @(transitive_le ??? (le_n (S x))) |
---|
723 | ] |
---|
724 | cases daemon (* XXX invariant *) |
---|
725 | ] |
---|
726 | >p in Hpi; whd in match (fetch_pseudo_instruction ??); >nth_append_second |
---|
727 | >nat_of_bitvector_bitvector_of_nat >(injective_S … Hx) |
---|
728 | [3: @le_n] |
---|
729 | [2,3: cases daemon (* XXX invariant *)] |
---|
730 | <minus_n_n cases (half_add ???) #x #y normalize nodelta -x -y #Heq <Heq |
---|
731 | whd in match (construct_costs ?????) in Hc; whd in match (assembly_1_pseudoinstruction ?????); |
---|
732 | cases ins in p Hc; normalize nodelta |
---|
733 | [1,2,4,5: #x #p >Hind #H <(pair_eq1 ?????? H) >commutative_plus >nat_of_bitvector_bitvector_of_nat |
---|
734 | [1,3,5,7: @refl |
---|
735 | |2,4,6,8: cases daemon (* XXX invariant *) |
---|
736 | ] |
---|
737 | |3: #c #p >Hind #H <(pair_eq1 ?????? H) >nat_of_bitvector_bitvector_of_nat |
---|
738 | [2: cases daemon (* XXX invariant *) ] |
---|
739 | whd in match (expand_pseudo_instruction ?????); normalize <plus_n_O @refl |
---|
740 | |6: #x #y #p >Hind #H <(pair_eq1 ?????? H) >commutative_plus >nat_of_bitvector_bitvector_of_nat |
---|
741 | [ @refl |
---|
742 | | cases daemon (* XXX invariant *) |
---|
743 | ] |
---|
744 | ] |
---|
745 | ] |
---|
746 | ] |
---|
747 | ] |
---|
748 | qed. |
---|
749 | |
---|
750 | definition sigma0: pseudo_assembly_program → policy_type2 → (nat × (nat × (BitVectorTrie Word 16))) ≝ |
---|
751 | λprog. |
---|
752 | λjump_expansion. |
---|
753 | sigma00 jump_expansion (\snd prog) |
---|
754 | 〈0, 〈0, (insert … (bitvector_of_nat ? 0) (bitvector_of_nat ? 0) (Stub …))〉〉. |
---|
755 | normalize nodelta @conj |
---|
756 | [ / by refl/ |
---|
757 | | #H @conj |
---|
758 | [ >lookup_insert_hit @refl |
---|
759 | | #x #Hx @⊥ @(absurd … Hx) @le_to_not_lt @le_O_n |
---|
760 | ] |
---|
761 | ] |
---|
762 | qed. |
---|
763 | |
---|
764 | definition tech_pc_sigma00: pseudo_assembly_program → policy_type2 → |
---|
765 | list labelled_instruction → (nat × nat) ≝ |
---|
766 | λprogram,jump_expansion,instr_list. |
---|
767 | let 〈ppc,pc_sigma_map〉 ≝ sigma00 jump_expansion instr_list |
---|
768 | 〈0, 〈0, (insert … (bitvector_of_nat ? 0) (bitvector_of_nat ? 0) (Stub ? ?))〉〉 in |
---|
769 | (* acc copied from sigma0 *) |
---|
770 | let 〈pc,map〉 ≝ pc_sigma_map in |
---|
771 | 〈ppc,pc〉. |
---|
772 | normalize nodelta @conj |
---|
773 | [ / by refl/ |
---|
774 | | #H @conj |
---|
775 | [ >lookup_insert_hit @refl |
---|
776 | | #x #Hx @⊥ @(absurd … Hx) @le_to_not_lt @le_O_n |
---|
777 | ] |
---|
778 | ] |
---|
779 | qed. |
---|
780 | |
---|
781 | definition sigma_safe: pseudo_assembly_program → policy_type2 → |
---|
782 | option (Word → Word) ≝ |
---|
783 | λinstr_list,jump_expansion. |
---|
784 | let 〈ppc,pc_sigma_map〉 ≝ sigma0 instr_list jump_expansion in |
---|
785 | let 〈pc, sigma_map〉 ≝ pc_sigma_map in |
---|
786 | if gtb pc (2^16) then |
---|
787 | None ? |
---|
788 | else |
---|
789 | Some ? (λx. lookup … x sigma_map (zero …)). *) |
---|
790 | |
---|
791 | (* stuff about policy *) |
---|
792 | |
---|
793 | (*definition policy_ok ≝ λjump_expansion,p. sigma_safe p jump_expansion ≠ None ….*) |
---|
794 | |
---|
795 | (*definition policy ≝ λp. Σjump_expansion:policy_type2. policy_ok jump_expansion p.*) |
---|
796 | |
---|
797 | (*lemma eject_policy: ∀p. policy p → policy_type2. |
---|
798 | #p #pol @(pi1 … pol) |
---|
799 | qed. |
---|
800 | |
---|
801 | coercion eject_policy nocomposites: ∀p.∀pol:policy p. policy_type2 ≝ eject_policy on _pol:(policy ?) to policy_type2. |
---|
802 | |
---|
803 | definition sigma: ∀p:pseudo_assembly_program. policy p → Word → Word ≝ |
---|
804 | λp,policy. |
---|
805 | match sigma_safe p (pi1 … policy) return λr:option (Word → Word). r ≠ None … → Word → Word with |
---|
806 | [ None ⇒ λabs. ⊥ |
---|
807 | | Some r ⇒ λ_.r] (pi2 … policy). |
---|
808 | cases abs /2 by / |
---|
809 | qed.*) |
---|
810 | |
---|
811 | (*CSC: Main axiom here, needs to be proved soon! *) |
---|
812 | (*lemma snd_assembly_1_pseudoinstruction_ok: |
---|
813 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
814 | ∀ppc:Word.∀pi,lookup_labels,lookup_datalabels. |
---|
815 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
816 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
---|
817 | (nat_of_bitvector 16 ppc) < |\snd program| → |
---|
818 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
819 | let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels (pol lookup_labels) (sigma program pol ppc) lookup_datalabels pi) in |
---|
820 | sigma program pol (add ? ppc (bitvector_of_nat ? 1)) = |
---|
821 | bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len). |
---|
822 | #program #pol #ppc #pi #lookup_labels #lookup_datalabels #Hll #Hldl #Hppc |
---|
823 | lapply (refl … (sigma0 program pol)) whd in match (sigma0 ??) in ⊢ (??%? → ?); |
---|
824 | cases (sigma00 ???) #x #Hpmap #EQ |
---|
825 | whd in match (sigma ???); |
---|
826 | whd in match (sigma program pol (add ???)); |
---|
827 | whd in match sigma_safe; normalize nodelta |
---|
828 | (*Problem1: backtracking cases (sigma0 program pol)*) |
---|
829 | generalize in match (pi2 ???); whd in match policy_ok; normalize nodelta |
---|
830 | whd in match sigma_safe; normalize nodelta <EQ cases x in Hpmap EQ; -x #final_ppc #x |
---|
831 | cases x -x #final_pc #smap normalize nodelta #Hpmap #EQ #Heq #Hfetch cases (gtb final_pc (2^16)) in Heq; |
---|
832 | normalize nodelta |
---|
833 | [ #abs @⊥ @(absurd ?? abs) @refl |
---|
834 | | #_ lapply (proj1 ?? ((proj2 ?? Hpmap) (proj1 ?? Hpmap))) #Hpmap1 |
---|
835 | lapply ((proj2 ?? ((proj2 ?? Hpmap) (proj1 ?? Hpmap))) (nat_of_bitvector 16 ppc) Hppc) #Hpmap2 -Hpmap |
---|
836 | <(bitvector_of_nat_nat_of_bitvector 16 ppc) >add_SO |
---|
837 | |
---|
838 | >(Hpmap2 ? (refl …)) @eq_f @eq_f2 [%] |
---|
839 | >bitvector_of_nat_nat_of_bitvector |
---|
840 | >Hfetch lapply Hfetch lapply pi |
---|
841 | |
---|
842 | |
---|
843 | whd in match assembly_1_pseudoinstruction; normalize nodelta |
---|
844 | |
---|
845 | qed.*) |
---|
846 | |
---|
847 | |
---|
848 | (*example sigma_0: ∀p,pol. sigma p pol (bitvector_of_nat ? 0) = bitvector_of_nat ? 0. |
---|
849 | cases daemon. |
---|
850 | qed.*) |
---|
851 | |
---|
852 | (*CSC: FALSE!!!*) |
---|
853 | axiom fetch_pseudo_instruction_split: |
---|
854 | ∀instr_list,ppc. |
---|
855 | ∃pre,suff,lbl. |
---|
856 | (pre @ [〈lbl,\fst (fetch_pseudo_instruction instr_list ppc)〉]) @ suff = instr_list. |
---|
857 | |
---|
858 | (*lemma sigma00_append: |
---|
859 | ∀jump_expansion,l1,l2. |
---|
860 | ∀acc:ℕ×ℕ×(BitVectorTrie Word 16). |
---|
861 | sigma00 jump_expansion (l1@l2) acc = |
---|
862 | sigma00 jump_expansion |
---|
863 | l2 (pi1 ?? (sigma00 jump_expansion l1 acc)).*) |
---|
864 | |
---|
865 | (* lemma sigma00_strict: |
---|
866 | ∀jump_expansion,l,acc. acc = None ? → |
---|
867 | sigma00 jump_expansion l acc = None …. |
---|
868 | #jump_expansion #l elim l |
---|
869 | [ #acc #H >H % |
---|
870 | | #hd #tl #IH #acc #H >H change with (sigma00 ? tl ? = ?) @IH % ] |
---|
871 | qed. |
---|
872 | |
---|
873 | lemma policy_ok_prefix_ok: |
---|
874 | ∀program.∀pol:policy program.∀suffix,prefix. |
---|
875 | prefix@suffix = \snd program → |
---|
876 | sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) ≠ None …. |
---|
877 | * #preamble #instr_list #pol #suffix #prefix #prf whd in prf:(???%); |
---|
878 | generalize in match (pi2 ?? pol); whd in prf:(???%); <prf in pol; #pol |
---|
879 | whd in match policy_ok; whd in match sigma_safe; whd in match sigma0; |
---|
880 | normalize nodelta >sigma00_append |
---|
881 | cases (sigma00 ?? prefix ?) |
---|
882 | [2: #x #_ % #abs destruct(abs) |
---|
883 | | * #abs @⊥ @abs >sigma00_strict % ] |
---|
884 | qed. |
---|
885 | |
---|
886 | lemma policy_ok_prefix_hd_ok: |
---|
887 | ∀program.∀pol:policy program.∀suffix,hd,prefix,ppc_pc_map. |
---|
888 | (prefix@[hd])@suffix = \snd program → |
---|
889 | Some ? ppc_pc_map = sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) → |
---|
890 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
891 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
892 | let 〈label, i〉 ≝ hd in |
---|
893 | construct_costs_safe program pol ppc program_counter (Stub …) i ≠ None …. |
---|
894 | * #preamble #instr_list #pol #suffix #hd #prefix #ppc_pc_map #EQ1 #EQ2 |
---|
895 | generalize in match (policy_ok_prefix_ok 〈preamble,instr_list〉 pol suffix |
---|
896 | (prefix@[hd]) EQ1) in ⊢ ?; >sigma00_append <EQ2 whd in ⊢ (?(??%?) → ?); |
---|
897 | @pair_elim #ppc #pc_map #EQ3 normalize nodelta |
---|
898 | @pair_elim #pc #map #EQ4 normalize nodelta |
---|
899 | @pair_elim #l' #i' #EQ5 normalize nodelta |
---|
900 | cases (construct_costs_safe ??????) normalize |
---|
901 | [* #abs @⊥ @abs % | #X #_ % #abs destruct(abs)] |
---|
902 | qed. *) |
---|
903 | |
---|
904 | (* JPB,CSC: this definition is now replaced by the expand_pseudo_instruction higher up |
---|
905 | definition expand_pseudo_instruction: |
---|
906 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
907 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pc. |
---|
908 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
909 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
---|
910 | let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in |
---|
911 | pc = sigma program pol ppc → |
---|
912 | Σres:list instruction. Some … res = expand_pseudo_instruction_safe pc (lookup_labels pi) lookup_datalabels (pol ppc) pi |
---|
913 | ≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pc,prf1,prf2,prf3. |
---|
914 | match expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc (pol ppc) (\fst (fetch_pseudo_instruction (\snd program) ppc)) with |
---|
915 | [ None ⇒ let dummy ≝ [ ] in dummy |
---|
916 | | Some res ⇒ res ]. |
---|
917 | [ @⊥ whd in p:(??%??); |
---|
918 | generalize in match (pi2 ?? pol); whd in ⊢ (% → ?); |
---|
919 | whd in ⊢ (?(??%?) → ?); change with (sigma00 ????) in ⊢ (?(??(match % with [_ ⇒ ? | _ ⇒ ?])?) → ?); |
---|
920 | generalize in match (refl … (sigma00 program pol (\snd program) (Some ? 〈O,〈O,Stub (BitVector 16) 16〉〉))); |
---|
921 | cases (sigma00 ????) in ⊢ (??%? → %); normalize nodelta [#_ * #abs @abs %] |
---|
922 | #res #K |
---|
923 | cases (fetch_pseudo_instruction_split (\snd program) ppc) #pre * #suff * #lbl #EQ1 |
---|
924 | generalize in match (policy_ok_prefix_hd_ok program pol … EQ1 ?) in ⊢ ?; |
---|
925 | cases daemon (* CSC: XXXXXXXX Ero qui |
---|
926 | |
---|
927 | [3: @policy_ok_prefix_ok ] |
---|
928 | | sigma00 program pol pre |
---|
929 | |
---|
930 | |
---|
931 | |
---|
932 | QUA USARE LEMMA policy_ok_prefix_hd_ok combinato a lemma da fare che |
---|
933 | fetch ppc = hd sse program = pre @ [hd] @ tl e |pre| = ppc |
---|
934 | per concludere construct_costs_safe ≠ None *) |
---|
935 | | >p %] |
---|
936 | qed. *) |
---|
937 | |
---|
938 | (* MAIN AXIOM HERE, HIDDEN USING cases daemon *) |
---|
939 | (* definition assembly_1_pseudoinstruction': |
---|
940 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
941 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
942 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
943 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
---|
944 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
945 | Σres:(nat × (list Byte)). |
---|
946 | res = assembly_1_pseudoinstruction program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi ∧ |
---|
947 | let 〈len,code〉 ≝ res in |
---|
948 | sigma program pol (add ? ppc (bitvector_of_nat ? 1)) = |
---|
949 | bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len) |
---|
950 | ≝ λprogram: pseudo_assembly_program. |
---|
951 | λpol: policy program. |
---|
952 | λppc: Word. |
---|
953 | λlookup_labels. |
---|
954 | λlookup_datalabels. |
---|
955 | λpi. |
---|
956 | λprf1,prf2,prf3. |
---|
957 | assembly_1_pseudoinstruction program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi. |
---|
958 | [ @⊥ elim pi in p; [*] |
---|
959 | try (#ARG1 #ARG2 #ARG3 #abs) try (#ARG1 #ARG2 #abs) try (#ARG1 #abs) try #abs |
---|
960 | generalize in match (jmeq_to_eq ??? abs); -abs; #abs whd in abs:(??%?); try destruct(abs) |
---|
961 | whd in abs:(??match % with [_ ⇒ ? | _ ⇒ ?]?); |
---|
962 | (* WRONG HERE, NEEDS LEMMA SAYING THAT THE POLICY DOES NOT RETURN MEDIUM! *) |
---|
963 | cases daemon |
---|
964 | | % [ >p %] |
---|
965 | cases res in p ⊢ %; -res; #len #code #EQ normalize nodelta; |
---|
966 | (* THIS SHOULD BE TRUE INSTEAD *) |
---|
967 | cases daemon] |
---|
968 | qed. |
---|
969 | |
---|
970 | definition assembly_1_pseudoinstruction: |
---|
971 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
972 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
973 | lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) → |
---|
974 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
---|
975 | \fst (fetch_pseudo_instruction (\snd program) ppc) = pi → |
---|
976 | nat × (list Byte) |
---|
977 | ≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pi,prf1,prf2,prf3. |
---|
978 | assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 |
---|
979 | prf2 prf3. |
---|
980 | |
---|
981 | lemma assembly_1_pseudoinstruction_ok1: |
---|
982 | ∀program:pseudo_assembly_program.∀pol: policy program. |
---|
983 | ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi. |
---|
984 | ∀prf1:lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)). |
---|
985 | ∀prf2:lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)). |
---|
986 | ∀prf3:\fst (fetch_pseudo_instruction (\snd program) ppc) = pi. |
---|
987 | Some … (assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3) |
---|
988 | = assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi. |
---|
989 | #program #pol #ppc #lookup_labels #lookup_datalabels #pi #prf1 #prf2 #prf3 |
---|
990 | cases (pi2 … (assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3)) |
---|
991 | #H1 #_ @H1 |
---|
992 | qed. *) |
---|
993 | |
---|
994 | (* MAIN AXIOM HERE, HIDDEN USING cases daemon *) |
---|
995 | (* definition construct_costs': |
---|
996 | ∀program. ∀pol:policy program. ∀ppc,pc,costs,i. |
---|
997 | Σres:(nat × (BitVectorTrie costlabel 16)). Some … res = construct_costs_safe program pol ppc pc costs i |
---|
998 | ≝ |
---|
999 | λprogram.λpol: policy program.λppc,pc,costs,i. |
---|
1000 | match construct_costs_safe program pol ppc pc costs i with |
---|
1001 | [ None ⇒ let dummy ≝ 〈0, Stub costlabel 16〉 in dummy |
---|
1002 | | Some res ⇒ res ]. |
---|
1003 | [ cases daemon |
---|
1004 | | >p %] |
---|
1005 | qed. |
---|
1006 | |
---|
1007 | definition construct_costs ≝ |
---|
1008 | λprogram,pol,ppc,pc,costs,i. pi1 … (construct_costs' program pol ppc pc costs i). *) |
---|
1009 | |
---|
1010 | (* |
---|
1011 | axiom suffix_of: ∀A:Type[0]. ∀l,prefix:list A. list A. |
---|
1012 | axiom suffix_of_ok: ∀A,l,prefix. prefix @ suffix_of A l prefix = l. |
---|
1013 | |
---|
1014 | axiom foldl_strong_step: |
---|
1015 | ∀A:Type[0]. |
---|
1016 | ∀P: list A → Type[0]. |
---|
1017 | ∀l: list A. |
---|
1018 | ∀H: ∀prefix,hd,tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]). |
---|
1019 | ∀acc: P [ ]. |
---|
1020 | ∀Q: ∀prefix. P prefix → Prop. |
---|
1021 | ∀HQ: ∀prefix,hd,tl.∀prf: l = prefix @ [hd] @ tl. |
---|
1022 | ∀acc: P prefix. Q prefix acc → Q (prefix @ [hd]) (H prefix hd tl prf acc). |
---|
1023 | Q [ ] acc → |
---|
1024 | Q l (foldl_strong A P l H acc). |
---|
1025 | (* |
---|
1026 | #A #P #l #H #acc #Q #HQ #Hacc normalize; |
---|
1027 | generalize in match |
---|
1028 | (foldl_strong ? |
---|
1029 | (λpre. Q pre (foldl_strong_internal A P l (suffix_of A l pre) ? [ ] pre acc ?)) |
---|
1030 | l ? Hacc) |
---|
1031 | [3: >suffix_of_ok % | 2: #prefix #hd #tl #EQ @(H prefix hd (tl@suffix_of A l pre) EQ) ] |
---|
1032 | [2: #prefix #hd #tl #prf #X whd in ⊢ (??%) |
---|
1033 | #K |
---|
1034 | |
---|
1035 | generalize in match |
---|
1036 | (foldl_strong ? |
---|
1037 | (λpre. Q pre (foldl_strong_internal A P l H pre (suffix_of A l pre) acc (suffix_of_ok A l pre)))) |
---|
1038 | [2: @H |
---|
1039 | *) |
---|
1040 | |
---|
1041 | axiom foldl_elim: |
---|
1042 | ∀A:Type[0]. |
---|
1043 | ∀B: Type[0]. |
---|
1044 | ∀H: A → B → A. |
---|
1045 | ∀acc: A. |
---|
1046 | ∀l: list B. |
---|
1047 | ∀Q: A → Prop. |
---|
1048 | (∀acc:A.∀b:B. Q acc → Q (H acc b)) → |
---|
1049 | Q acc → |
---|
1050 | Q (foldl A B H acc l). |
---|
1051 | *) |
---|
1052 | |
---|
1053 | (* |
---|
1054 | lemma tech_pc_sigma00_append_Some: |
---|
1055 | ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc. |
---|
1056 | tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 → |
---|
1057 | tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. |
---|
1058 | #program #pol #prefix #costs #label #i #ppc #pc #H |
---|
1059 | whd in match tech_pc_sigma00 in ⊢ %; normalize nodelta; |
---|
1060 | whd in match sigma00 in ⊢ %; normalize nodelta in ⊢ %; |
---|
1061 | generalize in match (pi2 … pol) whd in ⊢ (% → ?) whd in ⊢ (?(??%?) → ?) |
---|
1062 | whd in match sigma0; normalize nodelta; |
---|
1063 | >foldl_step |
---|
1064 | change with (? → match match sigma00 program pol prefix with [None ⇒ ? | Some res ⇒ ?] with [ None ⇒ ? | Some res ⇒ ? ] = ?) |
---|
1065 | whd in match tech_pc_sigma00 in H; normalize nodelta in H; |
---|
1066 | cases (sigma00 program pol prefix) in H ⊢ % |
---|
1067 | [ whd in ⊢ (??%% → ?) #abs destruct(abs) |
---|
1068 | | * #ppc' * #pc' #sigma_map normalize nodelta; #H generalize in match (option_destruct_Some ??? H) |
---|
1069 | |
---|
1070 | normalize nodelta; -H; |
---|
1071 | |
---|
1072 | |
---|
1073 | generalize in match H; -H; |
---|
1074 | generalize in match (foldl ?????); in H ⊢ (??match match % with [_ ⇒ ? | _ ⇒ ?] with [_ ⇒ ? | _ ⇒ ?]?) |
---|
1075 | [2: whd in ⊢ (??%%) |
---|
1076 | XXX |
---|
1077 | *) |
---|
1078 | |
---|
1079 | (* axiom construct_costs_sigma: |
---|
1080 | ∀p.∀pol:policy p.∀ppc,pc,costs,i. |
---|
1081 | bitvector_of_nat ? pc = sigma p pol (bitvector_of_nat ? ppc) → |
---|
1082 | bitvector_of_nat ? (\fst (construct_costs p pol ppc pc costs i)) = sigma p pol (bitvector_of_nat 16 (S ppc)). |
---|
1083 | |
---|
1084 | axiom tech_pc_sigma00_append_Some: |
---|
1085 | ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc. |
---|
1086 | tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 → |
---|
1087 | tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. *) |
---|
1088 | |
---|
1089 | axiom eq_identifier_eq: |
---|
1090 | ∀tag: String. |
---|
1091 | ∀l. |
---|
1092 | ∀r. |
---|
1093 | eq_identifier tag l r = true → l = r. |
---|
1094 | |
---|
1095 | axiom neq_identifier_neq: |
---|
1096 | ∀tag: String. |
---|
1097 | ∀l, r: identifier tag. |
---|
1098 | eq_identifier tag l r = false → (l = r → False). |
---|
1099 | |
---|
1100 | (* label_map: identifier ↦ pseudo program counter *) |
---|
1101 | definition label_map ≝ identifier_map ASMTag ℕ. |
---|
1102 | |
---|
1103 | (* Labels *) |
---|
1104 | definition is_label ≝ |
---|
1105 | λx:labelled_instruction.λl:Identifier. |
---|
1106 | let 〈lbl,instr〉 ≝ x in |
---|
1107 | match lbl with |
---|
1108 | [ Some l' ⇒ l' = l |
---|
1109 | | _ ⇒ False |
---|
1110 | ]. |
---|
1111 | |
---|
1112 | lemma label_does_not_occur: |
---|
1113 | ∀i:ℕ.∀p:list labelled_instruction.∀l:Identifier. |
---|
1114 | is_label (nth i ? p 〈None ?, Comment [ ]〉) l → does_not_occur ?? l p = false. |
---|
1115 | #i #p #l generalize in match i; elim p |
---|
1116 | [ #i >nth_nil #H cases H |
---|
1117 | | #h #t #IH #i cases i -i |
---|
1118 | [ cases h #hi #hp cases hi |
---|
1119 | [ normalize #H cases H |
---|
1120 | | #l' #Heq whd in ⊢ (??%?); change with (eq_identifier ? l' l) in match (instruction_matches_identifier ????); |
---|
1121 | whd in Heq; >Heq |
---|
1122 | >eq_identifier_refl / by refl/ |
---|
1123 | ] |
---|
1124 | | #i #H whd in match (does_not_occur ????); |
---|
1125 | whd in match (instruction_matches_identifier ????); |
---|
1126 | cases h #hi #hp cases hi normalize nodelta |
---|
1127 | [ @(IH i) @H |
---|
1128 | | #l' @eq_identifier_elim |
---|
1129 | [ normalize / by / |
---|
1130 | | normalize #_ @(IH i) @H |
---|
1131 | ] |
---|
1132 | ] |
---|
1133 | ] |
---|
1134 | ] |
---|
1135 | qed. |
---|
1136 | |
---|
1137 | (* The function that creates the label-to-address map *) |
---|
1138 | definition create_label_cost_map: ∀program:list labelled_instruction. |
---|
1139 | (Σlabels_costs:label_map × (BitVectorTrie costlabel 16). (* Both on ppcs *) |
---|
1140 | let 〈labels,costs〉 ≝ labels_costs in |
---|
1141 | (*∀i:ℕ.lt i (|program|) → *) |
---|
1142 | ∀l.occurs_exactly_once ?? l program → |
---|
1143 | (*is_label (nth i ? program 〈None ?, Comment [ ]〉) l → |
---|
1144 | lookup ?? labels l = Some ? i*) |
---|
1145 | bitvector_of_nat ? (lookup_def ?? labels l 0) = |
---|
1146 | address_of_word_labels_code_mem program l |
---|
1147 | ) ≝ |
---|
1148 | λprogram. |
---|
1149 | \fst (foldl_strong (option Identifier × pseudo_instruction) |
---|
1150 | (λprefix.Σlabels_costs_ppc:label_map × (BitVectorTrie costlabel 16) × nat. |
---|
1151 | let 〈labels,costs,ppc〉 ≝ labels_costs_ppc in |
---|
1152 | ppc = |prefix| ∧ |
---|
1153 | ∀l.occurs_exactly_once ?? l prefix → |
---|
1154 | bitvector_of_nat ? (lookup_def ?? labels l 0) = |
---|
1155 | address_of_word_labels_code_mem prefix l |
---|
1156 | (* |
---|
1157 | ∀i:ℕ.lt i (|prefix|) → |
---|
1158 | ∀l.occurs_exactly_once ?? l prefix → |
---|
1159 | is_label (nth i ? prefix 〈None ?, Comment [ ]〉) l → |
---|
1160 | lookup … labels l = Some ? i *) |
---|
1161 | ) |
---|
1162 | program |
---|
1163 | (λprefix.λx.λtl.λprf.λlabels_costs_ppc. |
---|
1164 | let 〈labels,costs,ppc〉 ≝ labels_costs_ppc in |
---|
1165 | let 〈label,instr〉 ≝ x in |
---|
1166 | let labels ≝ |
---|
1167 | match label with |
---|
1168 | [ None ⇒ labels |
---|
1169 | | Some l ⇒ add … labels l ppc |
---|
1170 | ] in |
---|
1171 | let costs ≝ |
---|
1172 | match instr with |
---|
1173 | [ Cost cost ⇒ insert … (bitvector_of_nat ? ppc) cost costs |
---|
1174 | | _ ⇒ costs ] in |
---|
1175 | 〈labels,costs,S ppc〉 |
---|
1176 | ) 〈(empty_map …),(Stub ??),0〉). |
---|
1177 | [2: normalize nodelta lapply (pi2 … labels_costs_ppc) >p >p1 normalize nodelta * #IH1 #IH2 |
---|
1178 | -labels_costs_ppc % [>IH1 >length_append <plus_n_Sm <plus_n_O %] |
---|
1179 | inversion label [#EQ | #l #EQ] |
---|
1180 | [ #lbl #Hocc <address_of_word_labels_code_mem_None [2: @Hocc] normalize nodelta |
---|
1181 | >occurs_exactly_once_None in Hocc; @(IH2 lbl) |
---|
1182 | | #lbl normalize nodelta inversion (eq_identifier ? lbl l) |
---|
1183 | [ #Heq #Hocc >(eq_identifier_eq … Heq) |
---|
1184 | >address_of_word_labels_code_mem_Some_hit |
---|
1185 | [ >IH1 >lookup_def_add_hit % |
---|
1186 | | <(eq_identifier_eq … Heq) in Hocc; // |
---|
1187 | ] |
---|
1188 | | #Hneq #Hocc |
---|
1189 | <address_of_word_labels_code_mem_Some_miss |
---|
1190 | [ >lookup_def_add_miss |
---|
1191 | [ @IH2 >occurs_exactly_once_Some_eq in Hocc; >eq_identifier_sym> Hneq // |
---|
1192 | | % @neq_identifier_neq @Hneq |
---|
1193 | ] |
---|
1194 | | @Hocc |
---|
1195 | | >eq_identifier_sym @Hneq |
---|
1196 | ] |
---|
1197 | ] |
---|
1198 | ] |
---|
1199 | (* | |
---|
1200 | [ @Hocc |
---|
1201 | | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) in Hlbl; / by / |
---|
1202 | ] |
---|
1203 | [1,2: normalize nodelta |
---|
1204 | (* #i >append_length >commutative_plus #Hi normalize in Hi; cases (le_to_or_lt_eq … Hi) -Hi; |
---|
1205 | [3: #Hi *) #lbl #Hocc (*#Hlbl*) lapply (occurs_exactly_once_Some_stronger ?????? Hocc) -Hocc |
---|
1206 | @eq_identifier_elim |
---|
1207 | [ #Heq #Hocc normalize in Hocc; >(nth_append_first ?? prefix ?? (le_S_S_to_le … Hi)) in Hlbl; #Hl |
---|
1208 | @⊥ @(absurd … Hocc) >(label_does_not_occur i … Hl) normalize nodelta /2 by nmk/ |
---|
1209 | | #Heq #Hocc normalize in Hocc; >lookup_add_miss |
---|
1210 | [ @(IH2 … i (le_S_S_to_le … Hi)) |
---|
1211 | [ @Hocc |
---|
1212 | | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) in Hlbl; / by / |
---|
1213 | ] |
---|
1214 | | @sym_neq @Heq |
---|
1215 | ] |
---|
1216 | ] |
---|
1217 | |4: #Hi #lbl #Hocc >(injective_S … Hi) >nth_append_second |
---|
1218 | [ <minus_n_n #Hl normalize in Hl; >Hl >IH1 @lookup_add_hit |
---|
1219 | | @le_n |
---|
1220 | ] |
---|
1221 | |1: #Hi #lbl >occurs_exactly_once_None #Hocc #Hlbl |
---|
1222 | @(IH2 … i (le_S_S_to_le … Hi)) |
---|
1223 | [ @Hocc |
---|
1224 | | >(nth_append_first ? ? prefix ? ? (le_S_S_to_le … Hi)) in Hlbl; / by / |
---|
1225 | ] |
---|
1226 | |2: #Hi #lbl #Hocc >(injective_S … Hi) >nth_append_second |
---|
1227 | [ <minus_n_n #Hl cases Hl |
---|
1228 | | @le_n |
---|
1229 | ] |
---|
1230 | ] |
---|
1231 | ] *) |
---|
1232 | | @pair_elim * #labels #costs #ppc #EQ destruct normalize nodelta % try % |
---|
1233 | #l #abs cases (abs) |
---|
1234 | | cases (foldl_strong ? (λ_.Σx.?) ???) * * #labels #costs #ppc normalize nodelta * |
---|
1235 | #_ #H @H |
---|
1236 | ] |
---|
1237 | qed. |
---|
1238 | |
---|
1239 | theorem create_label_cost_map_ok: |
---|
1240 | ∀pseudo_program: pseudo_assembly_program. |
---|
1241 | let 〈labels, costs〉 ≝ create_label_cost_map (\snd pseudo_program) in |
---|
1242 | ∀id. occurs_exactly_once ?? id (\snd pseudo_program) → |
---|
1243 | bitvector_of_nat ? (lookup_def ?? labels id 0) = address_of_word_labels_code_mem (\snd pseudo_program) id. |
---|
1244 | #p @pi2 |
---|
1245 | qed. |
---|
1246 | |
---|
1247 | definition assembly: |
---|
1248 | ∀p:pseudo_assembly_program.∀sigma:Word → Word × bool.list Byte × (BitVectorTrie costlabel 16) ≝ |
---|
1249 | λp.let 〈preamble, instr_list〉 ≝ p in |
---|
1250 | λsigma. |
---|
1251 | let 〈labels_to_ppc,ppc_to_costs〉 ≝ create_label_cost_map instr_list in |
---|
1252 | let datalabels ≝ construct_datalabels preamble in |
---|
1253 | let lookup_labels ≝ λx. \fst (sigma (bitvector_of_nat ? (lookup_def … labels_to_ppc x 0))) in |
---|
1254 | let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero ?) in |
---|
1255 | let result ≝ |
---|
1256 | foldl_strong |
---|
1257 | (option Identifier × pseudo_instruction) |
---|
1258 | (λpre. Σpc_ppc_code:(Word × (Word × (list Byte))). |
---|
1259 | let 〈pc,ppc_code〉 ≝ pc_ppc_code in |
---|
1260 | let 〈ppc,code〉 ≝ ppc_code in |
---|
1261 | ∀ppc'. |
---|
1262 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc' in |
---|
1263 | let 〈len,assembledi〉 ≝ |
---|
1264 | assembly_1_pseudoinstruction lookup_labels sigma ppc' lookup_datalabels pi in |
---|
1265 | True) |
---|
1266 | instr_list |
---|
1267 | (λprefix,hd,tl,prf,pc_ppc_code. |
---|
1268 | let 〈pc, ppc_code〉 ≝ pc_ppc_code in |
---|
1269 | let 〈ppc, code〉 ≝ ppc_code in |
---|
1270 | let 〈pc_delta, program〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma ppc lookup_datalabels (\snd hd) in |
---|
1271 | let 〈new_pc, flags〉 ≝ add_16_with_carry pc (bitvector_of_nat ? pc_delta) false in |
---|
1272 | let new_ppc ≝ add ? ppc (bitvector_of_nat ? 1) in |
---|
1273 | 〈new_pc, 〈new_ppc, (code @ program)〉〉) |
---|
1274 | 〈(zero ?), 〈(zero ?), [ ]〉〉 |
---|
1275 | in |
---|
1276 | 〈\snd (\snd result), |
---|
1277 | fold … (λppc.λcost.λpc_to_costs. insert … (\fst (sigma ppc)) cost pc_to_costs) ppc_to_costs (Stub ??)〉. |
---|
1278 | @pair_elim normalize nodelta #x #y #z @pair_elim normalize nodelta #w1 #w2 #w3 #w4 @pair_elim // |
---|
1279 | qed. |
---|
1280 | |
---|
1281 | definition assembly_unlabelled_program: assembly_program → option (list Byte × (BitVectorTrie Identifier 16)) ≝ |
---|
1282 | λp. Some ? (〈foldr ? ? (λi,l. assembly1 i @ l) [ ] p, Stub …〉). |
---|