source: src/ASM/Assembly.ma @ 1975

Last change on this file since 1975 was 1948, checked in by mulligan, 8 years ago

Weakened statements of ASM/Assembly.ma and ASM/AssemblyProof.ma, so all lemmas and functions that accepted a sigma before now accept a weakened sigma coupled with a policy. ASM/AssemblyProof.ma compiles until main_thm.

File size: 56.2 KB
Line 
1include "ASM/ASM.ma".
2include "ASM/Arithmetic.ma".
3include "ASM/Fetch.ma".
4include "ASM/Status.ma".
5include alias "basics/logic.ma".
6include alias "arithmetics/nat.ma".
7include "utilities/extralib.ma".
8
9(**************************************** START OF POLICY ABSTRACTION ********************)
10
11(* definition of & operations on jump length *)
12inductive jump_length: Type[0] ≝
13  | short_jump: jump_length
14  | medium_jump: jump_length
15  | long_jump: jump_length.
16
17definition 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
360definition 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
418qed.
419
420definition expand_relative_jump_internal:
421 ∀lookup_labels:Identifier → Word.∀sigma:Word → Word.
422 Identifier → Word → ([[relative]] → preinstruction [[relative]]) →
423 list instruction
424 ≝
425  λlookup_labels.λsigma.λlbl.λppc,i.
426   let lookup_address ≝ sigma (lookup_labels lbl) in
427   let pc ≝ 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  %
439qed.
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
448lemma eject_rel_jump_length: ∀x,y. rel_jump_length_ok x y → jump_length.
449 #x #y #p @(pi1 … p)
450qed.
451
452coercion 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/
466qed.*)
467
468definition 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.
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
518definition expand_pseudo_instruction:
519    ∀lookup_labels.
520    ∀sigma: Word → Word.
521    ∀policy: Word → bool.
522      Word → ? → pseudo_instruction → list instruction ≝
523  λlookup_labels: Identifier → Word.
524  λsigma: Word → Word.
525  λpolicy: Word → bool.
526  λppc.
527  λlookup_datalabels:Identifier → Word.
528  λi.
529  match i with
530  [ Cost cost ⇒ [ ]
531  | Comment comment ⇒ [ ]
532  | Call call ⇒
533    let 〈addr_5, resta〉 ≝ split ? 5 11 (sigma (lookup_labels call)) in
534    let pc ≝ sigma ppc in
535    let do_a_long ≝ policy ppc in
536    let 〈pc_5, restp〉 ≝ split ? 5 11 pc in
537    if eq_bv ? addr_5 pc_5 ∧ ¬ do_a_long then
538      let address ≝ ADDR11 resta in
539        [ ACALL address ]
540    else
541      let address ≝ ADDR16 (sigma (lookup_labels call)) in
542        [ LCALL address ]
543  | Mov d trgt ⇒
544    let address ≝ DATA16 (lookup_datalabels trgt) in
545      [ RealInstruction (MOV ? (inl ? ? (inl ? ? (inr ? ? 〈DPTR, address〉))))]
546  | Instruction instr ⇒ expand_relative_jump lookup_labels sigma ppc instr
547  | Jmp jmp ⇒
548    let pc ≝ sigma ppc in
549    let do_a_long ≝ policy ppc in
550    let lookup_address ≝ sigma (lookup_labels jmp) in
551    let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in
552    let 〈upper, lower〉 ≝ split ? 8 8 result in
553    if eq_bv ? upper (zero 8) ∧ ¬ do_a_long then
554      let address ≝ RELATIVE lower in
555        [ SJMP address ]
556    else
557      let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 (lookup_labels jmp) in
558      let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 pc in
559      if eq_bv ? fst_5_addr fst_5_pc ∧ ¬ do_a_long then
560        let address ≝ ADDR11 rest_addr in
561          [ AJMP address ]
562      else   
563        let address ≝ ADDR16 lookup_address in
564        [ LJMP address ]
565  ].
566  %
567qed.
568
569(*
570(*X?
571definition jump_length_ok ≝
572 λlookup_labels:Identifier → Word.
573 λpc:Word.
574 Σjump_len:jump_length.
575  (* CSC,JPB: Cheating here, use Jaap's better definition select_reljump_length *)
576  ∀x,y.expand_pseudo_instruction_safe lookup_labels pc jump_len x y ≠ None ?.
577*)
578
579lemma eject_jump_length: ∀x,y. jump_length_ok x y → jump_length.
580 #x #y #p @(pi1 … p)
581qed.
582
583coercion eject_jump_length nocomposites:
584 ∀x,y.∀pol:jump_length_ok x y. jump_length ≝
585 eject_jump_length on _pol:(jump_length_ok ??) to jump_length.
586
587definition expand_pseudo_instruction:
588 ∀lookup_labels:Identifier → Word. ∀pc:Word. jump_length_ok lookup_labels pc →
589 ? → pseudo_instruction → list instruction ≝
590 λlookup_labels,pc,jump_len,lookup_datalabels,i.
591   match expand_pseudo_instruction_safe lookup_labels pc jump_len lookup_datalabels i
592   return λres. res ≠ None ? → ?
593   with
594   [ None ⇒ λabs.⊥
595   | Some res ⇒ λ_.res ] (pi2 … jump_len lookup_datalabels i).
596 cases abs /2/
597qed.
598*)
599(*X?
600definition policy_type ≝
601 λlookup_labels:Identifier → Word.
602 ∀pc:Word. jump_length_ok lookup_labels pc.
603*)
604
605(*definition policy_type2 ≝
606 λprogram.
607  Σpol:Word → jump_length.
608   let lookup_labels ≝
609    (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) in
610   ∀pc:Word. let jump_len ≝ pol pc in
611    ∀x,y.expand_pseudo_instruction_safe lookup_labels pc jump_len x y ≠ None ?.*)
612 
613definition assembly_1_pseudoinstruction ≝
614  λlookup_labels.
615  λsigma: Word → Word.
616  λpolicy: Word → bool.
617  λppc: Word.
618  λlookup_datalabels.
619  λi.
620  let pseudos ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels i in
621  let mapped ≝ map ? ? assembly1 pseudos in
622  let flattened ≝ flatten ? mapped in
623  let pc_len ≝ length ? flattened in
624   〈pc_len, flattened〉.
625
626definition instruction_size ≝
627  λlookup_labels.
628  λsigma: Word → Word.
629  λpolicy: Word → bool.
630  λppc.
631  λi.
632    \fst (assembly_1_pseudoinstruction lookup_labels sigma policy ppc (λx.zero …) i).
633
634(* Jaap: never used
635lemma fetch_pseudo_instruction_prefix:
636  ∀prefix.∀x.∀ppc.ppc < (|prefix|) →
637  fetch_pseudo_instruction prefix (bitvector_of_nat ? ppc) =
638  fetch_pseudo_instruction (prefix@x) (bitvector_of_nat ? ppc).
639 #prefix #x #ppc elim prefix
640 [ #Hppc @⊥ @(absurd … Hppc) @le_to_not_lt @le_O_n
641 | #h #t #Hind #Hppc whd in match (fetch_pseudo_instruction ??);
642   whd in match (fetch_pseudo_instruction ((h::t)@x) ?);
643   >nth_append_first
644   [ //
645   | >nat_of_bitvector_bitvector_of_nat
646     [ @Hppc
647     | cases daemon (* XXX invariant *)
648     ]
649   ]
650 ]
651qed.
652*)
653
654(*
655(* This establishes the correspondence between pseudo program counters and
656   program counters. It is at the heart of the proof. *)
657(*CSC: code taken from build_maps *)
658definition sigma00:
659 ∀jump_expansion:policy_type2.∀l:list labelled_instruction.? →
660 (Σppc_pc_map:ℕ×(ℕ×(BitVectorTrie Word 16)).
661  let 〈ppc,pc_map〉 ≝ ppc_pc_map in
662  let 〈program_counter, sigma_map〉 ≝ pc_map in
663  ppc = |l| ∧
664  (ppc = |l| →
665   (bvt_lookup ?? (bitvector_of_nat ? ppc) sigma_map (zero ?) = (bitvector_of_nat ? program_counter)) ∧
666   (∀x.x < |l| →
667    ∀pi.\fst (fetch_pseudo_instruction l (bitvector_of_nat ? x)) = pi →
668   let pc_x ≝ bvt_lookup ?? (bitvector_of_nat 16 x) sigma_map (zero ?) in
669   bvt_lookup ?? (bitvector_of_nat 16 (S x)) sigma_map (zero ?) =
670   bitvector_of_nat 16 ((nat_of_bitvector ? pc_x) +
671   (\fst (assembly_1_pseudoinstruction lookup_labels(*X?(λx.pc_x)*) (jump_expansion (*?(λx.pc_x)*)) pc_x
672     (λx.zero ?) pi)))))
673 ) ≝
674 (*?*)λlookup_labels.
675 λjump_expansion(*X?: policy_type2*).
676 λl:list labelled_instruction.
677 λacc.
678  foldl_strong ?
679   (λprefix.(Σppc_pc_map:ℕ×(ℕ×(BitVectorTrie Word 16)).
680     let 〈ppc,pc_map〉 ≝ ppc_pc_map in
681     let 〈program_counter, sigma_map〉 ≝ pc_map in
682     (ppc = |prefix|) ∧
683     (ppc = |prefix| →
684      (bvt_lookup ?? (bitvector_of_nat ? ppc) sigma_map (zero ?) = (bitvector_of_nat ? program_counter)) ∧
685      (∀x.x < |prefix| →
686       ∀pi.\fst (fetch_pseudo_instruction l (bitvector_of_nat ? x)) = pi →
687       let pc_x ≝  bvt_lookup ?? (bitvector_of_nat 16 x) sigma_map (zero ?) in
688       bvt_lookup ?? (bitvector_of_nat 16 (S x)) sigma_map (zero ?) =
689       bitvector_of_nat 16 ((nat_of_bitvector ? pc_x) +
690       (\fst (assembly_1_pseudoinstruction (*X?(λx.pc_x)*)lookup_labels (jump_expansion (*X?(λx.pc_x)*)) pc_x
691        (λx.zero ?) pi))))))
692    )
693   l
694   (λhd.λi.λtl.λp.λppc_pc_map.
695     let 〈ppc,pc_map〉 ≝ ppc_pc_map in
696     let 〈program_counter, sigma_map〉 ≝ pc_map in
697     let 〈label, i〉 ≝ i in
698      let 〈pc,ignore〉 ≝ construct_costs lookup_labels program_counter (jump_expansion (*X?(λx.bitvector_of_nat ? program_counter)*)) ppc (Stub …) i in
699         〈S ppc, 〈pc, insert ?? (bitvector_of_nat 16 (S ppc)) (bitvector_of_nat 16 pc) sigma_map〉〉
700   ) acc.
701cases i in p; #label #ins #p @pair_elim #new_ppc #x normalize nodelta cases x -x #old_pc #old_map
702@pair_elim #new_pc #ignore #Hc #Heq normalize nodelta @conj
703[ lapply (pi2 ?? ppc_pc_map) >p1 >p2 normalize nodelta #Hind
704  <(pair_eq1 ?????? Heq) >(proj1 ?? Hind) >append_length <commutative_plus normalize @refl
705| #Hnew <(pair_eq2 ?????? (pair_eq2 ?????? Heq)) <(pair_eq1 ?????? Heq) @conj
706  [ >lookup_insert_hit >(pair_eq1 ?????? (pair_eq2 ?????? Heq)) @refl
707  | #x <(pair_eq1 ?????? Heq) >append_length <commutative_plus #Hx normalize in Hx;
708    #pi #Hpi <(pair_eq2 ?????? (pair_eq2 ?????? Heq)) <(pair_eq1 ?????? Heq) in Hnew;
709    >append_length <commutative_plus #Hnew normalize in Hnew; >(injective_S … Hnew)
710    elim (le_to_or_lt_eq … Hx) -Hx #Hx
711    [ lapply (pi2 ?? ppc_pc_map) >p1 >p2 normalize nodelta #Hind
712      lapply (proj2 ?? ((proj2 ?? Hind) (proj1 ?? Hind)) x (le_S_S_to_le … Hx) pi Hpi)
713      -Hind #Hind >lookup_insert_miss
714      [2: @bitvector_of_nat_abs
715        [3: @lt_to_not_eq @Hx
716        |1: @(transitive_le … Hx)
717        ]
718        cases daemon (* XXX invariant *)
719      ]
720      >lookup_insert_miss
721      [2: @bitvector_of_nat_abs
722        [3: @lt_to_not_eq @(transitive_le … (le_S_S_to_le … Hx)) @le_S @le_n
723        |1: @(transitive_le … (le_S_S_to_le … Hx))
724        ]
725        cases daemon (* XXX invariant *)
726      ]
727      @Hind
728    | lapply (pi2 ?? ppc_pc_map) >p1 >p2 normalize nodelta
729      #Hind lapply (proj1 ?? ((proj2 ?? Hind) (proj1 ?? Hind))) -Hind
730      >(injective_S … Hnew) #Hind <(injective_S … Hx) >lookup_insert_hit >lookup_insert_miss
731      [2: @bitvector_of_nat_abs
732        [3: @lt_to_not_eq @le_n
733        |1: @(transitive_le ??? (le_n (S x)))
734        ]
735        cases daemon (* XXX invariant *)
736      ]
737      >p in Hpi; whd in match (fetch_pseudo_instruction ??); >nth_append_second
738      >nat_of_bitvector_bitvector_of_nat >(injective_S … Hx)
739      [3: @le_n]
740      [2,3: cases daemon (* XXX invariant *)]
741      <minus_n_n cases (half_add ???) #x #y normalize nodelta -x -y #Heq <Heq
742      whd in match (construct_costs ?????) in Hc; whd in match (assembly_1_pseudoinstruction ?????);
743      cases ins in p Hc; normalize nodelta
744      [1,2,4,5: #x #p >Hind #H <(pair_eq1 ?????? H) >commutative_plus >nat_of_bitvector_bitvector_of_nat
745        [1,3,5,7: @refl
746        |2,4,6,8: cases daemon (* XXX invariant *)
747        ]
748      |3: #c #p >Hind #H <(pair_eq1 ?????? H) >nat_of_bitvector_bitvector_of_nat
749        [2: cases daemon (* XXX invariant *) ]
750        whd in match (expand_pseudo_instruction ?????); normalize <plus_n_O @refl
751      |6: #x #y #p >Hind #H <(pair_eq1 ?????? H) >commutative_plus >nat_of_bitvector_bitvector_of_nat
752        [ @refl
753        | cases daemon (* XXX invariant *)
754        ]
755      ]
756    ]
757  ]
758]
759qed.
760
761definition sigma0: pseudo_assembly_program → policy_type2 → (nat × (nat × (BitVectorTrie Word 16))) ≝
762  λprog.
763  λjump_expansion.
764    sigma00 jump_expansion (\snd prog)
765    〈0, 〈0, (insert … (bitvector_of_nat ? 0) (bitvector_of_nat ? 0) (Stub …))〉〉.
766 normalize nodelta @conj
767 [ / by refl/
768 | #H @conj
769   [ >lookup_insert_hit @refl
770   | #x #Hx @⊥ @(absurd … Hx) @le_to_not_lt @le_O_n
771   ]
772 ]
773qed.
774
775definition tech_pc_sigma00: pseudo_assembly_program → policy_type2 →
776 list labelled_instruction → (nat × nat) ≝
777 λprogram,jump_expansion,instr_list.
778   let 〈ppc,pc_sigma_map〉 ≝ sigma00 jump_expansion instr_list
779   〈0, 〈0, (insert … (bitvector_of_nat ? 0) (bitvector_of_nat ? 0) (Stub ? ?))〉〉 in
780   (* acc copied from sigma0 *)
781   let 〈pc,map〉 ≝ pc_sigma_map in
782     〈ppc,pc〉.
783 normalize nodelta @conj
784 [ / by refl/
785 | #H @conj
786   [ >lookup_insert_hit @refl
787   | #x #Hx @⊥ @(absurd … Hx) @le_to_not_lt @le_O_n
788   ]
789 ]
790qed.
791
792definition sigma_safe: pseudo_assembly_program → policy_type2 →
793 option (Word → Word) ≝
794 λinstr_list,jump_expansion.
795  let 〈ppc,pc_sigma_map〉 ≝ sigma0 instr_list jump_expansion in
796  let 〈pc, sigma_map〉 ≝ pc_sigma_map in
797    if gtb pc (2^16) then
798      None ?
799    else
800      Some ? (λx. lookup … x sigma_map (zero …)). *)
801
802(* stuff about policy *)
803
804(*definition policy_ok ≝ λjump_expansion,p. sigma_safe p jump_expansion ≠ None ….*)
805
806(*definition policy ≝ λp. Σjump_expansion:policy_type2. policy_ok jump_expansion p.*)
807
808(*lemma eject_policy: ∀p. policy p → policy_type2.
809 #p #pol @(pi1 … pol)
810qed.
811
812coercion eject_policy nocomposites: ∀p.∀pol:policy p. policy_type2 ≝ eject_policy on _pol:(policy ?) to policy_type2.
813
814definition sigma: ∀p:pseudo_assembly_program. policy p → Word → Word ≝
815 λp,policy.
816  match sigma_safe p (pi1 … policy) return λr:option (Word → Word). r ≠ None … → Word → Word with
817   [ None ⇒ λabs. ⊥
818   | Some r ⇒ λ_.r] (pi2 … policy).
819 cases abs /2 by /
820qed.*)
821
822(*CSC: Main axiom here, needs to be proved soon! *)
823(*lemma snd_assembly_1_pseudoinstruction_ok:
824 ∀program:pseudo_assembly_program.∀pol: policy program.
825 ∀ppc:Word.∀pi,lookup_labels,lookup_datalabels.
826  lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) →
827  lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
828  (nat_of_bitvector 16 ppc) < |\snd program| →
829  \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
830   let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels (pol lookup_labels) (sigma program pol ppc) lookup_datalabels  pi) in
831    sigma program pol (add ? ppc (bitvector_of_nat ? 1)) =
832     bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len).
833 #program #pol #ppc #pi #lookup_labels #lookup_datalabels #Hll #Hldl #Hppc
834 lapply (refl … (sigma0 program pol)) whd in match (sigma0 ??) in ⊢ (??%? → ?);
835 cases (sigma00 ???) #x #Hpmap #EQ
836 whd in match (sigma ???);
837 whd in match (sigma program pol (add ???));
838 whd in match sigma_safe; normalize nodelta
839 (*Problem1: backtracking cases (sigma0 program pol)*)
840 generalize in match (pi2 ???); whd in match policy_ok; normalize nodelta
841 whd in match sigma_safe; normalize nodelta <EQ cases x in Hpmap EQ; -x #final_ppc #x
842 cases x -x #final_pc #smap normalize nodelta #Hpmap #EQ #Heq #Hfetch cases (gtb final_pc (2^16)) in Heq;
843 normalize nodelta
844 [ #abs @⊥ @(absurd ?? abs) @refl
845 | #_ lapply (proj1 ?? ((proj2 ?? Hpmap) (proj1 ?? Hpmap))) #Hpmap1
846   lapply ((proj2 ?? ((proj2 ?? Hpmap) (proj1 ?? Hpmap))) (nat_of_bitvector 16 ppc) Hppc) #Hpmap2 -Hpmap
847   <(bitvector_of_nat_nat_of_bitvector 16 ppc) >add_SO
848   
849   >(Hpmap2 ? (refl …)) @eq_f @eq_f2 [%]
850   >bitvector_of_nat_nat_of_bitvector
851   >Hfetch lapply Hfetch lapply pi
852
853   
854   whd in match assembly_1_pseudoinstruction; normalize nodelta
855 
856qed.*)
857
858
859(*example sigma_0: ∀p,pol. sigma p pol (bitvector_of_nat ? 0) = bitvector_of_nat ? 0.
860 cases daemon.
861qed.*)
862
863(*CSC: FALSE!!!*)
864axiom fetch_pseudo_instruction_split:
865 ∀instr_list,ppc.
866  ∃pre,suff,lbl.
867   (pre @ [〈lbl,\fst (fetch_pseudo_instruction instr_list ppc)〉]) @ suff = instr_list.
868
869(*lemma sigma00_append:
870 ∀jump_expansion,l1,l2.
871 ∀acc:ℕ×ℕ×(BitVectorTrie Word 16).
872  sigma00 jump_expansion (l1@l2) acc =
873  sigma00 jump_expansion
874    l2 (pi1 ?? (sigma00 jump_expansion l1 acc)).*)
875
876(* lemma sigma00_strict:
877 ∀jump_expansion,l,acc. acc = None ? →
878  sigma00 jump_expansion l acc = None ….
879 #jump_expansion #l elim l
880  [ #acc #H >H %
881  | #hd #tl #IH #acc #H >H change with (sigma00 ? tl ? = ?) @IH % ]
882qed.
883
884lemma policy_ok_prefix_ok:
885 ∀program.∀pol:policy program.∀suffix,prefix.
886  prefix@suffix = \snd program →
887   sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) ≠ None ….
888 * #preamble #instr_list #pol #suffix #prefix #prf whd in prf:(???%);
889 generalize in match (pi2 ?? pol); whd in prf:(???%); <prf in pol; #pol
890 whd in match policy_ok; whd in match sigma_safe; whd in match sigma0;
891 normalize nodelta >sigma00_append
892 cases (sigma00 ?? prefix ?)
893  [2: #x #_ % #abs destruct(abs)
894  | * #abs @⊥ @abs >sigma00_strict % ]
895qed.
896
897lemma policy_ok_prefix_hd_ok:
898 ∀program.∀pol:policy program.∀suffix,hd,prefix,ppc_pc_map.
899  (prefix@[hd])@suffix = \snd program →
900   Some ? ppc_pc_map = sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) →
901    let 〈ppc,pc_map〉 ≝ ppc_pc_map in
902    let 〈program_counter, sigma_map〉 ≝ pc_map in
903    let 〈label, i〉 ≝ hd in
904     construct_costs_safe program pol ppc program_counter (Stub …) i ≠ None ….
905 * #preamble #instr_list #pol #suffix #hd #prefix #ppc_pc_map #EQ1 #EQ2
906 generalize in match (policy_ok_prefix_ok 〈preamble,instr_list〉 pol suffix
907  (prefix@[hd]) EQ1) in ⊢ ?; >sigma00_append <EQ2 whd in ⊢ (?(??%?) → ?);
908 @pair_elim #ppc #pc_map #EQ3 normalize nodelta
909 @pair_elim #pc #map #EQ4 normalize nodelta
910 @pair_elim #l' #i' #EQ5 normalize nodelta
911 cases (construct_costs_safe ??????) normalize
912  [* #abs @⊥ @abs % | #X #_ % #abs destruct(abs)]
913qed. *)
914
915(* JPB,CSC: this definition is now replaced by the expand_pseudo_instruction higher up
916definition expand_pseudo_instruction:
917 ∀program:pseudo_assembly_program.∀pol: policy program.
918  ∀ppc:Word.∀lookup_labels,lookup_datalabels,pc.
919  lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) →
920  lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
921  let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in
922  pc = sigma program pol ppc →
923  Σres:list instruction. Some … res = expand_pseudo_instruction_safe pc (lookup_labels pi) lookup_datalabels (pol ppc) pi
924≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pc,prf1,prf2,prf3.
925   match expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc (pol ppc) (\fst (fetch_pseudo_instruction (\snd program) ppc)) with
926    [ None ⇒ let dummy ≝ [ ] in dummy
927    | Some res ⇒ res ].
928 [ @⊥ whd in p:(??%??);
929   generalize in match (pi2 ?? pol); whd in ⊢ (% → ?);
930   whd in ⊢ (?(??%?) → ?); change with (sigma00 ????) in ⊢ (?(??(match % with [_ ⇒ ? | _ ⇒ ?])?) → ?);
931   generalize in match (refl … (sigma00 program pol (\snd program) (Some ? 〈O,〈O,Stub (BitVector 16) 16〉〉)));
932   cases (sigma00 ????) in ⊢ (??%? → %); normalize nodelta [#_ * #abs @abs %]
933   #res #K
934   cases (fetch_pseudo_instruction_split (\snd program) ppc) #pre * #suff * #lbl #EQ1
935   generalize in match (policy_ok_prefix_hd_ok program pol … EQ1 ?) in ⊢ ?;
936   cases daemon (* CSC: XXXXXXXX Ero qui
937   
938    [3: @policy_ok_prefix_ok ]
939    | sigma00 program pol pre
940
941
942
943   QUA USARE LEMMA policy_ok_prefix_hd_ok combinato a lemma da fare che
944   fetch ppc = hd sse program = pre @ [hd] @ tl e |pre| = ppc
945   per concludere construct_costs_safe ≠ None *)
946 | >p %]
947qed. *)
948
949(* MAIN AXIOM HERE, HIDDEN USING cases daemon *)
950(* definition assembly_1_pseudoinstruction':
951 ∀program:pseudo_assembly_program.∀pol: policy program.
952  ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi.
953  lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) →
954  lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
955  \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
956  Σres:(nat × (list Byte)).
957   res = assembly_1_pseudoinstruction program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi ∧
958   let 〈len,code〉 ≝ res in
959    sigma program pol (add ? ppc (bitvector_of_nat ? 1)) =
960     bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len)
961≝ λprogram: pseudo_assembly_program.
962  λpol: policy program.
963  λppc: Word.
964  λlookup_labels.
965  λlookup_datalabels.
966  λpi.
967  λprf1,prf2,prf3.
968   assembly_1_pseudoinstruction program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi.
969 [ @⊥ elim pi in p; [*]
970   try (#ARG1 #ARG2 #ARG3 #abs) try (#ARG1 #ARG2 #abs) try (#ARG1 #abs) try #abs
971   generalize in match (jmeq_to_eq ??? abs); -abs; #abs whd in abs:(??%?); try destruct(abs)
972   whd in abs:(??match % with [_ ⇒ ? | _ ⇒ ?]?);
973   (* WRONG HERE, NEEDS LEMMA SAYING THAT THE POLICY DOES NOT RETURN MEDIUM! *)
974   cases daemon
975 | % [ >p %]
976   cases res in p ⊢ %; -res; #len #code #EQ normalize nodelta;
977   (* THIS SHOULD BE TRUE INSTEAD *)
978   cases daemon]
979qed.
980
981definition assembly_1_pseudoinstruction:
982 ∀program:pseudo_assembly_program.∀pol: policy program.
983  ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi.
984  lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) →
985  lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
986  \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
987   nat × (list Byte)
988≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pi,prf1,prf2,prf3.
989   assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1
990    prf2 prf3.
991
992lemma assembly_1_pseudoinstruction_ok1:
993 ∀program:pseudo_assembly_program.∀pol: policy program.
994  ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi.
995  ∀prf1:lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)).
996  ∀prf2:lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)).
997  ∀prf3:\fst (fetch_pseudo_instruction (\snd program) ppc) = pi.
998     Some … (assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3)
999   = assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi.
1000 #program #pol #ppc #lookup_labels #lookup_datalabels #pi #prf1 #prf2 #prf3
1001 cases (pi2 … (assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3))
1002 #H1 #_ @H1
1003qed. *)
1004
1005(* MAIN AXIOM HERE, HIDDEN USING cases daemon *)
1006(* definition construct_costs':
1007 ∀program. ∀pol:policy program. ∀ppc,pc,costs,i.
1008  Σres:(nat × (BitVectorTrie costlabel 16)). Some … res = construct_costs_safe program pol ppc pc costs i
1009
1010  λprogram.λpol: policy program.λppc,pc,costs,i.
1011   match construct_costs_safe program pol ppc pc costs i with
1012    [ None ⇒ let dummy ≝ 〈0, Stub costlabel 16〉 in dummy
1013    | Some res ⇒ res ].
1014 [ cases daemon
1015 | >p %]
1016qed.
1017
1018definition construct_costs ≝
1019 λprogram,pol,ppc,pc,costs,i. pi1 … (construct_costs' program pol ppc pc costs i). *)
1020
1021(*
1022axiom suffix_of: ∀A:Type[0]. ∀l,prefix:list A. list A.
1023axiom suffix_of_ok: ∀A,l,prefix. prefix @ suffix_of A l prefix = l.
1024
1025axiom foldl_strong_step:
1026 ∀A:Type[0].
1027  ∀P: list A → Type[0].
1028   ∀l: list A.
1029    ∀H: ∀prefix,hd,tl. l =  prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]).
1030     ∀acc: P [ ].
1031      ∀Q: ∀prefix. P prefix → Prop.
1032       ∀HQ: ∀prefix,hd,tl.∀prf: l = prefix @ [hd] @ tl.
1033        ∀acc: P prefix. Q prefix acc → Q (prefix @ [hd]) (H prefix hd tl prf acc).
1034       Q [ ] acc →
1035        Q l (foldl_strong A P l H acc).
1036(*
1037 #A #P #l #H #acc #Q #HQ #Hacc normalize;
1038 generalize in match
1039  (foldl_strong ?
1040   (λpre. Q pre (foldl_strong_internal A P l (suffix_of A l pre) ? [ ] pre acc ?))
1041   l ? Hacc)
1042 [3: >suffix_of_ok % | 2: #prefix #hd #tl #EQ @(H prefix hd (tl@suffix_of A l pre) EQ) ]
1043 [2: #prefix #hd #tl #prf #X whd in ⊢ (??%)
1044 #K
1045
1046 generalize in match
1047  (foldl_strong ?
1048   (λpre. Q pre (foldl_strong_internal A P l H pre (suffix_of A l pre) acc (suffix_of_ok A l pre))))
1049 [2: @H
1050*)
1051
1052axiom foldl_elim:
1053 ∀A:Type[0].
1054  ∀B: Type[0].
1055   ∀H: A → B → A.
1056    ∀acc: A.
1057     ∀l: list B.
1058      ∀Q: A → Prop.
1059       (∀acc:A.∀b:B. Q acc → Q (H acc b)) →
1060         Q acc →
1061          Q (foldl A B H acc l).
1062*)
1063
1064(*
1065lemma tech_pc_sigma00_append_Some:
1066 ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc.
1067  tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 →
1068   tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉.
1069 #program #pol #prefix #costs #label #i #ppc #pc #H
1070  whd in match tech_pc_sigma00 in ⊢ %; normalize nodelta;
1071  whd in match sigma00 in ⊢ %; normalize nodelta in ⊢ %;
1072  generalize in match (pi2 … pol) whd in ⊢ (% → ?) whd in ⊢ (?(??%?) → ?)
1073  whd in match sigma0; normalize nodelta;
1074  >foldl_step
1075  change with (? → match match sigma00 program pol prefix with [None ⇒ ? | Some res ⇒ ?] with [ None ⇒ ? | Some res ⇒ ? ] = ?)
1076  whd in match tech_pc_sigma00 in H; normalize nodelta in H;
1077  cases (sigma00 program pol prefix) in H ⊢ %
1078   [ whd in ⊢ (??%% → ?) #abs destruct(abs)
1079   | * #ppc' * #pc' #sigma_map normalize nodelta; #H generalize in match (option_destruct_Some ??? H)
1080     
1081     normalize nodelta; -H;
1082     
1083 
1084   generalize in match H; -H;
1085  generalize in match (foldl ?????); in H ⊢ (??match match % with [_ ⇒ ? | _ ⇒ ?] with [_ ⇒ ? | _ ⇒ ?]?)
1086   [2: whd in ⊢ (??%%)
1087XXX
1088*)
1089
1090(* axiom construct_costs_sigma:
1091 ∀p.∀pol:policy p.∀ppc,pc,costs,i.
1092  bitvector_of_nat ? pc = sigma p pol (bitvector_of_nat ? ppc) →
1093   bitvector_of_nat ? (\fst (construct_costs p pol ppc pc costs i)) = sigma p pol (bitvector_of_nat 16 (S ppc)).
1094
1095axiom tech_pc_sigma00_append_Some:
1096 ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc.
1097  tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 →
1098   tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. *)
1099
1100axiom eq_identifier_eq:
1101  ∀tag: String.
1102  ∀l.
1103  ∀r.
1104    eq_identifier tag l r = true → l = r.
1105
1106axiom neq_identifier_neq:
1107  ∀tag: String.
1108  ∀l, r: identifier tag.
1109    eq_identifier tag l r = false → (l = r → False).
1110
1111(* label_map: identifier ↦ pseudo program counter *)
1112definition label_map ≝ identifier_map ASMTag ℕ.
1113
1114(* Labels *)
1115definition is_label ≝
1116  λx:labelled_instruction.λl:Identifier.
1117  let 〈lbl,instr〉 ≝ x in
1118  match lbl with
1119  [ Some l' ⇒ l' = l
1120  | _       ⇒ False
1121  ].
1122
1123lemma label_does_not_occur:
1124  ∀i:ℕ.∀p:list labelled_instruction.∀l:Identifier.
1125  is_label (nth i ? p 〈None ?, Comment [ ]〉) l → does_not_occur ?? l p = false.
1126 #i #p #l generalize in match i; elim p
1127 [ #i >nth_nil #H cases H
1128 | #h #t #IH #i cases i -i
1129   [ cases h #hi #hp cases hi
1130     [ normalize #H cases H
1131     | #l' #Heq whd in ⊢ (??%?); change with (eq_identifier ? l' l) in match (instruction_matches_identifier ????);
1132       whd in Heq; >Heq
1133       >eq_identifier_refl / by refl/
1134     ]
1135   | #i #H whd in match (does_not_occur ????);
1136     whd in match (instruction_matches_identifier ????);
1137     cases h #hi #hp cases hi normalize nodelta
1138     [ @(IH i) @H
1139     | #l' @eq_identifier_elim
1140       [ normalize / by /
1141       | normalize #_ @(IH i) @H
1142       ]
1143     ]
1144   ]
1145 ]
1146qed.
1147
1148(* The function that creates the label-to-address map *)
1149definition create_label_cost_map0: ∀program:list labelled_instruction.
1150  (Σlabels_costs:label_map × (BitVectorTrie costlabel 16). (* Both on ppcs *)
1151    let 〈labels,costs〉 ≝ labels_costs in
1152    ∀l.occurs_exactly_once ?? l program →
1153    bitvector_of_nat ? (lookup_def ?? labels l 0) =
1154     address_of_word_labels_code_mem program l
1155  ) ≝
1156  λprogram.
1157  \fst (foldl_strong (option Identifier × pseudo_instruction)
1158  (λprefix.Σlabels_costs_ppc:label_map × (BitVectorTrie costlabel 16) × nat.
1159    let 〈labels,costs,ppc〉 ≝ labels_costs_ppc in
1160    ppc = |prefix| ∧
1161    ∀l.occurs_exactly_once ?? l prefix →
1162    bitvector_of_nat ? (lookup_def ?? labels l 0) =
1163     address_of_word_labels_code_mem prefix l)
1164  program
1165  (λprefix.λx.λtl.λprf.λlabels_costs_ppc.
1166   let 〈labels,costs,ppc〉 ≝ labels_costs_ppc in
1167   let 〈label,instr〉 ≝ x in
1168   let labels ≝
1169     match label with
1170     [ None   ⇒ labels
1171     | Some l ⇒ add … labels l ppc
1172     ] in
1173   let costs ≝
1174     match instr with
1175     [ Cost cost ⇒ insert … (bitvector_of_nat ? ppc) cost costs
1176     | _ ⇒ costs ] in
1177      〈labels,costs,S ppc〉
1178   ) 〈(empty_map …),(Stub ??),0〉).
1179[2: normalize nodelta lapply (pi2 … labels_costs_ppc) >p >p1 normalize nodelta * #IH1 #IH2
1180  -labels_costs_ppc % [>IH1 >length_append <plus_n_Sm <plus_n_O %]
1181 inversion label [#EQ | #l #EQ]
1182 [ #lbl #Hocc <address_of_word_labels_code_mem_None [2: @Hocc] normalize nodelta
1183   >occurs_exactly_once_None in Hocc; @(IH2 lbl)
1184 | #lbl normalize nodelta inversion (eq_identifier ? lbl l)
1185   [ #Heq #Hocc >(eq_identifier_eq … Heq)
1186     >address_of_word_labels_code_mem_Some_hit
1187     [ >IH1 >lookup_def_add_hit %
1188     | <(eq_identifier_eq … Heq) in Hocc; //
1189     ]
1190   | #Hneq #Hocc
1191     <address_of_word_labels_code_mem_Some_miss
1192     [ >lookup_def_add_miss
1193       [ @IH2 >occurs_exactly_once_Some_eq in Hocc; >eq_identifier_sym> Hneq //
1194       | % @neq_identifier_neq @Hneq
1195       ]
1196     | @Hocc
1197     | >eq_identifier_sym @Hneq
1198     ]
1199   ]
1200 ]
1201| @pair_elim * #labels #costs #ppc #EQ destruct normalize nodelta % try %
1202  #l #abs cases (abs)
1203| cases (foldl_strong ? (λ_.Σx.?) ???) * * #labels #costs #ppc normalize nodelta *
1204  #_ #H @H
1205]
1206qed.
1207
1208(* The function that creates the label-to-address map *)
1209definition create_label_cost_map: ∀program:list labelled_instruction.
1210  label_map × (BitVectorTrie costlabel 16) ≝
1211    λprogram.
1212      pi1 … (create_label_cost_map0 program).
1213
1214theorem create_label_cost_map_ok:
1215 ∀pseudo_program: pseudo_assembly_program.
1216   let 〈labels, costs〉 ≝ create_label_cost_map (\snd pseudo_program) in
1217    ∀id. occurs_exactly_once ??  id (\snd pseudo_program) →
1218     bitvector_of_nat ? (lookup_def ?? labels id 0) = address_of_word_labels_code_mem (\snd pseudo_program) id.
1219 #p change with (pi1 … (create_label_cost_map0 ?)) in match (create_label_cost_map ?); @pi2
1220qed.
1221 
1222definition assembly:
1223    ∀p: pseudo_assembly_program.
1224    ∀sigma: Word → Word.
1225    ∀policy: Word → bool.
1226      list Byte × (BitVectorTrie costlabel 16) ≝
1227  λp.
1228  let 〈preamble, instr_list〉 ≝ p in
1229  λsigma.
1230  λpolicy.
1231  let 〈labels_to_ppc,ppc_to_costs〉 ≝ create_label_cost_map instr_list in
1232  let datalabels ≝ construct_datalabels preamble in
1233  let lookup_labels ≝ λx. sigma (bitvector_of_nat ? (lookup_def … labels_to_ppc x 0)) in
1234  let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero ?) in
1235  let result ≝
1236     foldl_strong
1237      (option Identifier × pseudo_instruction)
1238      (λpre. Σpc_ppc_code:(Word × (Word × (list Byte))).
1239        let 〈pc,ppc_code〉 ≝ pc_ppc_code in
1240        let 〈ppc,code〉 ≝ ppc_code in
1241         ∀ppc'.
1242          let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc' in
1243          let 〈len,assembledi〉 ≝
1244           assembly_1_pseudoinstruction lookup_labels sigma policy ppc' lookup_datalabels pi in
1245           True)
1246      instr_list
1247      (λprefix,hd,tl,prf,pc_ppc_code.
1248        let 〈pc, ppc_code〉 ≝ pc_ppc_code in
1249        let 〈ppc, code〉 ≝ ppc_code in
1250        let 〈pc_delta, program〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels (\snd hd) in
1251        let 〈new_pc, flags〉 ≝ add_16_with_carry pc (bitvector_of_nat ? pc_delta) false in
1252        let new_ppc ≝ add ? ppc (bitvector_of_nat ? 1) in
1253         〈new_pc, 〈new_ppc, (code @ program)〉〉)
1254      〈(zero ?), 〈(zero ?), [ ]〉〉
1255    in
1256     〈\snd (\snd result),
1257      fold … (λppc.λcost.λpc_to_costs. insert … (sigma ppc) cost pc_to_costs) ppc_to_costs (Stub ??)〉.
1258  @pair_elim normalize nodelta #x #y #z
1259  @pair_elim normalize nodelta #w1 #w2 #w3 #w4
1260  @pair_elim //
1261qed.
1262
1263definition assembly_unlabelled_program:
1264    assembly_program → option (list Byte × (BitVectorTrie Identifier 16)) ≝
1265  λp.
1266    Some … (〈foldr … (λi,l. assembly1 i @ l) [ ] p, Stub …〉).
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