source: src/ASM/Assembly.ma @ 2021

Last change on this file since 2021 was 2021, checked in by sacerdot, 7 years ago

Proof skeleton in place. Several daemons to be closed adding invariants.

File size: 61.0 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
420(* XXX: pc_plus_sjmp_length used to be just sigma of ppc.  This is incorrect
421        as relative lengths are computed from the *end* of the SJMP, not from
422        the beginning.  This caused a mismatch with Jaap's code which was
423        computing the lengths correctly, and has not been fixed.
424*)
425definition expand_relative_jump_internal:
426 ∀lookup_labels:Identifier → Word.∀sigma:Word → Word.
427 Identifier → Word → ([[relative]] → preinstruction [[relative]]) →
428 list instruction
429 ≝
430  λlookup_labels.λsigma.λlbl.λppc,i.
431   let lookup_address ≝ sigma (lookup_labels lbl) in
432   let pc_plus_sjmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in
433   let 〈result, flags〉 ≝ sub_16_with_carry pc_plus_sjmp_length lookup_address false in
434   let 〈upper, lower〉 ≝ split ? 8 8 result in
435   if eq_bv ? upper (zero 8) then
436     let address ≝ RELATIVE lower in
437       [ RealInstruction (i address) ]
438   else
439    [ RealInstruction (i (RELATIVE (bitvector_of_nat ? 2)));
440      SJMP (RELATIVE (bitvector_of_nat ? 3)); (* LJMP size? *)
441      LJMP (ADDR16 lookup_address)
442    ].
443  %
444qed.
445
446(*definition rel_jump_length_ok ≝
447 λlookup_address:Word.
448 λpc:Word.
449 Σjump_len:jump_length.
450  (* CSC,JPB: Cheating here, use Jaap's better definition select_reljump_length *)
451  ∀(*x,*)y. expand_relative_jump_internal_safe lookup_address jump_len (*x*) pc y ≠ None ?.
452
453lemma eject_rel_jump_length: ∀x,y. rel_jump_length_ok x y → jump_length.
454 #x #y #p @(pi1 … p)
455qed.
456
457coercion eject_rel_jump_length nocomposites:
458 ∀x,y.∀pol:rel_jump_length_ok x y. jump_length ≝
459 eject_rel_jump_length on _pol:(rel_jump_length_ok ??) to jump_length.*)
460
461(*definition expand_relative_jump_internal:
462 ∀lookup_address:Word. ∀pc:Word. ([[relative]] → preinstruction [[relative]]) →
463 list instruction
464≝ λlookup_address,pc,i.
465   match expand_relative_jump_internal_safe lookup_address pc i
466   return λres. res ≠ None ? → ?
467   with
468   [ None ⇒ λabs.⊥
469   | Some res ⇒ λ_.res ] (pi2 … jump_len i).
470 cases abs /2/
471qed.*)
472
473definition expand_relative_jump:
474  ∀lookup_labels.∀sigma.
475  Word → (*jump_length →*)
476  preinstruction Identifier → list instruction ≝
477  λlookup_labels: Identifier → Word.
478  λsigma:Word → Word.
479  λppc: Word.
480  (*λjmp_len: jump_length.*)
481  λi: preinstruction Identifier.
482  (*let rel_jmp ≝ RELATIVE (bitvector_of_nat ? 2) in*)
483  match i with
484  [ JC jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JC ?)
485  | JNC jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JNC ?)
486  | JB baddr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JB ? baddr)
487  | JZ jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JZ ?)
488  | JNZ jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JNZ ?)
489  | JBC baddr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JBC ? baddr)
490  | JNB baddr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (JNB ? baddr)
491  | CJNE addr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (CJNE ? addr)
492  | DJNZ addr jmp ⇒ expand_relative_jump_internal lookup_labels sigma jmp ppc (DJNZ ? addr)
493  | ADD arg1 arg2 ⇒ [ ADD ? arg1 arg2 ]
494  | ADDC arg1 arg2 ⇒ [ ADDC ? arg1 arg2 ]
495  | SUBB arg1 arg2 ⇒ [ SUBB ? arg1 arg2 ]
496  | INC arg ⇒ [ INC ? arg ]
497  | DEC arg ⇒ [ DEC ? arg ]
498  | MUL arg1 arg2 ⇒ [ MUL ? arg1 arg2 ]
499  | DIV arg1 arg2 ⇒ [ DIV ? arg1 arg2 ]
500  | DA arg ⇒ [ DA ? arg ]
501  | ANL arg ⇒ [ ANL ? arg ]
502  | ORL arg ⇒ [ ORL ? arg ]
503  | XRL arg ⇒ [ XRL ? arg ]
504  | CLR arg ⇒ [ CLR ? arg ]
505  | CPL arg ⇒ [ CPL ? arg ]
506  | RL arg ⇒ [ RL ? arg ]
507  | RR arg ⇒ [ RR ? arg ]
508  | RLC arg ⇒ [ RLC ? arg ]
509  | RRC arg ⇒ [ RRC ? arg ]
510  | SWAP arg ⇒ [ SWAP ? arg ]
511  | MOV arg ⇒ [ MOV ? arg ]
512  | MOVX arg ⇒ [ MOVX ? arg ]
513  | SETB arg ⇒ [ SETB ? arg ]
514  | PUSH arg ⇒ [ PUSH ? arg ]
515  | POP arg ⇒ [ POP ? arg ]
516  | XCH arg1 arg2 ⇒ [ XCH ? arg1 arg2 ]
517  | XCHD arg1 arg2 ⇒ [ XCHD ? arg1 arg2 ]
518  | RET ⇒ [ RET ? ]
519  | RETI ⇒ [ RETI ? ]
520  | NOP ⇒ [ RealInstruction (NOP ?) ]
521  ].
522
523definition expand_pseudo_instruction:
524    ∀lookup_labels.
525    ∀sigma: Word → Word.
526    ∀policy: Word → bool.
527      Word → ? → pseudo_instruction → list instruction ≝
528  λlookup_labels: Identifier → Word.
529  λsigma: Word → Word.
530  λpolicy: Word → bool.
531  λppc.
532  λlookup_datalabels:Identifier → Word.
533  λi.
534  match i with
535  [ Cost cost ⇒ [ ]
536  | Comment comment ⇒ [ ]
537  | Call call ⇒
538    let 〈addr_5, resta〉 ≝ split ? 5 11 (sigma (lookup_labels call)) in
539    let pc ≝ sigma ppc in
540    let do_a_long ≝ policy ppc in
541    let 〈pc_5, restp〉 ≝ split ? 5 11 pc in
542    if eq_bv ? addr_5 pc_5 ∧ ¬ do_a_long then
543      let address ≝ ADDR11 resta in
544        [ ACALL address ]
545    else
546      let address ≝ ADDR16 (sigma (lookup_labels call)) in
547        [ LCALL address ]
548  | Mov d trgt ⇒
549    let address ≝ DATA16 (lookup_datalabels trgt) in
550      [ RealInstruction (MOV ? (inl ? ? (inl ? ? (inr ? ? 〈DPTR, address〉))))]
551  | Instruction instr ⇒ expand_relative_jump lookup_labels sigma ppc instr
552  | Jmp jmp ⇒
553    let pc ≝ sigma ppc in
554    let do_a_long ≝ policy ppc in
555    let lookup_address ≝ sigma (lookup_labels jmp) in
556    let 〈result, flags〉 ≝ sub_16_with_carry pc lookup_address false in
557    let 〈upper, lower〉 ≝ split ? 8 8 result in
558    if eq_bv ? upper (zero 8) ∧ ¬ do_a_long then
559      let address ≝ RELATIVE lower in
560        [ SJMP address ]
561    else
562      let 〈fst_5_addr, rest_addr〉 ≝ split ? 5 11 (lookup_labels jmp) in
563      let 〈fst_5_pc, rest_pc〉 ≝ split ? 5 11 pc in
564      if eq_bv ? fst_5_addr fst_5_pc ∧ ¬ do_a_long then
565        let address ≝ ADDR11 rest_addr in
566          [ AJMP address ]
567      else   
568        let address ≝ ADDR16 lookup_address in
569        [ LJMP address ]
570  ].
571  %
572qed.
573
574(*
575(*X?
576definition jump_length_ok ≝
577 λlookup_labels:Identifier → Word.
578 λpc:Word.
579 Σjump_len:jump_length.
580  (* CSC,JPB: Cheating here, use Jaap's better definition select_reljump_length *)
581  ∀x,y.expand_pseudo_instruction_safe lookup_labels pc jump_len x y ≠ None ?.
582*)
583
584lemma eject_jump_length: ∀x,y. jump_length_ok x y → jump_length.
585 #x #y #p @(pi1 … p)
586qed.
587
588coercion eject_jump_length nocomposites:
589 ∀x,y.∀pol:jump_length_ok x y. jump_length ≝
590 eject_jump_length on _pol:(jump_length_ok ??) to jump_length.
591
592definition expand_pseudo_instruction:
593 ∀lookup_labels:Identifier → Word. ∀pc:Word. jump_length_ok lookup_labels pc →
594 ? → pseudo_instruction → list instruction ≝
595 λlookup_labels,pc,jump_len,lookup_datalabels,i.
596   match expand_pseudo_instruction_safe lookup_labels pc jump_len lookup_datalabels i
597   return λres. res ≠ None ? → ?
598   with
599   [ None ⇒ λabs.⊥
600   | Some res ⇒ λ_.res ] (pi2 … jump_len lookup_datalabels i).
601 cases abs /2/
602qed.
603*)
604(*X?
605definition policy_type ≝
606 λlookup_labels:Identifier → Word.
607 ∀pc:Word. jump_length_ok lookup_labels pc.
608*)
609
610(*definition policy_type2 ≝
611 λprogram.
612  Σpol:Word → jump_length.
613   let lookup_labels ≝
614    (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) in
615   ∀pc:Word. let jump_len ≝ pol pc in
616    ∀x,y.expand_pseudo_instruction_safe lookup_labels pc jump_len x y ≠ None ?.*)
617 
618definition assembly_1_pseudoinstruction ≝
619  λlookup_labels.
620  λsigma: Word → Word.
621  λpolicy: Word → bool.
622  λppc: Word.
623  λlookup_datalabels.
624  λi.
625  let pseudos ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels i in
626  let mapped ≝ map ? ? assembly1 pseudos in
627  let flattened ≝ flatten ? mapped in
628  let pc_len ≝ length ? flattened in
629   〈pc_len, flattened〉.
630
631definition instruction_size ≝
632  λlookup_labels.
633  λsigma: Word → Word.
634  λpolicy: Word → bool.
635  λppc.
636  λi.
637    \fst (assembly_1_pseudoinstruction lookup_labels sigma policy ppc (λx.zero …) i).
638
639(* Jaap: never used
640lemma fetch_pseudo_instruction_prefix:
641  ∀prefix.∀x.∀ppc.ppc < (|prefix|) →
642  fetch_pseudo_instruction prefix (bitvector_of_nat ? ppc) =
643  fetch_pseudo_instruction (prefix@x) (bitvector_of_nat ? ppc).
644 #prefix #x #ppc elim prefix
645 [ #Hppc @⊥ @(absurd … Hppc) @le_to_not_lt @le_O_n
646 | #h #t #Hind #Hppc whd in match (fetch_pseudo_instruction ??);
647   whd in match (fetch_pseudo_instruction ((h::t)@x) ?);
648   >nth_append_first
649   [ //
650   | >nat_of_bitvector_bitvector_of_nat
651     [ @Hppc
652     | cases daemon (* XXX invariant *)
653     ]
654   ]
655 ]
656qed.
657*)
658
659(*
660(* This establishes the correspondence between pseudo program counters and
661   program counters. It is at the heart of the proof. *)
662(*CSC: code taken from build_maps *)
663definition sigma00:
664 ∀jump_expansion:policy_type2.∀l:list labelled_instruction.? →
665 (Σppc_pc_map:ℕ×(ℕ×(BitVectorTrie Word 16)).
666  let 〈ppc,pc_map〉 ≝ ppc_pc_map in
667  let 〈program_counter, sigma_map〉 ≝ pc_map in
668  ppc = |l| ∧
669  (ppc = |l| →
670   (bvt_lookup ?? (bitvector_of_nat ? ppc) sigma_map (zero ?) = (bitvector_of_nat ? program_counter)) ∧
671   (∀x.x < |l| →
672    ∀pi.\fst (fetch_pseudo_instruction l (bitvector_of_nat ? x)) = pi →
673   let pc_x ≝ bvt_lookup ?? (bitvector_of_nat 16 x) sigma_map (zero ?) in
674   bvt_lookup ?? (bitvector_of_nat 16 (S x)) sigma_map (zero ?) =
675   bitvector_of_nat 16 ((nat_of_bitvector ? pc_x) +
676   (\fst (assembly_1_pseudoinstruction lookup_labels(*X?(λx.pc_x)*) (jump_expansion (*?(λx.pc_x)*)) pc_x
677     (λx.zero ?) pi)))))
678 ) ≝
679 (*?*)λlookup_labels.
680 λjump_expansion(*X?: policy_type2*).
681 λl:list labelled_instruction.
682 λacc.
683  foldl_strong ?
684   (λprefix.(Σppc_pc_map:ℕ×(ℕ×(BitVectorTrie Word 16)).
685     let 〈ppc,pc_map〉 ≝ ppc_pc_map in
686     let 〈program_counter, sigma_map〉 ≝ pc_map in
687     (ppc = |prefix|) ∧
688     (ppc = |prefix| →
689      (bvt_lookup ?? (bitvector_of_nat ? ppc) sigma_map (zero ?) = (bitvector_of_nat ? program_counter)) ∧
690      (∀x.x < |prefix| →
691       ∀pi.\fst (fetch_pseudo_instruction l (bitvector_of_nat ? x)) = pi →
692       let pc_x ≝  bvt_lookup ?? (bitvector_of_nat 16 x) sigma_map (zero ?) in
693       bvt_lookup ?? (bitvector_of_nat 16 (S x)) sigma_map (zero ?) =
694       bitvector_of_nat 16 ((nat_of_bitvector ? pc_x) +
695       (\fst (assembly_1_pseudoinstruction (*X?(λx.pc_x)*)lookup_labels (jump_expansion (*X?(λx.pc_x)*)) pc_x
696        (λx.zero ?) pi))))))
697    )
698   l
699   (λhd.λi.λtl.λp.λppc_pc_map.
700     let 〈ppc,pc_map〉 ≝ ppc_pc_map in
701     let 〈program_counter, sigma_map〉 ≝ pc_map in
702     let 〈label, i〉 ≝ i in
703      let 〈pc,ignore〉 ≝ construct_costs lookup_labels program_counter (jump_expansion (*X?(λx.bitvector_of_nat ? program_counter)*)) ppc (Stub …) i in
704         〈S ppc, 〈pc, insert ?? (bitvector_of_nat 16 (S ppc)) (bitvector_of_nat 16 pc) sigma_map〉〉
705   ) acc.
706cases i in p; #label #ins #p @pair_elim #new_ppc #x normalize nodelta cases x -x #old_pc #old_map
707@pair_elim #new_pc #ignore #Hc #Heq normalize nodelta @conj
708[ lapply (pi2 ?? ppc_pc_map) >p1 >p2 normalize nodelta #Hind
709  <(pair_eq1 ?????? Heq) >(proj1 ?? Hind) >append_length <commutative_plus normalize @refl
710| #Hnew <(pair_eq2 ?????? (pair_eq2 ?????? Heq)) <(pair_eq1 ?????? Heq) @conj
711  [ >lookup_insert_hit >(pair_eq1 ?????? (pair_eq2 ?????? Heq)) @refl
712  | #x <(pair_eq1 ?????? Heq) >append_length <commutative_plus #Hx normalize in Hx;
713    #pi #Hpi <(pair_eq2 ?????? (pair_eq2 ?????? Heq)) <(pair_eq1 ?????? Heq) in Hnew;
714    >append_length <commutative_plus #Hnew normalize in Hnew; >(injective_S … Hnew)
715    elim (le_to_or_lt_eq … Hx) -Hx #Hx
716    [ lapply (pi2 ?? ppc_pc_map) >p1 >p2 normalize nodelta #Hind
717      lapply (proj2 ?? ((proj2 ?? Hind) (proj1 ?? Hind)) x (le_S_S_to_le … Hx) pi Hpi)
718      -Hind #Hind >lookup_insert_miss
719      [2: @bitvector_of_nat_abs
720        [3: @lt_to_not_eq @Hx
721        |1: @(transitive_le … Hx)
722        ]
723        cases daemon (* XXX invariant *)
724      ]
725      >lookup_insert_miss
726      [2: @bitvector_of_nat_abs
727        [3: @lt_to_not_eq @(transitive_le … (le_S_S_to_le … Hx)) @le_S @le_n
728        |1: @(transitive_le … (le_S_S_to_le … Hx))
729        ]
730        cases daemon (* XXX invariant *)
731      ]
732      @Hind
733    | lapply (pi2 ?? ppc_pc_map) >p1 >p2 normalize nodelta
734      #Hind lapply (proj1 ?? ((proj2 ?? Hind) (proj1 ?? Hind))) -Hind
735      >(injective_S … Hnew) #Hind <(injective_S … Hx) >lookup_insert_hit >lookup_insert_miss
736      [2: @bitvector_of_nat_abs
737        [3: @lt_to_not_eq @le_n
738        |1: @(transitive_le ??? (le_n (S x)))
739        ]
740        cases daemon (* XXX invariant *)
741      ]
742      >p in Hpi; whd in match (fetch_pseudo_instruction ??); >nth_append_second
743      >nat_of_bitvector_bitvector_of_nat >(injective_S … Hx)
744      [3: @le_n]
745      [2,3: cases daemon (* XXX invariant *)]
746      <minus_n_n cases (half_add ???) #x #y normalize nodelta -x -y #Heq <Heq
747      whd in match (construct_costs ?????) in Hc; whd in match (assembly_1_pseudoinstruction ?????);
748      cases ins in p Hc; normalize nodelta
749      [1,2,4,5: #x #p >Hind #H <(pair_eq1 ?????? H) >commutative_plus >nat_of_bitvector_bitvector_of_nat
750        [1,3,5,7: @refl
751        |2,4,6,8: cases daemon (* XXX invariant *)
752        ]
753      |3: #c #p >Hind #H <(pair_eq1 ?????? H) >nat_of_bitvector_bitvector_of_nat
754        [2: cases daemon (* XXX invariant *) ]
755        whd in match (expand_pseudo_instruction ?????); normalize <plus_n_O @refl
756      |6: #x #y #p >Hind #H <(pair_eq1 ?????? H) >commutative_plus >nat_of_bitvector_bitvector_of_nat
757        [ @refl
758        | cases daemon (* XXX invariant *)
759        ]
760      ]
761    ]
762  ]
763]
764qed.
765
766definition sigma0: pseudo_assembly_program → policy_type2 → (nat × (nat × (BitVectorTrie Word 16))) ≝
767  λprog.
768  λjump_expansion.
769    sigma00 jump_expansion (\snd prog)
770    〈0, 〈0, (insert … (bitvector_of_nat ? 0) (bitvector_of_nat ? 0) (Stub …))〉〉.
771 normalize nodelta @conj
772 [ / by refl/
773 | #H @conj
774   [ >lookup_insert_hit @refl
775   | #x #Hx @⊥ @(absurd … Hx) @le_to_not_lt @le_O_n
776   ]
777 ]
778qed.
779
780definition tech_pc_sigma00: pseudo_assembly_program → policy_type2 →
781 list labelled_instruction → (nat × nat) ≝
782 λprogram,jump_expansion,instr_list.
783   let 〈ppc,pc_sigma_map〉 ≝ sigma00 jump_expansion instr_list
784   〈0, 〈0, (insert … (bitvector_of_nat ? 0) (bitvector_of_nat ? 0) (Stub ? ?))〉〉 in
785   (* acc copied from sigma0 *)
786   let 〈pc,map〉 ≝ pc_sigma_map in
787     〈ppc,pc〉.
788 normalize nodelta @conj
789 [ / by refl/
790 | #H @conj
791   [ >lookup_insert_hit @refl
792   | #x #Hx @⊥ @(absurd … Hx) @le_to_not_lt @le_O_n
793   ]
794 ]
795qed.
796
797definition sigma_safe: pseudo_assembly_program → policy_type2 →
798 option (Word → Word) ≝
799 λinstr_list,jump_expansion.
800  let 〈ppc,pc_sigma_map〉 ≝ sigma0 instr_list jump_expansion in
801  let 〈pc, sigma_map〉 ≝ pc_sigma_map in
802    if gtb pc (2^16) then
803      None ?
804    else
805      Some ? (λx. lookup … x sigma_map (zero …)). *)
806
807(* stuff about policy *)
808
809(*definition policy_ok ≝ λjump_expansion,p. sigma_safe p jump_expansion ≠ None ….*)
810
811(*definition policy ≝ λp. Σjump_expansion:policy_type2. policy_ok jump_expansion p.*)
812
813(*lemma eject_policy: ∀p. policy p → policy_type2.
814 #p #pol @(pi1 … pol)
815qed.
816
817coercion eject_policy nocomposites: ∀p.∀pol:policy p. policy_type2 ≝ eject_policy on _pol:(policy ?) to policy_type2.
818
819definition sigma: ∀p:pseudo_assembly_program. policy p → Word → Word ≝
820 λp,policy.
821  match sigma_safe p (pi1 … policy) return λr:option (Word → Word). r ≠ None … → Word → Word with
822   [ None ⇒ λabs. ⊥
823   | Some r ⇒ λ_.r] (pi2 … policy).
824 cases abs /2 by /
825qed.*)
826
827(*CSC: Main axiom here, needs to be proved soon! *)
828(*lemma snd_assembly_1_pseudoinstruction_ok:
829 ∀program:pseudo_assembly_program.∀pol: policy program.
830 ∀ppc:Word.∀pi,lookup_labels,lookup_datalabels.
831  lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) →
832  lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
833  (nat_of_bitvector 16 ppc) < |\snd program| →
834  \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
835   let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels (pol lookup_labels) (sigma program pol ppc) lookup_datalabels  pi) in
836    sigma program pol (add ? ppc (bitvector_of_nat ? 1)) =
837     bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len).
838 #program #pol #ppc #pi #lookup_labels #lookup_datalabels #Hll #Hldl #Hppc
839 lapply (refl … (sigma0 program pol)) whd in match (sigma0 ??) in ⊢ (??%? → ?);
840 cases (sigma00 ???) #x #Hpmap #EQ
841 whd in match (sigma ???);
842 whd in match (sigma program pol (add ???));
843 whd in match sigma_safe; normalize nodelta
844 (*Problem1: backtracking cases (sigma0 program pol)*)
845 generalize in match (pi2 ???); whd in match policy_ok; normalize nodelta
846 whd in match sigma_safe; normalize nodelta <EQ cases x in Hpmap EQ; -x #final_ppc #x
847 cases x -x #final_pc #smap normalize nodelta #Hpmap #EQ #Heq #Hfetch cases (gtb final_pc (2^16)) in Heq;
848 normalize nodelta
849 [ #abs @⊥ @(absurd ?? abs) @refl
850 | #_ lapply (proj1 ?? ((proj2 ?? Hpmap) (proj1 ?? Hpmap))) #Hpmap1
851   lapply ((proj2 ?? ((proj2 ?? Hpmap) (proj1 ?? Hpmap))) (nat_of_bitvector 16 ppc) Hppc) #Hpmap2 -Hpmap
852   <(bitvector_of_nat_nat_of_bitvector 16 ppc) >add_SO
853   
854   >(Hpmap2 ? (refl …)) @eq_f @eq_f2 [%]
855   >bitvector_of_nat_nat_of_bitvector
856   >Hfetch lapply Hfetch lapply pi
857
858   
859   whd in match assembly_1_pseudoinstruction; normalize nodelta
860 
861qed.*)
862
863
864(*example sigma_0: ∀p,pol. sigma p pol (bitvector_of_nat ? 0) = bitvector_of_nat ? 0.
865 cases daemon.
866qed.*)
867
868(*CSC: FALSE!!!*)
869axiom fetch_pseudo_instruction_split:
870 ∀instr_list,ppc.
871  ∃pre,suff,lbl.
872   (pre @ [〈lbl,\fst (fetch_pseudo_instruction instr_list ppc)〉]) @ suff = instr_list.
873
874(*lemma sigma00_append:
875 ∀jump_expansion,l1,l2.
876 ∀acc:ℕ×ℕ×(BitVectorTrie Word 16).
877  sigma00 jump_expansion (l1@l2) acc =
878  sigma00 jump_expansion
879    l2 (pi1 ?? (sigma00 jump_expansion l1 acc)).*)
880
881(* lemma sigma00_strict:
882 ∀jump_expansion,l,acc. acc = None ? →
883  sigma00 jump_expansion l acc = None ….
884 #jump_expansion #l elim l
885  [ #acc #H >H %
886  | #hd #tl #IH #acc #H >H change with (sigma00 ? tl ? = ?) @IH % ]
887qed.
888
889lemma policy_ok_prefix_ok:
890 ∀program.∀pol:policy program.∀suffix,prefix.
891  prefix@suffix = \snd program →
892   sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) ≠ None ….
893 * #preamble #instr_list #pol #suffix #prefix #prf whd in prf:(???%);
894 generalize in match (pi2 ?? pol); whd in prf:(???%); <prf in pol; #pol
895 whd in match policy_ok; whd in match sigma_safe; whd in match sigma0;
896 normalize nodelta >sigma00_append
897 cases (sigma00 ?? prefix ?)
898  [2: #x #_ % #abs destruct(abs)
899  | * #abs @⊥ @abs >sigma00_strict % ]
900qed.
901
902lemma policy_ok_prefix_hd_ok:
903 ∀program.∀pol:policy program.∀suffix,hd,prefix,ppc_pc_map.
904  (prefix@[hd])@suffix = \snd program →
905   Some ? ppc_pc_map = sigma00 program pol prefix (Some … 〈0, 〈0, Stub …〉〉) →
906    let 〈ppc,pc_map〉 ≝ ppc_pc_map in
907    let 〈program_counter, sigma_map〉 ≝ pc_map in
908    let 〈label, i〉 ≝ hd in
909     construct_costs_safe program pol ppc program_counter (Stub …) i ≠ None ….
910 * #preamble #instr_list #pol #suffix #hd #prefix #ppc_pc_map #EQ1 #EQ2
911 generalize in match (policy_ok_prefix_ok 〈preamble,instr_list〉 pol suffix
912  (prefix@[hd]) EQ1) in ⊢ ?; >sigma00_append <EQ2 whd in ⊢ (?(??%?) → ?);
913 @pair_elim #ppc #pc_map #EQ3 normalize nodelta
914 @pair_elim #pc #map #EQ4 normalize nodelta
915 @pair_elim #l' #i' #EQ5 normalize nodelta
916 cases (construct_costs_safe ??????) normalize
917  [* #abs @⊥ @abs % | #X #_ % #abs destruct(abs)]
918qed. *)
919
920(* JPB,CSC: this definition is now replaced by the expand_pseudo_instruction higher up
921definition expand_pseudo_instruction:
922 ∀program:pseudo_assembly_program.∀pol: policy program.
923  ∀ppc:Word.∀lookup_labels,lookup_datalabels,pc.
924  lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) →
925  lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
926  let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc) in
927  pc = sigma program pol ppc →
928  Σres:list instruction. Some … res = expand_pseudo_instruction_safe pc (lookup_labels pi) lookup_datalabels (pol ppc) pi
929≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pc,prf1,prf2,prf3.
930   match expand_pseudo_instruction_safe lookup_labels lookup_datalabels pc (pol ppc) (\fst (fetch_pseudo_instruction (\snd program) ppc)) with
931    [ None ⇒ let dummy ≝ [ ] in dummy
932    | Some res ⇒ res ].
933 [ @⊥ whd in p:(??%??);
934   generalize in match (pi2 ?? pol); whd in ⊢ (% → ?);
935   whd in ⊢ (?(??%?) → ?); change with (sigma00 ????) in ⊢ (?(??(match % with [_ ⇒ ? | _ ⇒ ?])?) → ?);
936   generalize in match (refl … (sigma00 program pol (\snd program) (Some ? 〈O,〈O,Stub (BitVector 16) 16〉〉)));
937   cases (sigma00 ????) in ⊢ (??%? → %); normalize nodelta [#_ * #abs @abs %]
938   #res #K
939   cases (fetch_pseudo_instruction_split (\snd program) ppc) #pre * #suff * #lbl #EQ1
940   generalize in match (policy_ok_prefix_hd_ok program pol … EQ1 ?) in ⊢ ?;
941   cases daemon (* CSC: XXXXXXXX Ero qui
942   
943    [3: @policy_ok_prefix_ok ]
944    | sigma00 program pol pre
945
946
947
948   QUA USARE LEMMA policy_ok_prefix_hd_ok combinato a lemma da fare che
949   fetch ppc = hd sse program = pre @ [hd] @ tl e |pre| = ppc
950   per concludere construct_costs_safe ≠ None *)
951 | >p %]
952qed. *)
953
954(* MAIN AXIOM HERE, HIDDEN USING cases daemon *)
955(* definition assembly_1_pseudoinstruction':
956 ∀program:pseudo_assembly_program.∀pol: policy program.
957  ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi.
958  lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) →
959  lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
960  \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
961  Σres:(nat × (list Byte)).
962   res = assembly_1_pseudoinstruction program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi ∧
963   let 〈len,code〉 ≝ res in
964    sigma program pol (add ? ppc (bitvector_of_nat ? 1)) =
965     bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len)
966≝ λprogram: pseudo_assembly_program.
967  λpol: policy program.
968  λppc: Word.
969  λlookup_labels.
970  λlookup_datalabels.
971  λpi.
972  λprf1,prf2,prf3.
973   assembly_1_pseudoinstruction program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi.
974 [ @⊥ elim pi in p; [*]
975   try (#ARG1 #ARG2 #ARG3 #abs) try (#ARG1 #ARG2 #abs) try (#ARG1 #abs) try #abs
976   generalize in match (jmeq_to_eq ??? abs); -abs; #abs whd in abs:(??%?); try destruct(abs)
977   whd in abs:(??match % with [_ ⇒ ? | _ ⇒ ?]?);
978   (* WRONG HERE, NEEDS LEMMA SAYING THAT THE POLICY DOES NOT RETURN MEDIUM! *)
979   cases daemon
980 | % [ >p %]
981   cases res in p ⊢ %; -res; #len #code #EQ normalize nodelta;
982   (* THIS SHOULD BE TRUE INSTEAD *)
983   cases daemon]
984qed.
985
986definition assembly_1_pseudoinstruction:
987 ∀program:pseudo_assembly_program.∀pol: policy program.
988  ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi.
989  lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) →
990  lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
991  \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
992   nat × (list Byte)
993≝ λprogram,pol,ppc,lookup_labels,lookup_datalabels,pi,prf1,prf2,prf3.
994   assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1
995    prf2 prf3.
996
997lemma assembly_1_pseudoinstruction_ok1:
998 ∀program:pseudo_assembly_program.∀pol: policy program.
999  ∀ppc:Word.∀lookup_labels,lookup_datalabels,pi.
1000  ∀prf1:lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)).
1001  ∀prf2:lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)).
1002  ∀prf3:\fst (fetch_pseudo_instruction (\snd program) ppc) = pi.
1003     Some … (assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3)
1004   = assembly_1_pseudoinstruction_safe program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi.
1005 #program #pol #ppc #lookup_labels #lookup_datalabels #pi #prf1 #prf2 #prf3
1006 cases (pi2 … (assembly_1_pseudoinstruction' program pol ppc lookup_labels lookup_datalabels pi prf1 prf2 prf3))
1007 #H1 #_ @H1
1008qed. *)
1009
1010(* MAIN AXIOM HERE, HIDDEN USING cases daemon *)
1011(* definition construct_costs':
1012 ∀program. ∀pol:policy program. ∀ppc,pc,costs,i.
1013  Σres:(nat × (BitVectorTrie costlabel 16)). Some … res = construct_costs_safe program pol ppc pc costs i
1014
1015  λprogram.λpol: policy program.λppc,pc,costs,i.
1016   match construct_costs_safe program pol ppc pc costs i with
1017    [ None ⇒ let dummy ≝ 〈0, Stub costlabel 16〉 in dummy
1018    | Some res ⇒ res ].
1019 [ cases daemon
1020 | >p %]
1021qed.
1022
1023definition construct_costs ≝
1024 λprogram,pol,ppc,pc,costs,i. pi1 … (construct_costs' program pol ppc pc costs i). *)
1025
1026(*
1027axiom suffix_of: ∀A:Type[0]. ∀l,prefix:list A. list A.
1028axiom suffix_of_ok: ∀A,l,prefix. prefix @ suffix_of A l prefix = l.
1029
1030axiom foldl_strong_step:
1031 ∀A:Type[0].
1032  ∀P: list A → Type[0].
1033   ∀l: list A.
1034    ∀H: ∀prefix,hd,tl. l =  prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]).
1035     ∀acc: P [ ].
1036      ∀Q: ∀prefix. P prefix → Prop.
1037       ∀HQ: ∀prefix,hd,tl.∀prf: l = prefix @ [hd] @ tl.
1038        ∀acc: P prefix. Q prefix acc → Q (prefix @ [hd]) (H prefix hd tl prf acc).
1039       Q [ ] acc →
1040        Q l (foldl_strong A P l H acc).
1041(*
1042 #A #P #l #H #acc #Q #HQ #Hacc normalize;
1043 generalize in match
1044  (foldl_strong ?
1045   (λpre. Q pre (foldl_strong_internal A P l (suffix_of A l pre) ? [ ] pre acc ?))
1046   l ? Hacc)
1047 [3: >suffix_of_ok % | 2: #prefix #hd #tl #EQ @(H prefix hd (tl@suffix_of A l pre) EQ) ]
1048 [2: #prefix #hd #tl #prf #X whd in ⊢ (??%)
1049 #K
1050
1051 generalize in match
1052  (foldl_strong ?
1053   (λpre. Q pre (foldl_strong_internal A P l H pre (suffix_of A l pre) acc (suffix_of_ok A l pre))))
1054 [2: @H
1055*)
1056
1057axiom foldl_elim:
1058 ∀A:Type[0].
1059  ∀B: Type[0].
1060   ∀H: A → B → A.
1061    ∀acc: A.
1062     ∀l: list B.
1063      ∀Q: A → Prop.
1064       (∀acc:A.∀b:B. Q acc → Q (H acc b)) →
1065         Q acc →
1066          Q (foldl A B H acc l).
1067*)
1068
1069(*
1070lemma tech_pc_sigma00_append_Some:
1071 ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc.
1072  tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 →
1073   tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉.
1074 #program #pol #prefix #costs #label #i #ppc #pc #H
1075  whd in match tech_pc_sigma00 in ⊢ %; normalize nodelta;
1076  whd in match sigma00 in ⊢ %; normalize nodelta in ⊢ %;
1077  generalize in match (pi2 … pol) whd in ⊢ (% → ?) whd in ⊢ (?(??%?) → ?)
1078  whd in match sigma0; normalize nodelta;
1079  >foldl_step
1080  change with (? → match match sigma00 program pol prefix with [None ⇒ ? | Some res ⇒ ?] with [ None ⇒ ? | Some res ⇒ ? ] = ?)
1081  whd in match tech_pc_sigma00 in H; normalize nodelta in H;
1082  cases (sigma00 program pol prefix) in H ⊢ %
1083   [ whd in ⊢ (??%% → ?) #abs destruct(abs)
1084   | * #ppc' * #pc' #sigma_map normalize nodelta; #H generalize in match (option_destruct_Some ??? H)
1085     
1086     normalize nodelta; -H;
1087     
1088 
1089   generalize in match H; -H;
1090  generalize in match (foldl ?????); in H ⊢ (??match match % with [_ ⇒ ? | _ ⇒ ?] with [_ ⇒ ? | _ ⇒ ?]?)
1091   [2: whd in ⊢ (??%%)
1092XXX
1093*)
1094
1095(* axiom construct_costs_sigma:
1096 ∀p.∀pol:policy p.∀ppc,pc,costs,i.
1097  bitvector_of_nat ? pc = sigma p pol (bitvector_of_nat ? ppc) →
1098   bitvector_of_nat ? (\fst (construct_costs p pol ppc pc costs i)) = sigma p pol (bitvector_of_nat 16 (S ppc)).
1099
1100axiom tech_pc_sigma00_append_Some:
1101 ∀program.∀pol:policy program.∀prefix,costs,label,i,ppc,pc.
1102  tech_pc_sigma00 program pol prefix = Some … 〈ppc,pc〉 →
1103   tech_pc_sigma00 program pol (prefix@[〈label,i〉]) = Some … 〈S ppc,\fst (construct_costs program pol … ppc pc costs i)〉. *)
1104
1105axiom eq_identifier_eq:
1106  ∀tag: String.
1107  ∀l.
1108  ∀r.
1109    eq_identifier tag l r = true → l = r.
1110
1111axiom neq_identifier_neq:
1112  ∀tag: String.
1113  ∀l, r: identifier tag.
1114    eq_identifier tag l r = false → (l = r → False).
1115
1116(* label_map: identifier ↦ pseudo program counter *)
1117definition label_map ≝ identifier_map ASMTag ℕ.
1118
1119(* Labels *)
1120definition is_label ≝
1121  λx:labelled_instruction.λl:Identifier.
1122  let 〈lbl,instr〉 ≝ x in
1123  match lbl with
1124  [ Some l' ⇒ l' = l
1125  | _       ⇒ False
1126  ].
1127
1128lemma label_does_not_occur:
1129  ∀i:ℕ.∀p:list labelled_instruction.∀l:Identifier.
1130  is_label (nth i ? p 〈None ?, Comment [ ]〉) l → does_not_occur ?? l p = false.
1131 #i #p #l generalize in match i; elim p
1132 [ #i >nth_nil #H cases H
1133 | #h #t #IH #i cases i -i
1134   [ cases h #hi #hp cases hi
1135     [ normalize #H cases H
1136     | #l' #Heq whd in ⊢ (??%?); change with (eq_identifier ? l' l) in match (instruction_matches_identifier ????);
1137       whd in Heq; >Heq
1138       >eq_identifier_refl / by refl/
1139     ]
1140   | #i #H whd in match (does_not_occur ????);
1141     whd in match (instruction_matches_identifier ????);
1142     cases h #hi #hp cases hi normalize nodelta
1143     [ @(IH i) @H
1144     | #l' @eq_identifier_elim
1145       [ normalize / by /
1146       | normalize #_ @(IH i) @H
1147       ]
1148     ]
1149   ]
1150 ]
1151qed.
1152
1153(* The function that creates the label-to-address map *)
1154definition create_label_cost_map0: ∀program:list labelled_instruction.
1155  (Σlabels_costs:label_map × (BitVectorTrie costlabel 16). (* Both on ppcs *)
1156    let 〈labels,costs〉 ≝ labels_costs in
1157    ∀l.occurs_exactly_once ?? l program →
1158    bitvector_of_nat ? (lookup_def ?? labels l 0) =
1159     address_of_word_labels_code_mem program l
1160  ) ≝
1161  λprogram.
1162  \fst (pi1 ?? (foldl_strong (option Identifier × pseudo_instruction)
1163  (λprefix.Σlabels_costs_ppc:label_map × (BitVectorTrie costlabel 16) × ℕ.
1164    let 〈labels,costs,ppc〉 ≝ labels_costs_ppc in
1165    ppc = |prefix| ∧
1166    ∀l.occurs_exactly_once ?? l prefix →
1167    bitvector_of_nat ? (lookup_def ?? labels l 0) =
1168     address_of_word_labels_code_mem prefix l)
1169  program
1170  (λprefix.λx.λtl.λprf.λlabels_costs_ppc.
1171   let 〈labels,costs,ppc〉 ≝ pi1 ?? labels_costs_ppc in
1172   let 〈label,instr〉 ≝ x in
1173   let labels ≝
1174     match label with
1175     [ None   ⇒ labels
1176     | Some l ⇒ add … labels l ppc
1177     ] in
1178   let costs ≝
1179     match instr with
1180     [ Cost cost ⇒ insert … (bitvector_of_nat ? ppc) cost costs
1181     | _ ⇒ costs ] in
1182      〈labels,costs,S ppc〉
1183   ) 〈(empty_map …),(Stub ??),0〉)).
1184[ normalize nodelta lapply (pi2 … labels_costs_ppc) >p >p1 normalize nodelta * #IH1 #IH2
1185  -labels_costs_ppc % [>IH1 >length_append <plus_n_Sm <plus_n_O %]
1186 inversion label [#EQ | #l #EQ]
1187 [ #lbl #Hocc <address_of_word_labels_code_mem_None [2: @Hocc] normalize nodelta
1188   >occurs_exactly_once_None in Hocc; @(IH2 lbl)
1189 | #lbl normalize nodelta inversion (eq_identifier ? lbl l)
1190   [ #Heq #Hocc >(eq_identifier_eq … Heq)
1191     >address_of_word_labels_code_mem_Some_hit
1192     [ >IH1 >lookup_def_add_hit %
1193     | <(eq_identifier_eq … Heq) in Hocc; //
1194     ]
1195   | #Hneq #Hocc
1196     <address_of_word_labels_code_mem_Some_miss
1197     [ >lookup_def_add_miss
1198       [ @IH2 >occurs_exactly_once_Some_eq in Hocc; >eq_identifier_sym> Hneq //
1199       | % @neq_identifier_neq @Hneq
1200       ]
1201     | @Hocc
1202     | >eq_identifier_sym @Hneq
1203     ]
1204   ]
1205 ]
1206| @pair_elim * #labels #costs #ppc #EQ destruct normalize nodelta % try %
1207  #l #abs cases (abs)
1208| cases (foldl_strong ? (λ_.Σx.?) ???) * * #labels #costs #ppc normalize nodelta *
1209  #_ #H @H
1210]
1211qed.
1212
1213(* The function that creates the label-to-address map *)
1214definition create_label_cost_map: ∀program:list labelled_instruction.
1215  label_map × (BitVectorTrie costlabel 16) ≝
1216    λprogram.
1217      pi1 … (create_label_cost_map0 program).
1218
1219theorem create_label_cost_map_ok:
1220 ∀pseudo_program: pseudo_assembly_program.
1221   let 〈labels, costs〉 ≝ create_label_cost_map (\snd pseudo_program) in
1222    ∀id. occurs_exactly_once ??  id (\snd pseudo_program) →
1223     bitvector_of_nat ? (lookup_def ?? labels id 0) = address_of_word_labels_code_mem (\snd pseudo_program) id.
1224 #p change with (pi1 … (create_label_cost_map0 ?)) in match (create_label_cost_map ?); @pi2
1225qed.
1226
1227(*CSC: move elsewhere; also proved in CostProofs as shift_nth_safe *)
1228lemma nth_safe_prepend:
1229 ∀A,l1,l2,j.∀H:j<|l2|.∀K:|l1|+j<|(l1@l2)|.
1230  nth_safe A j l2 H =nth_safe A (|l1|+j) (l1@l2) K.
1231 #A #l1 elim l1 normalize //
1232qed.
1233
1234(*CSC: move elsewhere; also proved in CostProofs as shift_nth_prefix *)
1235lemma shift_nth_prefix:
1236 ∀T,l1,i,l2,K1,K2.
1237  nth_safe T i l1 K1 = nth_safe T i (l1@l2) K2.
1238  #T #l1 elim l1 normalize
1239  [
1240    #i #l1 #K1 cases(lt_to_not_zero … K1)
1241  |
1242    #hd #tl #IH #i #l2
1243    cases i
1244    [
1245      //
1246    |
1247      #i' #K1 #K2 whd in ⊢ (??%%);
1248      @IH
1249    ]
1250  ]
1251qed.
1252
1253lemma nth_cons:
1254 ∀A,hd,tl,l2,j,d.
1255  nth j A (tl@l2) d =nth (1+j) A (hd::tl@l2) d.
1256//
1257qed.
1258
1259(*CSC: move elsewhere *)
1260lemma fetch_pseudo_instruction_append:
1261 ∀l1,l2,ppc.
1262  let code_newppc ≝ fetch_pseudo_instruction l2 ppc in
1263  fetch_pseudo_instruction (l1@l2) (add … (bitvector_of_nat … (|l1|)) (ppc)) =
1264  〈\fst code_newppc, add … (bitvector_of_nat … (|l1|)) (\snd code_newppc)〉.
1265 #l1 elim l1
1266 [ #l2 #ppc >add_commutative <add_zero >add_commutative <add_zero //
1267 | #hd #tl #IH #l2 #ppc whd whd in match fetch_pseudo_instruction in ⊢ (??%?); normalize nodelta
1268   (*CSC: FALSE, NEED INVARIANT? *)
1269   > (?: nat_of_bitvector … (add 16 (bitvector_of_nat 16 (|hd::tl|)) ppc)
1270       = 1 + nat_of_bitvector … (add … (bitvector_of_nat … (|tl|)) ppc)) [2: cases daemon]
1271   <nth_cons lapply (IH l2 ppc) -IH normalize nodelta cases (fetch_pseudo_instruction l2 ppc)
1272   #i #newppc whd in match fetch_pseudo_instruction; normalize nodelta
1273   cases (nth ? labelled_instruction ??) #i' #newppc' normalize nodelta #EQ
1274   destruct -EQ change with (add ??? = ?) in e0;
1275   (*CSC: TRUE, NEEDS TRIVIAL ARITHMETICS *) cases daemon
1276 ]
1277qed.
1278
1279definition assembly:
1280    ∀p: pseudo_assembly_program.
1281    ∀sigma: Word → Word.
1282    ∀policy: Word → bool.
1283      Σres:list Byte × (BitVectorTrie costlabel 16).
1284       let 〈preamble,instr_list〉 ≝ p in
1285       let 〈assembled,costs〉 ≝ res in
1286       let 〈labels_to_ppc,ppc_to_costs〉 ≝ create_label_cost_map instr_list in
1287       let datalabels ≝ construct_datalabels preamble in
1288       let lookup_labels ≝ λx. sigma (bitvector_of_nat ? (lookup_def … labels_to_ppc x 0)) in
1289       let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero ?) in
1290       ∀ppc.
1291        nat_of_bitvector … ppc < |instr_list| →
1292         let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc in
1293         let 〈len,assembledi〉 ≝
1294          assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in
1295         ∀j:nat. ∀H: j < |assembledi|. ∀K.
1296          nth_safe ? j assembledi H =
1297           nth_safe ? (nat_of_bitvector … (add … (sigma ppc) (bitvector_of_nat ? j)))
1298            assembled K
1299
1300  λp.
1301  λsigma.
1302  λpolicy.
1303  deplet 〈preamble, instr_list〉 as p_refl ≝ p in
1304  let 〈labels_to_ppc,ppc_to_costs〉 ≝ create_label_cost_map instr_list in
1305  let datalabels ≝ construct_datalabels preamble in
1306  let lookup_labels ≝ λx. sigma (bitvector_of_nat ? (lookup_def … labels_to_ppc x 0)) in
1307  let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero ?) in
1308  let 〈ignore,revcode〉 ≝ pi1 … (
1309     foldl_strong
1310      (option Identifier × pseudo_instruction)
1311      (λpre. Σppc_code:(Word × (list Byte)).
1312        let 〈ppc,code〉 ≝ ppc_code in
1313         ppc = bitvector_of_nat … (|pre|) ∧
1314         ∀ppc'.
1315          nat_of_bitvector … ppc' < nat_of_bitvector … ppc →
1316           let 〈pi,newppc〉 ≝ fetch_pseudo_instruction instr_list ppc' in
1317           let 〈len,assembledi〉 ≝
1318            assembly_1_pseudoinstruction lookup_labels sigma policy ppc' lookup_datalabels pi in
1319           ∀j:nat. ∀H: j < |assembledi|. ∀K.
1320            nth_safe ? j assembledi H =
1321             nth_safe ? (nat_of_bitvector … (add … (sigma ppc') (bitvector_of_nat ? j))) (reverse … code) K)
1322      instr_list
1323      (λprefix,hd,tl,prf,ppc_code.
1324        let 〈ppc, code〉 ≝ pi1 … ppc_code in
1325        let 〈pc_delta, program〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels (\snd hd) in
1326        let new_ppc ≝ add ? ppc (bitvector_of_nat ? 1) in
1327         〈new_ppc, (reverse … program @ code)〉)
1328      〈(zero ?), [ ]〉)
1329    in
1330     〈reverse … revcode,
1331      fold … (λppc.λcost.λpc_to_costs. insert … (sigma ppc) cost pc_to_costs) ppc_to_costs (Stub ??)〉.
1332  [ cases (foldl_strong ? (λx.Σy.?) ???) in p2; #ignore_revcode #Hfold #EQignore_revcode
1333    >EQignore_revcode in Hfold; * #_ #Hfold whd >p1 whd #ppc #LTppc @Hfold
1334    (* CSC: ??? *) cases daemon
1335  | % // #ppc' #abs @⊥ cases (not_le_Sn_O ?) [#H @(H abs) | skip]
1336  | cases ppc_code in p1; -ppc_code #ppc_code #IH #EQppc_code >EQppc_code in IH; -EQppc_code
1337    * #IH1 #IH2 % [ normalize nodelta >IH1 >length_append cases daemon (*CSC: TRUE, LEMMA NEEDED *)]
1338    whd #ppc' #LTppc' cases hd in prf p2; #label #pi #prf #p2
1339    cases (le_to_or_lt_eq … LTppc')
1340    [2: #S_S_eq normalize nodelta in S_S_eq;
1341        (*CSC: FALSE, NEEDS INVARIANT *)
1342        cut (ppc' = ppc) [cases daemon] -S_S_eq #EQppc' >EQppc' in LTppc'; -ppc' #LTppc
1343        >prf >IH1 in ⊢ match % with [_ ⇒ ?]; >(add_zero … (bitvector_of_nat 16 (|prefix|))) in ⊢ match % with [_ ⇒ ?];
1344        @pair_elim #pi' #newppc' >fetch_pseudo_instruction_append #EQpair destruct(EQpair)
1345        >p2
1346        #j #LTj
1347        (* CSC: FALSE, NEEDS INVARIANT *)
1348        >(?: nat_of_bitvector … (add … (sigma ppc) (bitvector_of_nat … j)) =
1349             nat_of_bitvector … (sigma ppc) + j) [2: cases daemon]
1350        >reverse_append >reverse_reverse
1351        (* CSC: TRUE, NEEDS INVARIANT *)
1352        >(? : nat_of_bitvector … (sigma ppc) = |reverse … code|) [2: cases daemon]
1353        @nth_safe_prepend
1354    | #LTppc' lapply (IH2 ppc' ?) [ (*CSC: EASY, FINISH*) cases daemon ]
1355      @pair_elim #pi' #newppc' #eq_fetch_pseudoinstruction
1356      @pair_elim #len' #assembledi' #eq_assembly_1_pseudoinstruction #IH
1357      change with (let 〈len,assembledi〉 ≝ assembly_1_pseudoinstruction ????? pi' in ∀j:ℕ. ∀H:j<|assembledi|.?)
1358      >eq_assembly_1_pseudoinstruction #j #LTj >reverse_append >reverse_reverse #K
1359      >IH
1360      [2: (*CSC: FALSE, NEEDS INVARIANT? *) cases daemon
1361      | @shift_nth_prefix
1362      ]
1363    ]
1364  ] 
1365qed.
1366
1367definition assembly_unlabelled_program:
1368    assembly_program → option (list Byte × (BitVectorTrie Identifier 16)) ≝
1369  λp.
1370    Some … (〈foldr … (λi,l. assembly1 i @ l) [ ] p, Stub …〉).
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