source: src/ASM/AssemblyProof.ma @ 1870

Last change on this file since 1870 was 1668, checked in by boender, 9 years ago
  • split build_maps into build_maps and build_maps_ok
  • work with CSC on partial proof of main_thm (cases Comment and Cost)
File size: 88.2 KB
Line 
1include "ASM/Assembly.ma".
2include "ASM/Interpret.ma".
3include "ASM/StatusProofs.ma".
4include alias "arithmetics/nat.ma".
5
6definition bit_elim_prop: ∀P: bool → Prop. Prop ≝
7  λP.
8    P true ∧ P false.
9 
10let rec bitvector_elim_prop_internal
11  (n: nat) (P: BitVector n → Prop) (m: nat) on m: m ≤ n → BitVector (n - m) → Prop ≝
12  match m return λm. m ≤ n → BitVector (n - m) → Prop with
13  [ O    ⇒ λprf1. λprefix. P ?
14  | S n' ⇒ λprf2. λprefix. bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' ? ?)
15  ].
16  [ applyS prefix
17  | letin res ≝ (bit ::: prefix)
18    < (minus_S_S ? ?)
19    > (minus_Sn_m ? ?)
20  [ @ res
21  | @ prf2
22  ]
23  | /2/
24  ].
25qed.
26
27definition bitvector_elim_prop ≝
28  λn: nat.
29  λP: BitVector n → Prop.
30    bitvector_elim_prop_internal n P n ? ?.
31  [ @ (le_n ?)
32  | < (minus_n_n ?)
33    @ [[ ]]
34  ]
35qed.
36
37lemma bool_eq_internal_eq:
38  ∀b, c.
39    (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c.
40  #b #c
41  cases b
42  [ normalize //
43  | normalize
44    cases c
45    [ normalize //
46    | normalize //
47    ]
48  ]
49qed.
50
51definition bit_elim: ∀P: bool → bool. bool ≝
52  λP.
53    P true ∧ P false.
54
55let rec bitvector_elim_internal
56  (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝
57  match m return λm. m ≤ n → BitVector (n - m) → bool with
58  [ O    ⇒ λprf1. λprefix. P ?
59  | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?)
60  ].
61  [ applyS prefix
62  | letin res ≝ (bit ::: prefix)
63    < (minus_S_S ? ?)
64    > (minus_Sn_m ? ?)
65    [ @ res
66    | @ prf2
67    ]
68  | /2/
69  ].
70qed.
71
72definition bitvector_elim ≝
73  λn: nat.
74  λP: BitVector n → bool.
75    bitvector_elim_internal n P n ? ?.
76  [ @ (le_n ?)
77  | < (minus_n_n ?)
78    @ [[ ]]
79  ]
80qed.
81
82lemma super_rewrite2:
83 ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m.
84  ∀P: ∀m. Vector A m → Prop.
85   n=m → v1 ≃ v2 → P n v1 → P m v2.
86 #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ //
87qed.
88
89lemma vector_cons_append:
90  ∀A: Type[0].
91  ∀n: nat.
92  ∀e: A.
93  ∀v: Vector A n.
94    e ::: v = [[ e ]] @@ v.
95  # A # N # E # V
96  elim V
97  [ normalize %
98  | # NN # AA # VV # IH
99    normalize
100    %
101qed.
102
103lemma vector_cons_append2:
104  ∀A: Type[0].
105  ∀n, m: nat.
106  ∀v: Vector A n.
107  ∀q: Vector A m.
108  ∀hd: A.
109    hd:::(v@@q) = (hd:::v)@@q.
110  #A #n #m #v #q
111  elim v
112  [ #hd %
113  | #n' #hd' #tl' #ih #hd' <ih %
114  ]
115qed.
116
117lemma jmeq_cons_vector_monotone:
118  ∀A: Type[0].
119  ∀m, n: nat.
120  ∀v: Vector A m.
121  ∀q: Vector A n.
122  ∀prf: m = n.
123  ∀hd: A.
124    v ≃ q → hd:::v ≃ hd:::q.
125  #A #m #n #v #q #prf #hd #E
126  @(super_rewrite2 A … E)
127  [ assumption | % ]
128qed.
129
130lemma vector_associative_append:
131  ∀A: Type[0].
132  ∀n, m, o:  nat.
133  ∀v: Vector A n.
134  ∀q: Vector A m.
135  ∀r: Vector A o.
136    ((v @@ q) @@ r)
137    ≃
138    (v @@ (q @@ r)).
139  #A #n #m #o #v #q #r
140  elim v
141  [ %
142  | #n' #hd #tl #ih
143    <(vector_cons_append2 A … hd)
144    @jmeq_cons_vector_monotone
145    //
146  ]
147qed.
148
149lemma mem_middle_vector:
150  ∀A: Type[0].
151  ∀m, o: nat.
152  ∀eq: A → A → bool.
153  ∀reflex: ∀a. eq a a = true.
154  ∀p: Vector A m.
155  ∀a: A.
156  ∀r: Vector A o.
157    mem A eq ? (p@@(a:::r)) a = true.
158  # A # M # O # EQ # REFLEX # P # A
159  elim P
160  [ normalize
161    > (REFLEX A)
162    normalize
163    # H
164    %
165  | # NN # AA # PP # IH
166    normalize
167    cases (EQ A AA) //
168     @ IH
169  ]
170qed.
171
172lemma mem_monotonic_wrt_append:
173  ∀A: Type[0].
174  ∀m, o: nat.
175  ∀eq: A → A → bool.
176  ∀reflex: ∀a. eq a a = true.
177  ∀p: Vector A m.
178  ∀a: A.
179  ∀r: Vector A o.
180    mem A eq ? r a = true → mem A eq ? (p @@ r) a = true.
181  # A # M # O # EQ # REFLEX # P # A
182  elim P
183  [ #R #H @H
184  | #NN #AA # PP # IH #R #H
185    normalize
186    cases (EQ A AA)
187    [ normalize %
188    | @ IH @ H
189    ]
190  ]
191qed.
192
193lemma subvector_multiple_append:
194  ∀A: Type[0].
195  ∀o, n: nat.
196  ∀eq: A → A → bool.
197  ∀refl: ∀a. eq a a = true.
198  ∀h: Vector A o.
199  ∀v: Vector A n.
200  ∀m: nat.
201  ∀q: Vector A m.
202    bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)).
203  # A # O # N # EQ # REFLEX # H # V
204  elim V
205  [ normalize
206    # M # V %
207  | # NN # AA # VV # IH # MM # QQ
208    change with (bool_to_Prop (andb ??))
209    cut ((mem A EQ (O + (MM + S NN)) (H@@QQ@@AA:::VV) AA) = true)
210    [
211    | # HH > HH
212      > (vector_cons_append ? ? AA VV)
213      change with (bool_to_Prop (subvector_with ??????))
214      @(super_rewrite2 A ((MM + 1)+ NN) (MM+S NN) ??
215        (λSS.λVS.bool_to_Prop (subvector_with ?? (O+SS) ?? (H@@VS)))
216        ?
217        (vector_associative_append A ? ? ? QQ [[AA]] VV))
218      [ >associative_plus //
219      | @IH ]
220    ]
221    @(mem_monotonic_wrt_append)
222    [ @ REFLEX
223    | @(mem_monotonic_wrt_append)
224      [ @ REFLEX
225      | normalize
226        > REFLEX
227        normalize
228        %
229      ]
230    ]
231qed.
232
233lemma vector_cons_empty:
234  ∀A: Type[0].
235  ∀n: nat.
236  ∀v: Vector A n.
237    [[ ]] @@ v = v.
238  # A # N # V
239  elim V
240  [ normalize %
241  | # NN # HH # VV #H %
242  ]
243qed.
244
245corollary subvector_hd_tl:
246  ∀A: Type[0].
247  ∀o: nat.
248  ∀eq: A → A → bool.
249  ∀refl: ∀a. eq a a = true.
250  ∀h: A.
251  ∀v: Vector A o.
252    bool_to_Prop (subvector_with A ? ? eq v (h ::: v)).
253  # A # O # EQ # REFLEX # H # V
254  > (vector_cons_append A ? H V)
255  < (vector_cons_empty A ? ([[H]] @@ V))
256  @ (subvector_multiple_append A ? ? EQ REFLEX [[]] V ? [[ H ]])
257qed.
258
259lemma eq_a_reflexive:
260  ∀a. eq_a a a = true.
261  # A
262  cases A
263  %
264qed.
265
266lemma is_in_monotonic_wrt_append:
267  ∀m, n: nat.
268  ∀p: Vector addressing_mode_tag m.
269  ∀q: Vector addressing_mode_tag n.
270  ∀to_search: addressing_mode.
271    bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search).
272  # M # N # P # Q # TO_SEARCH
273  # H
274  elim Q
275  [ normalize
276    @ H
277  | # NN # PP # QQ # IH
278    normalize
279    cases (is_a PP TO_SEARCH)
280    [ normalize
281      %
282    | normalize
283      normalize in IH;
284      @ IH
285    ]
286  ]
287qed.
288
289corollary is_in_hd_tl:
290  ∀to_search: addressing_mode.
291  ∀hd: addressing_mode_tag.
292  ∀n: nat.
293  ∀v: Vector addressing_mode_tag n.
294    bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search).
295  # TO_SEARCH # HD # N # V
296  elim V
297  [ # H
298    normalize in H;
299    cases H
300  | # NN # HHD # VV # IH # HH
301    > vector_cons_append
302    > (vector_cons_append ? ? HHD VV)
303    @ (is_in_monotonic_wrt_append ? 1 ([[HHD]]@@VV) [[HD]] TO_SEARCH)
304    @ HH
305  ]
306qed.
307 
308let rec list_addressing_mode_tags_elim
309  (n: nat) (l: Vector addressing_mode_tag (S n)) on l: (l → bool) → bool ≝
310  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
311   (l → bool) → bool ] with
312  [ VEmpty      ⇒  true 
313  | VCons len hd tl ⇒ λP.
314    let process_hd ≝
315      match hd return λhd. ∀P: hd:::tl → bool. bool with
316      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
317      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
318      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
319      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
320      | acc_a ⇒ λP.P ACC_A
321      | acc_b ⇒ λP.P ACC_B
322      | dptr ⇒ λP.P DPTR
323      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
324      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
325      | acc_dptr ⇒ λP.P ACC_DPTR
326      | acc_pc ⇒ λP.P ACC_PC
327      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
328      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
329      | carry ⇒ λP.P CARRY
330      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
331      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
332      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
333      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
334      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
335      ]
336    in
337      andb (process_hd P)
338       (match len return λx. x = len → bool with
339         [ O ⇒ λprf. true
340         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
341  ].
342  try %
343  [ 2: cases (sym_eq ??? prf); @tl
344  | generalize in match H; generalize in match tl; cases prf;
345    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
346    #tl
347    normalize in ⊢ (∀_: %. ?);
348    # H
349    whd
350    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]);
351    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
352qed.
353
354definition product_elim ≝
355  λm, n: nat.
356  λv: Vector addressing_mode_tag (S m).
357  λq: Vector addressing_mode_tag (S n).
358  λP: (v × q) → bool.
359    list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)).
360
361definition union_elim ≝
362  λA, B: Type[0].
363  λelimA: (A → bool) → bool.
364  λelimB: (B → bool) → bool.
365  λelimU: A ⊎ B → bool.
366    elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))).
367
368(*                           
369definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝
370  λP.
371    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧
372    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧
373    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧
374    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧
375    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧
376    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧
377    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧
378    list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
379    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧
380    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧
381    P (DA ? ACC_A) ∧
382    bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧
383    bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧
384    bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧
385    bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧
386    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧
387    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧
388    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧
389    list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
390    P (RL ? ACC_A) ∧
391    P (RLC ? ACC_A) ∧
392    P (RR ? ACC_A) ∧
393    P (RRC ? ACC_A) ∧
394    P (SWAP ? ACC_A) ∧
395    P (RET ?) ∧
396    P (RETI ?) ∧
397    P (NOP ?) ∧
398    bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧
399    list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧
400    bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧
401    bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧
402    union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]])
403                   (product_elim ? ? [[ registr; indirect ]] [[ data ]])
404                   (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧
405    list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧
406    union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]])
407                   (product_elim ? ? [[direct]] [[ acc_a ; data ]])
408                   (λd. P (XRL ? d)) ∧
409    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
410                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
411                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
412                   (λd. P (ANL ? d)) ∧
413    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]])
414                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
415                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
416                   (λd. P (ORL ? d)) ∧
417    union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]])
418                   (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]])
419                   (λd. P (MOVX ? d)) ∧
420    union_elim ? ? (
421      union_elim ? ? (
422        union_elim ? ? (
423          union_elim ? ? (
424            union_elim ? ?  (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
425                            (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]]))
426                            (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]]))
427                            (product_elim ? ? [[dptr]] [[data16]]))
428                            (product_elim ? ? [[carry]] [[bit_addr]]))
429                            (product_elim ? ? [[bit_addr]] [[carry]])
430                            (λd. P (MOV ? d)).
431  %
432qed.
433 
434definition instruction_elim: ∀P: instruction → bool. bool ≝
435  λP. (*
436    bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧
437    bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧
438    bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧
439    bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *)
440    bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (*  ∧
441    P (JMP INDIRECT_DPTR) ∧
442    list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧
443    preinstruction_elim (λp. P (RealInstruction p)). *)
444  %
445qed.
446
447
448axiom instruction_elim_complete:
449 ∀P. instruction_elim P = true → ∀i. P i = true.
450*)
451(*definition eq_instruction ≝
452  λi, j: instruction.
453    true.*)
454
455definition eq_addressing_mode: addressing_mode → addressing_mode → bool ≝
456  λa, b: addressing_mode.
457  match a with
458  [ DIRECT d ⇒
459    match b with
460    [ DIRECT e ⇒ eq_bv ? d e
461    | _ ⇒ false
462    ]
463  | INDIRECT b' ⇒
464    match b with
465    [ INDIRECT e ⇒ eq_b b' e
466    | _ ⇒ false
467    ]
468  | EXT_INDIRECT b' ⇒
469    match b with
470    [ EXT_INDIRECT e ⇒ eq_b b' e
471    | _ ⇒ false
472    ]
473  | REGISTER bv ⇒
474    match b with
475    [ REGISTER bv' ⇒ eq_bv ? bv bv'
476    | _ ⇒ false
477    ]
478  | ACC_A ⇒ match b with [ ACC_A ⇒ true | _ ⇒ false ]
479  | ACC_B ⇒ match b with [ ACC_B ⇒ true | _ ⇒ false ]
480  | DPTR ⇒ match b with [ DPTR ⇒ true | _ ⇒ false ]
481  | DATA b' ⇒
482    match b with
483    [ DATA e ⇒ eq_bv ? b' e
484    | _ ⇒ false
485    ]
486  | DATA16 w ⇒
487    match b with
488    [ DATA16 e ⇒ eq_bv ? w e
489    | _ ⇒ false
490    ]
491  | ACC_DPTR ⇒ match b with [ ACC_DPTR ⇒ true | _ ⇒ false ]
492  | ACC_PC ⇒ match b with [ ACC_PC ⇒ true | _ ⇒ false ]
493  | EXT_INDIRECT_DPTR ⇒ match b with [ EXT_INDIRECT_DPTR ⇒ true | _ ⇒ false ]
494  | INDIRECT_DPTR ⇒ match b with [ INDIRECT_DPTR ⇒ true | _ ⇒ false ]
495  | CARRY ⇒ match b with [ CARRY ⇒ true | _ ⇒ false ]
496  | BIT_ADDR b' ⇒
497    match b with
498    [ BIT_ADDR e ⇒ eq_bv ? b' e
499    | _ ⇒ false
500    ]
501  | N_BIT_ADDR b' ⇒
502    match b with
503    [ N_BIT_ADDR e ⇒ eq_bv ? b' e
504    | _ ⇒ false
505    ]
506  | RELATIVE n ⇒
507    match b with
508    [ RELATIVE e ⇒ eq_bv ? n e
509    | _ ⇒ false
510    ]
511  | ADDR11 w ⇒
512    match b with
513    [ ADDR11 e ⇒ eq_bv ? w e
514    | _ ⇒ false
515    ]
516  | ADDR16 w ⇒
517    match b with
518    [ ADDR16 e ⇒ eq_bv ? w e
519    | _ ⇒ false
520    ]
521  ].
522
523lemma eq_bv_refl:
524  ∀n, b.
525    eq_bv n b b = true.
526  #n #b cases b //
527qed.
528
529lemma eq_b_refl:
530  ∀b.
531    eq_b b b = true.
532  #b cases b //
533qed.
534
535lemma eq_addressing_mode_refl:
536  ∀a. eq_addressing_mode a a = true.
537  #a cases a try #arg1 try #arg2 try @eq_bv_refl try @eq_b_refl
538  try normalize %
539qed.
540 
541definition eq_sum: ∀A, B. (A → A → bool) → (B → B → bool) → (A ⊎ B) → (A ⊎ B) → bool ≝
542  λlt, rt, leq, req, left, right.
543    match left with
544    [ inl l ⇒
545      match right with
546      [ inl l' ⇒ leq l l'
547      | _ ⇒ false
548      ]
549    | inr r ⇒
550      match right with
551      [ inr r' ⇒ req r r'
552      | _ ⇒ false
553      ]
554    ].
555
556definition eq_prod: ∀A, B. (A → A → bool) → (B → B → bool) → (A × B) → (A × B) → bool ≝
557  λlt, rt, leq, req, left, right.
558    let 〈l, r〉 ≝ left in
559    let 〈l', r'〉 ≝ right in
560      leq l l' ∧ req r r'.
561
562definition eq_preinstruction: preinstruction [[relative]] → preinstruction [[relative]] → bool ≝
563  λi, j.
564  match i with
565  [ ADD arg1 arg2 ⇒
566    match j with
567    [ ADD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
568    | _ ⇒ false
569    ]
570  | ADDC arg1 arg2 ⇒
571    match j with
572    [ ADDC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
573    | _ ⇒ false
574    ]
575  | SUBB arg1 arg2 ⇒
576    match j with
577    [ SUBB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
578    | _ ⇒ false
579    ]
580  | INC arg ⇒
581    match j with
582    [ INC arg' ⇒ eq_addressing_mode arg arg'
583    | _ ⇒ false
584    ]
585  | DEC arg ⇒
586    match j with
587    [ DEC arg' ⇒ eq_addressing_mode arg arg'
588    | _ ⇒ false
589    ]
590  | MUL arg1 arg2 ⇒
591    match j with
592    [ MUL arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
593    | _ ⇒ false
594    ]
595  | DIV arg1 arg2 ⇒
596    match j with
597    [ DIV arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
598    | _ ⇒ false
599    ]
600  | DA arg ⇒
601    match j with
602    [ DA arg' ⇒ eq_addressing_mode arg arg'
603    | _ ⇒ false
604    ]
605  | JC arg ⇒
606    match j with
607    [ JC arg' ⇒ eq_addressing_mode arg arg'
608    | _ ⇒ false
609    ]
610  | JNC arg ⇒
611    match j with
612    [ JNC arg' ⇒ eq_addressing_mode arg arg'
613    | _ ⇒ false
614    ]
615  | JB arg1 arg2 ⇒
616    match j with
617    [ JB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
618    | _ ⇒ false
619    ]
620  | JNB arg1 arg2 ⇒
621    match j with
622    [ JNB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
623    | _ ⇒ false
624    ]
625  | JBC arg1 arg2 ⇒
626    match j with
627    [ JBC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
628    | _ ⇒ false
629    ]
630  | JZ arg ⇒
631    match j with
632    [ JZ arg' ⇒ eq_addressing_mode arg arg'
633    | _ ⇒ false
634    ]
635  | JNZ arg ⇒
636    match j with
637    [ JNZ arg' ⇒ eq_addressing_mode arg arg'
638    | _ ⇒ false
639    ]
640  | CJNE arg1 arg2 ⇒
641    match j with
642    [ CJNE arg1' arg2' ⇒
643      let prod_eq_left ≝ eq_prod [[acc_a]] [[direct; data]] eq_addressing_mode eq_addressing_mode in
644      let prod_eq_right ≝ eq_prod [[registr; indirect]] [[data]] eq_addressing_mode eq_addressing_mode in
645      let arg1_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
646        arg1_eq arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
647    | _ ⇒ false
648    ]
649  | DJNZ arg1 arg2 ⇒
650    match j with
651    [ DJNZ arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
652    | _ ⇒ false
653    ]
654  | CLR arg ⇒
655    match j with
656    [ CLR arg' ⇒ eq_addressing_mode arg arg'
657    | _ ⇒ false
658    ]
659  | CPL arg ⇒
660    match j with
661    [ CPL arg' ⇒ eq_addressing_mode arg arg'
662    | _ ⇒ false
663    ]
664  | RL arg ⇒
665    match j with
666    [ RL arg' ⇒ eq_addressing_mode arg arg'
667    | _ ⇒ false
668    ]
669  | RLC arg ⇒
670    match j with
671    [ RLC arg' ⇒ eq_addressing_mode arg arg'
672    | _ ⇒ false
673    ]
674  | RR arg ⇒
675    match j with
676    [ RR arg' ⇒ eq_addressing_mode arg arg'
677    | _ ⇒ false
678    ]
679  | RRC arg ⇒
680    match j with
681    [ RRC arg' ⇒ eq_addressing_mode arg arg'
682    | _ ⇒ false
683    ]
684  | SWAP arg ⇒
685    match j with
686    [ SWAP arg' ⇒ eq_addressing_mode arg arg'
687    | _ ⇒ false
688    ]
689  | SETB arg ⇒
690    match j with
691    [ SETB arg' ⇒ eq_addressing_mode arg arg'
692    | _ ⇒ false
693    ]
694  | PUSH arg ⇒
695    match j with
696    [ PUSH arg' ⇒ eq_addressing_mode arg arg'
697    | _ ⇒ false
698    ]
699  | POP arg ⇒
700    match j with
701    [ POP arg' ⇒ eq_addressing_mode arg arg'
702    | _ ⇒ false
703    ]
704  | XCH arg1 arg2 ⇒
705    match j with
706    [ XCH arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
707    | _ ⇒ false
708    ]
709  | XCHD arg1 arg2 ⇒
710    match j with
711    [ XCHD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
712    | _ ⇒ false
713    ]
714  | RET ⇒ match j with [ RET ⇒ true | _ ⇒ false ]
715  | RETI ⇒ match j with [ RETI ⇒ true | _ ⇒ false ]
716  | NOP ⇒ match j with [ NOP ⇒ true | _ ⇒ false ]
717  | MOVX arg ⇒
718    match j with
719    [ MOVX arg' ⇒
720      let prod_eq_left ≝ eq_prod [[acc_a]] [[ext_indirect; ext_indirect_dptr]] eq_addressing_mode eq_addressing_mode in
721      let prod_eq_right ≝ eq_prod [[ext_indirect; ext_indirect_dptr]] [[acc_a]] eq_addressing_mode eq_addressing_mode in
722      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
723        sum_eq arg arg'
724    | _ ⇒ false
725    ]
726  | XRL arg ⇒
727    match j with
728    [ XRL arg' ⇒
729      let prod_eq_left ≝ eq_prod [[acc_a]] [[ data ; registr ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in
730      let prod_eq_right ≝ eq_prod [[direct]] [[ acc_a ; data ]] eq_addressing_mode eq_addressing_mode in
731      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
732        sum_eq arg arg'
733    | _ ⇒ false
734    ]
735  | ORL arg ⇒
736    match j with
737    [ ORL arg' ⇒
738      let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; data ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in
739      let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in
740      let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in
741      let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in
742      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
743        sum_eq arg arg'
744    | _ ⇒ false
745    ]
746  | ANL arg ⇒
747    match j with
748    [ ANL arg' ⇒
749      let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; direct ; indirect ; data ]] eq_addressing_mode eq_addressing_mode in
750      let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in
751      let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in
752      let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in
753      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
754        sum_eq arg arg'
755    | _ ⇒ false
756    ]
757  | MOV arg ⇒
758    match j with
759    [ MOV arg' ⇒
760      let prod_eq_6 ≝ eq_prod [[acc_a]] [[registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in
761      let prod_eq_5 ≝ eq_prod [[registr; indirect]] [[acc_a; direct; data]] eq_addressing_mode eq_addressing_mode in
762      let prod_eq_4 ≝ eq_prod [[direct]] [[acc_a; registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in
763      let prod_eq_3 ≝ eq_prod [[dptr]] [[data16]] eq_addressing_mode eq_addressing_mode in
764      let prod_eq_2 ≝ eq_prod [[carry]] [[bit_addr]] eq_addressing_mode eq_addressing_mode in
765      let prod_eq_1 ≝ eq_prod [[bit_addr]] [[carry]] eq_addressing_mode eq_addressing_mode in
766      let sum_eq_1 ≝ eq_sum ? ? prod_eq_6 prod_eq_5 in
767      let sum_eq_2 ≝ eq_sum ? ? sum_eq_1 prod_eq_4 in
768      let sum_eq_3 ≝ eq_sum ? ? sum_eq_2 prod_eq_3 in
769      let sum_eq_4 ≝ eq_sum ? ? sum_eq_3 prod_eq_2 in
770      let sum_eq_5 ≝ eq_sum ? ? sum_eq_4 prod_eq_1 in
771        sum_eq_5 arg arg'
772    | _ ⇒ false
773    ]
774  ].
775
776lemma eq_sum_refl:
777  ∀A, B: Type[0].
778  ∀leq, req.
779  ∀s.
780  ∀leq_refl: (∀t. leq t t = true).
781  ∀req_refl: (∀u. req u u = true).
782    eq_sum A B leq req s s = true.
783  #A #B #leq #req #s #leq_refl #req_refl
784  cases s whd in ⊢ (? → ??%?); //
785qed.
786
787lemma eq_prod_refl:
788  ∀A, B: Type[0].
789  ∀leq, req.
790  ∀s.
791  ∀leq_refl: (∀t. leq t t = true).
792  ∀req_refl: (∀u. req u u = true).
793    eq_prod A B leq req s s = true.
794  #A #B #leq #req #s #leq_refl #req_refl
795  cases s whd in ⊢ (? → ? → ??%?); #l #r >leq_refl normalize @req_refl
796qed.
797
798lemma eq_preinstruction_refl:
799  ∀i.
800    eq_preinstruction i i = true.
801  #i cases i try #arg1 try #arg2
802  try @eq_addressing_mode_refl
803  [1,2,3,4,5,6,7,8,10,16,17,18,19,20:
804    whd in ⊢ (??%?);
805    try %
806    >eq_addressing_mode_refl
807    >eq_addressing_mode_refl %
808  |13,15:
809    whd in ⊢ (??%?);
810    cases arg1
811    [*:
812      #arg1_left normalize nodelta
813      >eq_prod_refl [*: try % #argr @eq_addressing_mode_refl]
814    ]
815  |11,12:
816    whd in ⊢ (??%?);
817    cases arg1
818    [1:
819      #arg1_left normalize nodelta
820      >(eq_sum_refl …)
821      [1: % | 2,3: #arg @eq_prod_refl ]
822      @eq_addressing_mode_refl
823    |2:
824      #arg1_left normalize nodelta
825      @eq_prod_refl [*: @eq_addressing_mode_refl ]
826    |3:
827      #arg1_left normalize nodelta
828      >(eq_sum_refl …) [1: %
829      |2,3: #arg @eq_prod_refl #arg @eq_addressing_mode_refl ]
830    |4:
831      #arg1_left normalize nodelta
832      @eq_prod_refl [*: #arg @eq_addressing_mode_refl ]
833    ]
834  |14:
835    whd in ⊢ (??%?);
836    cases arg1
837    [ #arg1_left normalize nodelta
838      @eq_sum_refl
839      [1: #arg @eq_sum_refl
840        [1: #arg @eq_sum_refl
841          [1: #arg @eq_sum_refl
842            [1: #arg @eq_prod_refl
843              [*: @eq_addressing_mode_refl ]
844            |2: #arg @eq_prod_refl
845              [*: #arg @eq_addressing_mode_refl ]
846            ]
847          |2: #arg @eq_prod_refl
848            [*: #arg @eq_addressing_mode_refl ]
849          ]
850        |2: #arg @eq_prod_refl
851          [*: #arg @eq_addressing_mode_refl ]
852        ]
853      |2: #arg @eq_prod_refl
854        [*: #arg @eq_addressing_mode_refl ]
855      ]
856    |2: #arg1_right normalize nodelta
857        @eq_prod_refl [*: #arg @eq_addressing_mode_refl ]
858    ]
859  |*:
860    whd in ⊢ (??%?);
861    cases arg1
862    [*: #arg1 >eq_sum_refl
863      [1,4: normalize @eq_addressing_mode_refl
864      |2,3,5,6: #arg @eq_prod_refl
865        [*: #arg @eq_addressing_mode_refl ]
866      ]
867    ]
868  ]
869qed.
870
871definition eq_instruction: instruction → instruction → bool ≝
872  λi, j.
873  match i with
874  [ ACALL arg ⇒
875    match j with
876    [ ACALL arg' ⇒ eq_addressing_mode arg arg'
877    | _ ⇒ false
878    ]
879  | LCALL arg ⇒
880    match j with
881    [ LCALL arg' ⇒ eq_addressing_mode arg arg'
882    | _ ⇒ false
883    ]
884  | AJMP arg ⇒
885    match j with
886    [ AJMP arg' ⇒ eq_addressing_mode arg arg'
887    | _ ⇒ false
888    ]
889  | LJMP arg ⇒
890    match j with
891    [ LJMP arg' ⇒ eq_addressing_mode arg arg'
892    | _ ⇒ false
893    ]
894  | SJMP arg ⇒
895    match j with
896    [ SJMP arg' ⇒ eq_addressing_mode arg arg'
897    | _ ⇒ false
898    ]
899  | JMP arg ⇒
900    match j with
901    [ JMP arg' ⇒ eq_addressing_mode arg arg'
902    | _ ⇒ false
903    ]
904  | MOVC arg1 arg2 ⇒
905    match j with
906    [ MOVC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
907    | _ ⇒ false
908    ]
909  | RealInstruction instr ⇒
910    match j with
911    [ RealInstruction instr' ⇒ eq_preinstruction instr instr'
912    | _ ⇒ false
913    ]
914  ].
915 
916lemma eq_instruction_refl:
917  ∀i. eq_instruction i i = true.
918  #i cases i
919  [1,2,3,4,5,6: #arg1 @eq_addressing_mode_refl
920  |7: #arg1 #arg2
921      whd in ⊢ (??%?);
922      >eq_addressing_mode_refl
923      >eq_addressing_mode_refl
924      %
925  |8: #arg @eq_preinstruction_refl
926  ]
927qed.
928
929let rec vect_member
930  (A: Type[0]) (n: nat) (eq: A → A → bool)
931  (v: Vector A n) (a: A) on v: bool ≝
932  match v with
933  [ VEmpty          ⇒ false
934  | VCons len hd tl ⇒
935    eq hd a ∨ (vect_member A ? eq tl a)
936  ].
937
938let rec list_addressing_mode_tags_elim_prop
939  (n: nat)
940  (l: Vector addressing_mode_tag (S n))
941  on l:
942  ∀P: l → Prop.
943  ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a.
944  ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a.
945  ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a.
946  ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a.
947  ∀x: l. P x ≝
948  match l return
949    λy.
950      match y with
951      [ O    ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True
952      | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop.
953               ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True.
954               ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True.
955               ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True.
956               ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True.
957               ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True.
958               ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True.
959               ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True.
960               ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True.
961               ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True.
962               ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True.
963               ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True.
964               ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True.
965               ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True.
966               ∀carry_a: if vect_member … eq_a l carry then P CARRY else True.
967               ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True.
968               ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True.
969               ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True.
970               ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True.
971               ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True.
972               ∀x:l. P x
973      ] with
974  [ VEmpty          ⇒ λAbsurd. ⊥
975  | VCons len hd tl ⇒ λProof. ?
976  ] (refl ? (S n)). cases daemon. qed. (*
977  [ destruct(Absurd)
978  | # A1 # A2 # A3 # A4 # A5 # A6 # A7
979    # A8 # A9 # A10 # A11 # A12 # A13 # A14
980    # A15 # A16 # A17 # A18 # A19 # X
981    cases X
982    # SUB cases daemon ] qed.
983    cases SUB
984    [ # BYTE
985    normalize
986  ].
987 
988 
989(*    let prepare_hd ≝
990      match hd with
991      [ direct ⇒ λdirect_prf. ?
992      | indirect ⇒ λindirect_prf. ?
993      | ext_indirect ⇒ λext_indirect_prf. ?
994      | registr ⇒ λregistr_prf. ?
995      | acc_a ⇒ λacc_a_prf. ?
996      | acc_b ⇒ λacc_b_prf. ?
997      | dptr ⇒ λdptr_prf. ?
998      | data ⇒ λdata_prf. ?
999      | data16 ⇒ λdata16_prf. ?
1000      | acc_dptr ⇒ λacc_dptr_prf. ?
1001      | acc_pc ⇒ λacc_pc_prf. ?
1002      | ext_indirect_dptr ⇒ λext_indirect_prf. ?
1003      | indirect_dptr ⇒ λindirect_prf. ?
1004      | carry ⇒ λcarry_prf. ?
1005      | bit_addr ⇒ λbit_addr_prf. ?
1006      | n_bit_addr ⇒ λn_bit_addr_prf. ?
1007      | relative ⇒ λrelative_prf. ?
1008      | addr11 ⇒ λaddr11_prf. ?
1009      | addr16 ⇒ λaddr16_prf. ?
1010      ]
1011    in ? *)
1012  ].
1013  [ 1: destruct(absd)
1014  | 2: # A1 # A2 # A3 # A4 # A5 # A6
1015       # A7 # A8 # A9 # A10 # A11 # A12
1016       # A13 # A14 # A15 # A16 # A17 # A18
1017       # A19 *
1018  ].
1019
1020
1021  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
1022   (l → bool) → bool ] with
1023  [ VEmpty      ⇒  true 
1024  | VCons len hd tl ⇒ λP.
1025    let process_hd ≝
1026      match hd return λhd. ∀P: hd:::tl → bool. bool with
1027      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
1028      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
1029      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
1030      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
1031      | acc_a ⇒ λP.P ACC_A
1032      | acc_b ⇒ λP.P ACC_B
1033      | dptr ⇒ λP.P DPTR
1034      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
1035      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
1036      | acc_dptr ⇒ λP.P ACC_DPTR
1037      | acc_pc ⇒ λP.P ACC_PC
1038      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
1039      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
1040      | carry ⇒ λP.P CARRY
1041      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
1042      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
1043      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
1044      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
1045      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
1046      ]
1047    in
1048      andb (process_hd P)
1049       (match len return λx. x = len → bool with
1050         [ O ⇒ λprf. true
1051         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
1052  ].
1053  try %
1054  [ 2: cases (sym_eq ??? prf); @tl
1055  | generalize in match H; generalize in match tl; cases prf;
1056    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
1057    #tl
1058    normalize in ⊢ (∀_: %. ?)
1059    # H
1060    whd
1061    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1062    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
1063qed.
1064*)
1065
1066definition load_code_memory_aux ≝
1067 fold_left_i_aux … (
1068   λi, mem, v.
1069     insert … (bitvector_of_nat … i) v mem) (Stub Byte 16).
1070
1071lemma split_zero:
1072  ∀A,m.
1073  ∀v: Vector A m.
1074    〈[[]], v〉 = split A 0 m v.
1075  #A #m #v
1076  elim v
1077  [ %
1078  | #n #hd #tl #ih
1079    normalize in ⊢ (???%); %
1080  ]
1081qed.
1082
1083lemma Vector_O: ∀A. ∀v: Vector A 0. v ≃ VEmpty A.
1084 #A #v generalize in match (refl … 0); cases v in ⊢ (??%? → ?%%??); //
1085 #n #hd #tl #abs @⊥ destruct (abs)
1086qed.
1087
1088lemma Vector_Sn: ∀A. ∀n.∀v:Vector A (S n).
1089 ∃hd.∃tl.v ≃ VCons A n hd tl.
1090 #A #n #v generalize in match (refl … (S n)); cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??)));
1091 [ #abs @⊥ destruct (abs)
1092 | #m #hd #tl #EQ <(injective_S … EQ) %[@hd] %[@tl] // ]
1093qed.
1094
1095lemma vector_append_zero:
1096  ∀A,m.
1097  ∀v: Vector A m.
1098  ∀q: Vector A 0.
1099    v = q@@v.
1100  #A #m #v #q
1101  >(Vector_O A q) %
1102qed.
1103
1104lemma prod_eq_left:
1105  ∀A: Type[0].
1106  ∀p, q, r: A.
1107    p = q → 〈p, r〉 = 〈q, r〉.
1108  #A #p #q #r #hyp
1109  >hyp %
1110qed.
1111
1112lemma prod_eq_right:
1113  ∀A: Type[0].
1114  ∀p, q, r: A.
1115    p = q → 〈r, p〉 = 〈r, q〉.
1116  #A #p #q #r #hyp
1117  >hyp %
1118qed.
1119
1120corollary prod_vector_zero_eq_left:
1121  ∀A, n.
1122  ∀q: Vector A O.
1123  ∀r: Vector A n.
1124    〈q, r〉 = 〈[[ ]], r〉.
1125  #A #n #q #r
1126  generalize in match (Vector_O A q …);
1127  #hyp
1128  >hyp in ⊢ (??%?);
1129  %
1130qed.
1131
1132axiom split_succ:
1133  ∀A, m, n.
1134  ∀l: Vector A m.
1135  ∀r: Vector A n.
1136  ∀v: Vector A (m + n).
1137  ∀hd: A.
1138    v = l@@r → (〈l, r〉 = split A m n v → 〈hd:::l, r〉 = split A (S m) n (hd:::v)).
1139(*
1140  #A #m
1141  elim m
1142  [ #n #l #r #v #hd #equal #hyp
1143    normalize >(Vector_O A l) >equal
1144    >(Vector_O A l) %
1145  | #n' #ih #n #l #r #v #hd
1146    #equal #hyp
1147    cases(Vector_Sn A n' l)
1148    #hd' #exists
1149    cases exists #tl #jmeq
1150    >jmeq *)
1151
1152lemma split_prod:
1153  ∀A,m,n.
1154  ∀p: Vector A (m + n).
1155  ∀v: Vector A m.
1156  ∀q: Vector A n.
1157    p = v@@q → 〈v, q〉 = split A m n p.
1158  #A #m
1159  elim m
1160  [ #n #p #v #q #hyp
1161    >hyp <(vector_append_zero A n q v)
1162    >(prod_vector_zero_eq_left A …)
1163    @split_zero
1164  | #r #ih #n #p #v #q #hyp
1165    >hyp
1166    cases (Vector_Sn A r v)
1167    #hd #exists
1168    cases exists
1169    #tl #jmeq >jmeq
1170    @split_succ [1: % |2: @ih % ]
1171  ]
1172qed.
1173
1174(*
1175lemma split_prod_exists:
1176  ∀A, m, n.
1177  ∀p: Vector A (m + n).
1178  ∃v: Vector A m.
1179  ∃q: Vector A n.
1180    〈v, q〉 = split A m n p.
1181  #A #m #n #p
1182  elim m
1183  @ex_intro
1184  [1:
1185  |2: @ex_intro
1186      [1:
1187      |2:
1188      ]
1189  ]
1190*)
1191
1192axiom split_elim:
1193 ∀A,l,m,v.∀P: (Vector A l) × (Vector A m) → Prop.
1194  (∀vl,vm. v = vl@@vm → P 〈vl,vm〉) → P (split A l m v).
1195(*  #A #l #m #v #P #hyp
1196  normalize
1197  <(split_prod A l m v ? ? ?)
1198*)
1199
1200example half_add_SO:
1201 ∀pc.
1202 \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc).
1203 cases daemon.
1204qed.
1205
1206(*
1207axiom not_eqvb_S:
1208 ∀pc.
1209 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))).
1210
1211axiom not_eqvb_SS:
1212 ∀pc.
1213 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))).
1214 
1215axiom bitvector_elim_complete:
1216 ∀n,P. bitvector_elim n P = true → ∀bv. P bv.
1217
1218lemma bitvector_elim_complete':
1219 ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true.
1220 #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv
1221 generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ //
1222qed.
1223*)
1224
1225(*
1226lemma andb_elim':
1227 ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true.
1228 #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize //
1229qed.
1230*)
1231
1232let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word)
1233                       (encoding: list Byte) on encoding: Prop ≝
1234  match encoding with
1235  [ nil ⇒ final_pc = pc
1236  | cons hd tl ⇒
1237    let 〈new_pc, byte〉 ≝ next code_memory pc in
1238      hd = byte ∧ encoding_check code_memory new_pc final_pc tl
1239  ].
1240
1241lemma encoding_check_append: ∀code_memory,final_pc,l1,pc,l2.
1242 encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … final_pc) (l1@l2) →
1243  let intermediate_pc ≝ pc + length … l1 in
1244   encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … intermediate_pc) l1 ∧
1245    encoding_check code_memory (bitvector_of_nat … intermediate_pc) (bitvector_of_nat … final_pc) l2.
1246 #code_memory #final_pc #l1 elim l1
1247  [ #pc #l2 whd in ⊢ (????% → ?); #H <plus_n_O whd whd in ⊢ (?%?); /2/
1248  | #hd #tl #IH #pc #l2 * #H1 #H2 >half_add_SO in H2; #H2 cases (IH … H2) <plus_n_Sm
1249    #K1 #K2 % [2:@K2] whd % // >half_add_SO @K1 ]
1250qed.
1251
1252axiom bitvector_3_elim_prop:
1253 ∀P: BitVector 3 → Prop.
1254  P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] →
1255  P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] →
1256  P [[true;true;false]] → P [[true;true;true]] → ∀v. P v.
1257
1258definition ticks_of_instruction ≝
1259 λi.
1260  let trivial_code_memory ≝ assembly1 i in
1261  let trivial_status ≝ load_code_memory trivial_code_memory in
1262   \snd (fetch trivial_status (zero ?)).
1263
1264lemma fetch_assembly:
1265  ∀pc,i,code_memory,assembled.
1266    assembled = assembly1 i →
1267      let len ≝ length … assembled in
1268      encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
1269      let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in
1270      let 〈instr_pc, ticks〉 ≝ fetched in
1271      let 〈instr,pc'〉 ≝ instr_pc in
1272       (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' (bitvector_of_nat … (pc + len))) = true.
1273 #pc #i #code_memory #assembled cases i [8: *]
1274 [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]]
1275 [47,48,49:
1276 |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg
1277  [2,3,5,7,10,12,16,17,18,21,25,26,27,30,31,32,37,38,39,40,41,42,43,44,45,48,51,58,
1278   59,60,63,64,65,66,67: #ARG]]
1279 [4,5,6,7,8,9,10,11,12,13,22,23,24,27,28,39,40,41,42,43,44,45,46,47,48,49,50,51,52,
1280  56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2
1281  [1,2,4,7,9,10,12,13,15,16,17,18,20,22,23,24,25,26,27,28,29,30,31,32,33,36,37,38,
1282   39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,
1283   68,69,70,71: #ARG2]]
1284 [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3]
1285 normalize in ⊢ (???% → ?);
1286 [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)]
1287  normalize in ⊢ (???% → ?);]
1288 #H >H * #H1 try (whd in ⊢ (% → ?); * #H2)
1289 try (whd in ⊢ (% → ?); * #H3) whd in ⊢ (% → ?); #H4
1290 [ whd in match fetch; normalize nodelta <H1 ] cases daemon
1291 (*XXX
1292   
1293 whd in ⊢ (let fetched ≝ ??% in ?) <H1 whd in ⊢ (let fetched ≝ % in ?)
1294 [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3]
1295 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,
1296  30,31,32,33,34,35,36,37,38,39,40,43,45,48,49,52,53,54,55,56,57,60,61,62,65,66,
1297  69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2]
1298 whd >eq_instruction_refl >H4 @eq_bv_refl*)
1299qed.
1300
1301let rec fetch_many code_memory final_pc pc expected on expected: Prop ≝
1302 match expected with
1303  [ nil ⇒ eq_bv … pc final_pc = true
1304  | cons i tl ⇒
1305     let fetched ≝ fetch code_memory pc in
1306     let 〈instr_pc, ticks〉 ≝ fetched in
1307     let 〈instr,pc'〉 ≝ instr_pc in
1308      eq_instruction instr i = true ∧
1309      ticks = (ticks_of_instruction i) ∧
1310      fetch_many code_memory final_pc pc' tl].
1311
1312lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b.
1313 #A #a #b #EQ destruct //
1314qed.
1315
1316axiom eq_instruction_to_eq:
1317  ∀i1,i2. eq_instruction i1 i2 = true → i1 = i2.
1318               
1319lemma fetch_assembly_pseudo':
1320 ∀lookup_labels.∀pol:policy_type lookup_labels.∀ppc,lookup_datalabels.
1321  ∀pi,code_memory,len,assembled,instructions,pc.
1322   let expansion ≝ pol ppc in
1323   instructions = expand_pseudo_instruction lookup_labels ppc expansion lookup_datalabels pi →
1324    〈len,assembled〉 = assembly_1_pseudoinstruction lookup_labels pol ppc lookup_datalabels pi →
1325     encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
1326      fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions.
1327 #lookup_labels #pol #ppc #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc
1328 #EQ1 whd in ⊢ (???% → ?); <EQ1 whd in ⊢ (???% → ?); #EQ2
1329 cases (pair_destruct ?????? EQ2) -EQ2; #EQ2a #EQ2b
1330 >EQ2a >EQ2b -EQ2a -EQ2b;
1331  generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc
1332  generalize in match pc; elim instructions
1333  [ #pc whd in ⊢ (% → %); #H >H @eq_bv_refl
1334  | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd
1335    generalize in match (fetch_assembly pc i code_memory … (refl …) H1);
1336    cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %);
1337    cases newi_pc #newi #newpc whd in ⊢ (% → %); #K cases (conjunction_true … K) -K; #K1
1338    cases (conjunction_true … K1) -K1; #K1 #K2 #K3 % try %
1339    [ @K1 | @eqb_true_to_eq <(eq_instruction_to_eq … K1) @K2 | >(eq_bv_eq … K3) @IH @H2 ]
1340qed.
1341
1342axiom bitvector_of_nat_nat_of_bitvector:
1343  ∀n,v.
1344    bitvector_of_nat n (nat_of_bitvector n v) = v.
1345
1346(* CSC: soo long next one; can we merge with previous one now? *)
1347lemma fetch_assembly_pseudo:
1348  ∀program:pseudo_assembly_program.
1349  ∀pol:policy program.
1350  let lookup_labels ≝ λx:Identifier.sigma … pol (address_of_word_labels_code_mem (\snd  program) x) in
1351  ∀ppc.∀code_memory.
1352  let lookup_datalabels ≝ λx:Identifier.lookup_def … (construct_datalabels (\fst  program)) x (zero 16) in
1353  let pc ≝ sigma program pol ppc in
1354  let pi ≝  \fst  (fetch_pseudo_instruction (\snd program) ppc) in
1355  let instructions ≝ expand_pseudo_instruction lookup_labels ppc (pol lookup_labels ppc) lookup_datalabels pi in
1356  let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels (pol lookup_labels) ppc lookup_datalabels pi in
1357    encoding_check code_memory pc (bitvector_of_nat … (nat_of_bitvector ? pc + len)) assembled →
1358     fetch_many code_memory (bitvector_of_nat … (nat_of_bitvector ? pc + len)) pc instructions.
1359 #program #pol letin lookup_labels ≝ (λx.?) #ppc #code_memory
1360 letin lookup_datalabels ≝ (λx.?)
1361 letin pc ≝ (sigma ???) letin pi ≝ (fst ???) 
1362 letin instructions ≝ (expand_pseudo_instruction ?????)
1363 @pair_elim #len #assembled #EQ1 #H
1364 generalize in match
1365  (fetch_assembly_pseudo' lookup_labels ((pi1 ?? pol) lookup_labels) ppc lookup_datalabels pi code_memory len
1366   assembled instructions (nat_of_bitvector ? pc)) in ⊢ ?;
1367  >bitvector_of_nat_nat_of_bitvector >EQ1 normalize nodelta
1368  #X @X //
1369qed.
1370
1371(* This is a trivial consequence of fetch_assembly + the proof that the
1372   function that load the code in memory is correct. The latter is based
1373   on missing properties from the standard library on the BitVectorTrie
1374   data structrure. *)
1375axiom assembly_ok:
1376 ∀program,pol,assembled,costs'.
1377  let 〈labels,costs〉 ≝ build_maps program pol in
1378  〈assembled,costs'〉 = assembly program pol →
1379  costs = costs' ∧
1380  let code_memory ≝ load_code_memory assembled in
1381  let preamble ≝ \fst program in
1382  let datalabels ≝ construct_datalabels preamble in
1383  let lookup_labels ≝ λx.sigma … pol (address_of_word_labels_code_mem (\snd program) x) in 
1384  let lookup_datalabels ≝ λx. lookup_def ?? datalabels x (zero ?) in
1385  ∀ppc.
1386  let pi_newppc ≝ fetch_pseudo_instruction (\snd program) ppc in
1387   ∀len,assembledi.
1388     〈len,assembledi〉 = assembly_1_pseudoinstruction lookup_labels (pol lookup_labels) ppc lookup_datalabels (\fst pi_newppc) →
1389      encoding_check code_memory (sigma program pol ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len)) assembledi ∧
1390       sigma program pol (\snd pi_newppc) = bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len).
1391
1392lemma fetch_assembly_pseudo2:
1393 ∀program,pol,ppc.
1394(*  let 〈labels,costs〉 ≝ build_maps program pol in *)
1395  let assembled ≝ \fst (assembly program pol) in
1396  let code_memory ≝ load_code_memory assembled in
1397  let data_labels ≝ construct_datalabels (\fst program) in
1398  let lookup_labels ≝ λx.sigma … pol (address_of_word_labels_code_mem (\snd program) x) in
1399  let lookup_datalabels ≝ λx. lookup_def ? ? data_labels x (zero ?) in
1400  let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1401  let instructions ≝ expand_pseudo_instruction lookup_labels ppc (pol lookup_labels ppc) lookup_datalabels pi in
1402   fetch_many code_memory (sigma program pol newppc) (sigma program pol ppc) instructions.
1403 * #preamble #instr_list #pol #ppc
1404 letin assembled ≝ (\fst (assembly 〈preamble,instr_list〉 pol))
1405 letin code_memory ≝ (load_code_memory assembled)
1406 letin data_labels ≝ (construct_datalabels preamble)
1407 letin lookup_labels ≝ (λx. sigma … pol (address_of_word_labels_code_mem instr_list x))
1408 letin lookup_datalabels ≝ (λx. lookup_def ? ? data_labels x (zero ?))
1409 @pair_elim #pi #newppc #EQ1 change with (fetch_pseudo_instruction instr_list ? = ?) in EQ1;
1410 generalize in match (assembly_ok ? pol assembled (\snd (assembly 〈preamble,instr_list〉 pol)));
1411 @pair_elim #labels #costs #EQ2 <eq_pair_fst_snd
1412 #H cases (H (refl \ldots)) -H; #EQ3
1413 generalize in match (refl … (assembly_1_pseudoinstruction lookup_labels (pol lookup_labels) ppc lookup_datalabels ?));
1414 [ cases (assembly_1_pseudoinstruction ?????) in ⊢ (???% → ?); #len #assembledi #EQ4
1415   #H cases (H ppc len assembledi ?) [2: <EQ4 %] -H; #H1 #H2
1416   (* XXX JPB: attention to lookup_labels *)
1417   generalize in match (fetch_assembly_pseudo 〈preamble,instr_list〉 pol ppc (load_code_memory assembled)) in ⊢ ?;
1418   >EQ4; #H generalize in match (H H1) in ⊢ ?; -H >(pair_destruct_2 ????? (sym_eq … EQ1))
1419   >H2 >EQ1 #K @K
1420qed.
1421
1422(* OLD?
1423definition assembly_specification:
1424  ∀assembly_program: pseudo_assembly_program.
1425  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1426  λpseudo_assembly_program.
1427  λcode_mem.
1428    ∀pc: Word.
1429      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1430      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1431      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1432      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1433      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1434       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1435      match pre_assembled with
1436       [ None ⇒ True
1437       | Some pc_code ⇒
1438          let 〈new_pc,code〉 ≝ pc_code in
1439           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1440
1441axiom assembly_meets_specification:
1442  ∀pseudo_assembly_program.
1443    match assembly pseudo_assembly_program with
1444    [ None ⇒ True
1445    | Some code_mem_cost ⇒
1446      let 〈code_mem, cost〉 ≝ code_mem_cost in
1447        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1448    ].
1449(*
1450  # PROGRAM
1451  [ cases PROGRAM
1452    # PREAMBLE
1453    # INSTR_LIST
1454    elim INSTR_LIST
1455    [ whd
1456      whd in ⊢ (∀_. %)
1457      # PC
1458      whd
1459    | # INSTR
1460      # INSTR_LIST_TL
1461      # H
1462      whd
1463      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1464    ]
1465  | cases not_implemented
1466  ] *)
1467*)
1468
1469definition internal_pseudo_address_map ≝ list (BitVector 8).
1470
1471axiom low_internal_ram_of_pseudo_low_internal_ram:
1472 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1473
1474axiom high_internal_ram_of_pseudo_high_internal_ram:
1475 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1476
1477axiom low_internal_ram_of_pseudo_internal_ram_hit:
1478 ∀M:internal_pseudo_address_map.∀cm.∀s:PseudoStatus cm.∀pol:policy cm.∀addr:BitVector 7.
1479  member ? (eq_bv 8) (false:::addr) M = true →
1480   let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1481   let pbl ≝ lookup ? 7 addr (low_internal_ram … s) (zero 8) in
1482   let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram … s) (zero 8) in
1483   let bl ≝ lookup ? 7 addr ram (zero 8) in
1484   let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in
1485    bu@@bl = sigma cm pol (pbu@@pbl).
1486
1487(* changed from add to sub *)
1488axiom low_internal_ram_of_pseudo_internal_ram_miss:
1489 ∀T.∀M:internal_pseudo_address_map.∀cm.∀s:PreStatus T cm.∀addr:BitVector 7.
1490  let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1491  let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in
1492  let carr ≝ get_index_v ? ? flags 1 ? in
1493  carr = false →
1494  member ? (eq_bv 8) (false:::Saddr) M = false →
1495   member ? (eq_bv 8) (false:::addr) M = false →
1496    lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?).
1497  //
1498qed.
1499
1500definition addressing_mode_ok ≝
1501 λT.λM:internal_pseudo_address_map.λcm.λs:PreStatus T cm.
1502  λaddr:addressing_mode.
1503   match addr with
1504    [ DIRECT d ⇒
1505       ¬(member ? (eq_bv 8) d M) ∧
1506       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1507    | INDIRECT i ⇒
1508       let d ≝ get_register … s [[false;false;i]] in
1509       ¬(member ? (eq_bv 8) d M) ∧
1510       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1511    | EXT_INDIRECT _ ⇒ true
1512    | REGISTER _ ⇒ true
1513    | ACC_A ⇒ true
1514    | ACC_B ⇒ true
1515    | DPTR ⇒ true
1516    | DATA _ ⇒ true
1517    | DATA16 _ ⇒ true
1518    | ACC_DPTR ⇒ true
1519    | ACC_PC ⇒ true
1520    | EXT_INDIRECT_DPTR ⇒ true
1521    | INDIRECT_DPTR ⇒ true
1522    | CARRY ⇒ true
1523    | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1524    | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1525    | RELATIVE _ ⇒ true
1526    | ADDR11 _ ⇒ true
1527    | ADDR16 _ ⇒ true ].
1528   
1529definition next_internal_pseudo_address_map0 ≝
1530  λT.
1531  λfetched.
1532  λM: internal_pseudo_address_map.
1533  λcm:T.
1534  λs: PreStatus T cm.
1535   match fetched with
1536    [ Comment _ ⇒ Some ? M
1537    | Cost _ ⇒ Some … M
1538    | Jmp _ ⇒ Some … M
1539    | Call _ ⇒
1540       Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M)
1541    | Mov _ _ ⇒ Some … M
1542    | Instruction instr ⇒
1543       match instr with
1544        [ ADD addr1 addr2 ⇒
1545           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1546            Some ? M
1547           else
1548            None ?
1549        | ADDC addr1 addr2 ⇒
1550           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1551            Some ? M
1552           else
1553            None ?
1554        | SUBB addr1 addr2 ⇒
1555           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1556            Some ? M
1557           else
1558            None ?       
1559        | _ ⇒ (* TO BE COMPLETED *) Some ? M ]].
1560 
1561
1562definition next_internal_pseudo_address_map ≝
1563 λM:internal_pseudo_address_map.
1564 λcm.
1565  λs:PseudoStatus cm.
1566    next_internal_pseudo_address_map0 ?
1567     (\fst (fetch_pseudo_instruction (\snd cm) (program_counter … s))) M cm s.
1568
1569definition code_memory_of_pseudo_assembly_program:
1570 ∀pap:pseudo_assembly_program. policy pap → BitVectorTrie Byte 16
1571≝ λpap,pol.
1572   let p ≝ assembly pap pol in
1573    load_code_memory (\fst p).
1574
1575definition status_of_pseudo_status:
1576 internal_pseudo_address_map → ∀pap.∀ps:PseudoStatus pap. ∀pol:policy pap.
1577  Status (code_memory_of_pseudo_assembly_program … pol) ≝
1578 λM,pap,ps,pol.
1579  let cm ≝ code_memory_of_pseudo_assembly_program … pol in
1580  let pc ≝ sigma pap pol (program_counter … ps) in
1581  let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in
1582  let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in
1583     mk_PreStatus (BitVectorTrie Byte 16)
1584      cm
1585      lir
1586      hir
1587      (external_ram … ps)
1588      pc
1589      (special_function_registers_8051 … ps)
1590      (special_function_registers_8052 … ps)
1591      (p1_latch … ps)
1592      (p3_latch … ps)
1593      (clock … ps).
1594
1595(*
1596definition write_at_stack_pointer':
1597 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1598  λM: Type[0].
1599  λs: PreStatus M.
1600  λv: Byte.
1601    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1602    let bit_zero ≝ get_index_v… nu O ? in
1603    let bit_1 ≝ get_index_v… nu 1 ? in
1604    let bit_2 ≝ get_index_v… nu 2 ? in
1605    let bit_3 ≝ get_index_v… nu 3 ? in
1606      if bit_zero then
1607        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1608                              v (low_internal_ram ? s) in
1609          set_low_internal_ram ? s memory
1610      else
1611        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1612                              v (high_internal_ram ? s) in
1613          set_high_internal_ram ? s memory.
1614  [ cases l0 %
1615  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1616qed.
1617
1618definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1619 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1620≝
1621  λticks_of.
1622  λs.
1623  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1624  let ticks ≝ ticks_of (program_counter ? s) in
1625  let s ≝ set_clock ? s (clock ? s + ticks) in
1626  let s ≝ set_program_counter ? s pc in
1627    match instr with
1628    [ Instruction instr ⇒
1629       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1630    | Comment cmt ⇒ s
1631    | Cost cst ⇒ s
1632    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1633    | Call call ⇒
1634      let a ≝ address_of_word_labels s call in
1635      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1636      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1637      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1638      let s ≝ write_at_stack_pointer' ? s pc_bl in
1639      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1640      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1641      let s ≝ write_at_stack_pointer' ? s pc_bu in
1642        set_program_counter ? s a
1643    | Mov dptr ident ⇒
1644       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1645    ].
1646 [
1647 |2,3,4: %
1648 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1649 |
1650 | %
1651 ]
1652 cases not_implemented
1653qed.
1654*)
1655
1656(*
1657lemma execute_code_memory_unchanged:
1658 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1659 #ticks #ps whd in ⊢ (??? (??%))
1660 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1661  (program_counter pseudo_assembly_program ps)) #instr #pc
1662 whd in ⊢ (??? (??%)) cases instr
1663  [ #pre cases pre
1664     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1665       cases (split ????) #z1 #z2 %
1666     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1667       cases (split ????) #z1 #z2 %
1668     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1669       cases (split ????) #z1 #z2 %
1670     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1671       [ #x1 whd in ⊢ (??? (??%))
1672     | *: cases not_implemented
1673     ]
1674  | #comment %
1675  | #cost %
1676  | #label %
1677  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1678    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1679    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1680    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1681    (* CSC: ??? *)
1682  | #dptr #label (* CSC: ??? *)
1683  ]
1684  cases not_implemented
1685qed.
1686*)
1687
1688(* DEAD CODE?
1689lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1690 ∀M:internal_pseudo_address_map.
1691 ∀ps,ps': PseudoStatus.
1692 ∀pol.
1693  ∀prf:code_memory … ps = code_memory … ps'.
1694   let pol' ≝ ? in
1695   match status_of_pseudo_status M ps pol with
1696    [ None ⇒ status_of_pseudo_status M ps' pol' = None …
1697    | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w
1698    ].
1699 [2: <prf @pol]
1700 #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1701 generalize in match (refl … (assembly (code_memory … ps) pol))
1702 cases (assembly ??) in ⊢ (???% → %)
1703  [ #K whd whd in ⊢ (??%?) <H >K %
1704  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1705qed.
1706*)
1707
1708definition ticks_of0: ∀p:pseudo_assembly_program. policy p → (Identifier→Word) → Word → ? → nat × nat ≝
1709  λprogram: pseudo_assembly_program.λpol.λlookup_labels.
1710  λppc: Word.
1711  λfetched.
1712    match fetched with
1713    [ Instruction instr ⇒
1714      match instr with
1715      [ JC lbl ⇒
1716        match pol lookup_labels ppc with
1717        [ short_jump ⇒ 〈2, 2〉
1718        | medium_jump ⇒ ?
1719        | long_jump ⇒ 〈4, 4〉
1720        ]
1721      | JNC lbl ⇒
1722        match pol lookup_labels ppc with
1723        [ short_jump ⇒ 〈2, 2〉
1724        | medium_jump ⇒ ?
1725        | long_jump ⇒ 〈4, 4〉
1726        ]
1727      | JB bit lbl ⇒
1728        match pol lookup_labels ppc with
1729        [ short_jump ⇒ 〈2, 2〉
1730        | medium_jump ⇒ ?
1731        | long_jump ⇒ 〈4, 4〉
1732        ]
1733      | JNB bit lbl ⇒
1734        match pol lookup_labels ppc with
1735        [ short_jump ⇒ 〈2, 2〉
1736        | medium_jump ⇒ ?
1737        | long_jump ⇒ 〈4, 4〉
1738        ]
1739      | JBC bit lbl ⇒
1740        match pol lookup_labels ppc with
1741        [ short_jump ⇒ 〈2, 2〉
1742        | medium_jump ⇒ ?
1743        | long_jump ⇒ 〈4, 4〉
1744        ]
1745      | JZ lbl ⇒
1746        match pol lookup_labels ppc with
1747        [ short_jump ⇒ 〈2, 2〉
1748        | medium_jump ⇒ ?
1749        | long_jump ⇒ 〈4, 4〉
1750        ]
1751      | JNZ lbl ⇒
1752        match pol lookup_labels  ppc with
1753        [ short_jump ⇒ 〈2, 2〉
1754        | medium_jump ⇒ ?
1755        | long_jump ⇒ 〈4, 4〉
1756        ]
1757      | CJNE arg lbl ⇒
1758        match pol lookup_labels ppc with
1759        [ short_jump ⇒ 〈2, 2〉
1760        | medium_jump ⇒ ?
1761        | long_jump ⇒ 〈4, 4〉
1762        ]
1763      | DJNZ arg lbl ⇒
1764        match pol lookup_labels ppc with
1765        [ short_jump ⇒ 〈2, 2〉
1766        | medium_jump ⇒ ?
1767        | long_jump ⇒ 〈4, 4〉
1768        ]
1769      | ADD arg1 arg2 ⇒
1770        let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in
1771         〈ticks, ticks〉
1772      | ADDC arg1 arg2 ⇒
1773        let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in
1774         〈ticks, ticks〉
1775      | SUBB arg1 arg2 ⇒
1776        let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in
1777         〈ticks, ticks〉
1778      | INC arg ⇒
1779        let ticks ≝ ticks_of_instruction (INC ? arg) in
1780         〈ticks, ticks〉
1781      | DEC arg ⇒
1782        let ticks ≝ ticks_of_instruction (DEC ? arg) in
1783         〈ticks, ticks〉
1784      | MUL arg1 arg2 ⇒
1785        let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in
1786         〈ticks, ticks〉
1787      | DIV arg1 arg2 ⇒
1788        let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in
1789         〈ticks, ticks〉
1790      | DA arg ⇒
1791        let ticks ≝ ticks_of_instruction (DA ? arg) in
1792         〈ticks, ticks〉
1793      | ANL arg ⇒
1794        let ticks ≝ ticks_of_instruction (ANL ? arg) in
1795         〈ticks, ticks〉
1796      | ORL arg ⇒
1797        let ticks ≝ ticks_of_instruction (ORL ? arg) in
1798         〈ticks, ticks〉
1799      | XRL arg ⇒
1800        let ticks ≝ ticks_of_instruction (XRL ? arg) in
1801         〈ticks, ticks〉
1802      | CLR arg ⇒
1803        let ticks ≝ ticks_of_instruction (CLR ? arg) in
1804         〈ticks, ticks〉
1805      | CPL arg ⇒
1806        let ticks ≝ ticks_of_instruction (CPL ? arg) in
1807         〈ticks, ticks〉
1808      | RL arg ⇒
1809        let ticks ≝ ticks_of_instruction (RL ? arg) in
1810         〈ticks, ticks〉
1811      | RLC arg ⇒
1812        let ticks ≝ ticks_of_instruction (RLC ? arg) in
1813         〈ticks, ticks〉
1814      | RR arg ⇒
1815        let ticks ≝ ticks_of_instruction (RR ? arg) in
1816         〈ticks, ticks〉
1817      | RRC arg ⇒
1818        let ticks ≝ ticks_of_instruction (RRC ? arg) in
1819         〈ticks, ticks〉
1820      | SWAP arg ⇒
1821        let ticks ≝ ticks_of_instruction (SWAP ? arg) in
1822         〈ticks, ticks〉
1823      | MOV arg ⇒
1824        let ticks ≝ ticks_of_instruction (MOV ? arg) in
1825         〈ticks, ticks〉
1826      | MOVX arg ⇒
1827        let ticks ≝ ticks_of_instruction (MOVX ? arg) in
1828         〈ticks, ticks〉
1829      | SETB arg ⇒
1830        let ticks ≝ ticks_of_instruction (SETB ? arg) in
1831         〈ticks, ticks〉
1832      | PUSH arg ⇒
1833        let ticks ≝ ticks_of_instruction (PUSH ? arg) in
1834         〈ticks, ticks〉
1835      | POP arg ⇒
1836        let ticks ≝ ticks_of_instruction (POP ? arg) in
1837         〈ticks, ticks〉
1838      | XCH arg1 arg2 ⇒
1839        let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in
1840         〈ticks, ticks〉
1841      | XCHD arg1 arg2 ⇒
1842        let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in
1843         〈ticks, ticks〉
1844      | RET ⇒
1845        let ticks ≝ ticks_of_instruction (RET ?) in
1846         〈ticks, ticks〉
1847      | RETI ⇒
1848        let ticks ≝ ticks_of_instruction (RETI ?) in
1849         〈ticks, ticks〉
1850      | NOP ⇒
1851        let ticks ≝ ticks_of_instruction (NOP ?) in
1852         〈ticks, ticks〉
1853      ]
1854    | Comment comment ⇒ 〈0, 0〉
1855    | Cost cost ⇒ 〈0, 0〉
1856    | Jmp jmp ⇒ 〈2, 2〉
1857    | Call call ⇒ 〈2, 2〉
1858    | Mov dptr tgt ⇒ 〈2, 2〉
1859    ].
1860  cases not_implemented (* policy returned medium_jump for conditional jumping = impossible *)
1861qed.
1862
1863definition ticks_of: ∀p:pseudo_assembly_program. policy p → (Identifier→Word) → Word → nat × nat ≝
1864  λprogram: pseudo_assembly_program.λpol.λlookup_labels.
1865  λppc: Word.
1866    let 〈preamble, pseudo〉 ≝ program in
1867    let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in
1868     ticks_of0 program pol lookup_labels ppc fetched.
1869
1870lemma eq_rect_Type1_r:
1871  ∀A: Type[1].
1872  ∀a:A.
1873  ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p.
1874  #A #a #P #H #x #p
1875  generalize in match H;
1876  generalize in match P;
1877  cases p
1878  //
1879qed.
1880
1881axiom split_append:
1882  ∀A: Type[0].
1883  ∀m, n: nat.
1884  ∀v, v': Vector A m.
1885  ∀q, q': Vector A n.
1886    let 〈v', q'〉 ≝ split A m n (v@@q) in
1887      v = v' ∧ q = q'.
1888
1889axiom split_vector_singleton:
1890  ∀A: Type[0].
1891  ∀n: nat.
1892  ∀v: Vector A (S n).
1893  ∀rest: Vector A n.
1894  ∀s: Vector A 1.
1895  ∀prf.
1896    v = s @@ rest →
1897    ((get_index_v A ? v 0 prf) ::: rest) = v.
1898
1899example sub_minus_one_seven_eight:
1900  ∀v: BitVector 7.
1901  false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) =
1902  \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false).
1903 cases daemon.
1904qed.
1905
1906(*
1907lemma blah:
1908  ∀m: internal_pseudo_address_map.
1909  ∀s: PseudoStatus.
1910  ∀arg: Byte.
1911  ∀b: bool.
1912    addressing_mode_ok m s (DIRECT arg) = true →
1913      get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) =
1914      get_arg_8 ? s b (DIRECT arg).
1915  [2, 3: normalize % ]
1916  #m #s #arg #b #hyp
1917  whd in ⊢ (??%%)
1918  @split_elim''
1919  #nu' #nl' #arg_nu_nl_eq
1920  normalize nodelta
1921  generalize in match (refl ? (get_index_v bool 4 nu' ? ?))
1922  cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %)
1923  #get_index_v_eq
1924  normalize nodelta
1925  [2:
1926    normalize nodelta
1927    @split_elim''
1928    #bit_one' #three_bits' #bit_one_three_bit_eq
1929    generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl'))
1930    normalize nodelta
1931    generalize in match (refl ? (sub_7_with_carry ? ? ?))
1932    cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %)
1933    #Saddr #carr' #Saddr_carr_eq
1934    normalize nodelta
1935    #carr_hyp'
1936    @carr_hyp'
1937    [1:
1938    |2: whd in hyp:(??%?); generalize in match hyp; -hyp;
1939        generalize in match (refl ? (¬(member (BitVector 8) ? arg m)))
1940        cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %)
1941        #member_eq
1942        normalize nodelta
1943        [2: #destr destruct(destr)
1944        |1: -carr_hyp';
1945            >arg_nu_nl_eq
1946            <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq)
1947            [1: >get_index_v_eq in ⊢ (??%? → ?)
1948            |2: @le_S @le_S @le_S @le_n
1949            ]
1950            cases (member (BitVector 8) ? (\fst ?) ?)
1951            [1: #destr normalize in destr; destruct(destr)
1952            |2:
1953            ]
1954        ]
1955    |3: >get_index_v_eq in ⊢ (??%?)
1956        change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl')
1957        >(split_vector_singleton … bit_one_three_bit_eq)
1958        <arg_nu_nl_eq
1959        whd in hyp:(??%?)
1960        cases (member (BitVector 8) (eq_bv 8) arg m) in hyp
1961        normalize nodelta [*: #ignore @sym_eq ]
1962    ]
1963  |
1964  ].
1965*)
1966(*
1967map_address0 ... (DIRECT arg) = Some .. →
1968  get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) =
1969  get_arg_8 (internal_ram ...) (DIRECT arg)
1970
1971((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) 
1972                     then Some internal_pseudo_address_map M 
1973                     else None internal_pseudo_address_map )
1974                    =Some internal_pseudo_address_map M')
1975*)
1976
1977axiom low_internal_ram_write_at_stack_pointer:
1978 ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
1979  get_8051_sfr ? cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP →
1980  low_internal_ram ? cm2 s2 = low_internal_ram T2 cm3 s3 →
1981  sp1 = \snd (half_add ? (get_8051_sfr … cm1 s1 SFR_SP) (bitvector_of_nat 8 1)) →
1982  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
1983  bu@@bl = sigma cm2 pol (pbu@@pbl) →
1984   low_internal_ram T1 cm1
1985     (write_at_stack_pointer …
1986       (set_8051_sfr …
1987         (write_at_stack_pointer …
1988           (set_8051_sfr …
1989             (set_low_internal_ram … s1
1990               (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s2)))
1991             SFR_SP sp1)
1992          bl)
1993        SFR_SP sp2)
1994      bu)
1995   = low_internal_ram_of_pseudo_low_internal_ram (sp1::M)
1996      (low_internal_ram …
1997       (write_at_stack_pointer …
1998         (set_8051_sfr …
1999           (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl)
2000          SFR_SP sp2)
2001        pbu)).
2002
2003axiom high_internal_ram_write_at_stack_pointer:
2004 ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3,pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2005  get_8051_sfr ? cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP →
2006  high_internal_ram ?? s2 = high_internal_ram T2 cm3 s3 →
2007  sp1 = \snd (half_add ? (get_8051_sfr ? cm1 s1 SFR_SP) (bitvector_of_nat 8 1)) →
2008  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
2009  bu@@bl = sigma cm2 pol (pbu@@pbl) →
2010   high_internal_ram T1 cm1
2011     (write_at_stack_pointer …
2012       (set_8051_sfr …
2013         (write_at_stack_pointer …
2014           (set_8051_sfr …
2015             (set_high_internal_ram … s1
2016               (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … s2)))
2017             SFR_SP sp1)
2018          bl)
2019        SFR_SP sp2)
2020      bu)
2021   = high_internal_ram_of_pseudo_high_internal_ram (sp1::M)
2022      (high_internal_ram …
2023       (write_at_stack_pointer …
2024         (set_8051_sfr …
2025           (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl)
2026          SFR_SP sp2)
2027        pbu)).
2028
2029lemma Some_Some_elim:
2030 ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P.
2031 #T #x #y #P #H #K @H @option_destruct_Some //
2032qed.
2033
2034(*usare snd_assembly_1_pseudoinstruction_ok:
2035 ∀program:pseudo_assembly_program.∀pol: policy program.
2036 ∀ppc:Word.∀pi,lookup_labels,lookup_datalabels.
2037  lookup_labels = (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) →
2038  lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
2039  \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
2040   let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels (pol lookup_labels) (sigma program pol ppc) lookup_datalabels  pi) in
2041    sigma program pol (\snd (half_add ? ppc (bitvector_of_nat ? 1))) =
2042     bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len).
2043*)
2044
2045lemma pose: ∀A:Type[0].∀B:A → Type[0].∀a:A. (∀a':A. a'=a → B a') → B a.
2046 /2/
2047qed.
2048
2049(* To be moved in ProofStatus *)
2050lemma program_counter_set_program_counter:
2051 ∀T,cm,s,x. program_counter T cm (set_program_counter T cm s x) = x.
2052 //
2053qed.
2054
2055theorem main_thm:
2056 ∀M,M',cm,ps,pol,lookup_labels.
2057  next_internal_pseudo_address_map M cm ps = Some … M' →
2058   ∃n.
2059      execute n … (status_of_pseudo_status M … ps pol)
2060    = status_of_pseudo_status M' … (execute_1_pseudo_instruction (ticks_of cm pol lookup_labels) cm ps) pol.
2061 #M #M' * #preamble #instr_list #ps #pol #lookup_labels
2062 change with (next_internal_pseudo_address_map0 ????? = ? → ?)
2063 @(pose … (program_counter ?? ps)) #ppc #EQppc
2064 generalize in match (fetch_assembly_pseudo2 ? pol ppc) in ⊢ ?;
2065 @(pose … (\fst (assembly ? pol))) #assembled #EQassembled
2066 @(pose … (λx.sigma … pol (address_of_word_labels_code_mem instr_list x))) #lookup_labels #EQlookup_labels
2067 @(pose … (λx. lookup_def … (construct_datalabels preamble) x (zero 16))) #lookup_datalabels #EQlookup_datalabels
2068 whd in match execute_1_pseudo_instruction; normalize nodelta
2069 @pair_elim #pi #newppc #H1
2070 whd in match ticks_of; normalize nodelta <EQppc #H2 >H1 normalize nodelta;
2071 lapply (snd_fetch_pseudo_instruction instr_list ppc) >H1 #EQnewppc >EQnewppc
2072 lapply (snd_assembly_1_pseudoinstruction_ok … ppc pi … EQlookup_labels EQlookup_datalabels ?)
2073 [>H1 %]
2074 inversion pi
2075  [2,3: (* Comment, Cost *) #ARG #EQ
2076   #H3 normalize nodelta in H3; normalize in match (assembly_1_pseudoinstruction ?????) in H3;
2077   whd in ⊢ (??%? → ?); @Some_Some_elim #MAP <MAP
2078   whd in match (execute_1_pseudo_instruction0 ?????);
2079   %{0} @split_eq_status //
2080  |4: (* Jmp *) #label whd in ⊢ (??%? → ???% → ?)
2081   @Some_Some_elim #MAP cases (pol ?) normalize nodelta
2082       [3: (* long *) #EQ3 @(Some_Some_elim ????? EQ3) #EQ3'
2083         whd in match eject normalize nodelta >EQ3' in ⊢ (% → ?) whd in ⊢ (% → ?)
2084         @pair_elim' * #instr #newppc' #ticks #EQ4       
2085         * * #H2a #H2b whd in ⊢ (% → ?) #H2
2086         >H2b >(eq_instruction_to_eq … H2a)
2087         #EQ %[@1]
2088         <MAP >(eq_bv_eq … H2) >EQ
2089         whd in ⊢ (??%?) >EQ4 whd in ⊢ (??%?)
2090         cases ps in EQ4; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXX >XXX %
2091         whd in ⊢ (??%?)
2092         whd in ⊢ (??(match ?%? with [_ ⇒ ?])?)
2093         cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
2094  |6: (* Mov *) #arg1 #arg2
2095       #H1 #H2 #EQ %[@1]
2096       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2097       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2098       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2099       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2100       >H2b >(eq_instruction_to_eq … H2a)
2101       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2102       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?)
2103       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2104       normalize nodelta;
2105       [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2106       #result #flags
2107       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
2108  |5: (* Call *) #label #MAP
2109      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP;
2110      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2111       [ (* short *) #abs @⊥ destruct (abs)
2112       |3: (* long *) #H1 #H2 #EQ %[@1]
2113           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2114           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2115           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2116           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2117           >H2b >(eq_instruction_to_eq … H2a)
2118           generalize in match EQ; -EQ;
2119           whd in ⊢ (???% → ??%?);
2120           generalize in match (refl … (half_add 8 (get_8051_sfr ? ps SFR_SP) (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry #new_sp #EQ1 normalize nodelta;
2121           >(eq_bv_eq … H2c)
2122           change with
2123            ((?=let 〈ppc_bu,ppc_bl〉 ≝ split bool 8 8 newppc in ?) →
2124                (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2125           generalize in match (refl … (split … 8 8 newppc)) cases (split bool 8 8 newppc) in ⊢ (??%? → %) #ppc_bu #ppc_bl #EQppc
2126           generalize in match (refl … (split … 8 8 (sigma 〈preamble,instr_list〉 pol newppc))) cases (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)) in ⊢ (??%? → %) #pc_bu #pc_bl #EQpc normalize nodelta;
2127           >get_8051_sfr_write_at_stack_pointer >get_8051_sfr_write_at_stack_pointer
2128           >get_8051_sfr_set_8051_sfr >get_8051_sfr_set_8051_sfr
2129           generalize in match (refl … (half_add ? new_sp (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry' #new_sp' #EQ2 normalize nodelta;
2130           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2131           @split_eq_status;
2132            [ >code_memory_write_at_stack_pointer whd in ⊢ (??%?)
2133              >code_memory_write_at_stack_pointer %
2134            | >set_program_counter_set_low_internal_ram
2135              >set_clock_set_low_internal_ram
2136              @low_internal_ram_write_at_stack_pointer
2137               [ >EQ0 @pol | % | %
2138               | @(pair_destruct_2 … EQ1)
2139               | @(pair_destruct_2 … EQ2)
2140               | >(pair_destruct_1 ????? EQpc)
2141                 >(pair_destruct_2 ????? EQpc)
2142                 @split_elim #x #y #H <H -x y H;
2143                 >(pair_destruct_1 ????? EQppc)
2144                 >(pair_destruct_2 ????? EQppc)
2145                 @split_elim #x #y #H <H -x y H;
2146                 >EQ0 % ]
2147            | >set_low_internal_ram_set_high_internal_ram
2148              >set_program_counter_set_high_internal_ram
2149              >set_clock_set_high_internal_ram
2150              @high_internal_ram_write_at_stack_pointer
2151               [ >EQ0 @pol | % | %
2152               | @(pair_destruct_2 … EQ1)
2153               | @(pair_destruct_2 … EQ2)
2154               | >(pair_destruct_1 ????? EQpc)
2155                 >(pair_destruct_2 ????? EQpc)
2156                 @split_elim #x #y #H <H -x y H;
2157                 >(pair_destruct_1 ????? EQppc)
2158                 >(pair_destruct_2 ????? EQppc)
2159                 @split_elim #x #y #H <H -x y H;
2160                 >EQ0 % ]           
2161            | >external_ram_write_at_stack_pointer whd in ⊢ (??%?)
2162              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
2163              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
2164              >external_ram_write_at_stack_pointer %
2165            | change with (? = sigma ?? (address_of_word_labels_code_mem (\snd (code_memory ? ps)) ?))
2166              >EQ0 %
2167            | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (??%?)
2168              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
2169              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
2170              >special_function_registers_8051_write_at_stack_pointer %
2171            | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (??%?)
2172              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
2173              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
2174              >special_function_registers_8052_write_at_stack_pointer %
2175            | >p1_latch_write_at_stack_pointer whd in ⊢ (??%?)
2176              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
2177              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
2178              >p1_latch_write_at_stack_pointer %
2179            | >p3_latch_write_at_stack_pointer whd in ⊢ (??%?)
2180              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
2181              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
2182              >p3_latch_write_at_stack_pointer %
2183            | >clock_write_at_stack_pointer whd in ⊢ (??%?)
2184              >clock_write_at_stack_pointer whd in ⊢ (???%)
2185              >clock_write_at_stack_pointer whd in ⊢ (???%)
2186              >clock_write_at_stack_pointer %]
2187       (*| (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2188         @pair_elim' #fst_5_addr #rest_addr #EQ1
2189         @pair_elim' #fst_5_pc #rest_pc #EQ2
2190         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2191         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2192         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2193         change in ⊢ (? →??%?) with (execute_1_0 ??)
2194         @pair_elim' * #instr #newppc' #ticks #EQn
2195          * * #H2a #H2b whd in ⊢ (% → ?) #H2c >H2b >(eq_instruction_to_eq … H2a) whd in ⊢ (??%?)
2196          generalize in match EQ; -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2197          @pair_elim' #carry #new_sp change with (half_add ? (get_8051_sfr ? ps ?) ? = ? → ?) #EQ4
2198          @split_elim' #pc_bu #pc_bl >program_counter_set_8051_sfr XXX change with (newppc = ?) #EQ5
2199          @pair_elim' #carry' #new_sp' #EQ6 normalize nodelta; #EQx >EQx -EQx;
2200          change in ⊢ (??(match ????% with [_ ⇒ ?])?) with (sigma … newppc)
2201          @split_elim' #pc_bu' #pc_bl' #EQ7 change with (newppc' = ? → ?)
2202          >get_8051_sfr_set_8051_sfr
2203         
2204          whd in EQ:(???%) ⊢ ? >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2205           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?)
2206           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc)))
2207           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2208           generalize in match (refl … (split bool 4 4 pc_bu))
2209           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2210           generalize in match (refl … (split bool 3 8 rest_addr))
2211           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2212           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2213           generalize in match
2214            (refl …
2215             (half_add 16 (sigma 〈preamble,instr_list〉 newppc)
2216             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2217           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7
2218           @split_eq_status try %
2219            [ change with (? = sigma ? (address_of_word_labels ps label))
2220              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2221            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2222              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] *)]
2223  |4: (* Jmp *) #label #MAP
2224      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP >MAP -MAP;
2225      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2226       [3: (* long *) #H1 #H2 #EQ %[@1]
2227           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2228           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2229           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2230           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2231           >H2b >(eq_instruction_to_eq … H2a)
2232           generalize in match EQ; -EQ;
2233           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2234           cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
2235       |1: (* short *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2236           generalize in match
2237            (refl ?
2238             (sub_16_with_carry
2239              (sigma 〈preamble,instr_list〉 pol (program_counter … ps))
2240              (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))
2241              false))
2242           cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta;
2243           generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta;
2244           generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta;
2245           #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)]
2246           generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2247           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2248           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2249           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2250           >H2b >(eq_instruction_to_eq … H2a)
2251           generalize in match EQ; -EQ;
2252           whd in ⊢ (???% → ?);
2253           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2254           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ???) ? in ?) = ?)
2255           generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) (sign_extension lower)))
2256           cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4
2257           @split_eq_status try % change with (newpc = sigma ?? (address_of_word_labels ps label))
2258           (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2259       | (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2260         generalize in match
2261          (refl …
2262            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))))
2263         cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1
2264         generalize in match
2265          (refl …
2266            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (program_counter … ps))))
2267         cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2
2268         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2269         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2270         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2271         change in ⊢ (? →??%?) with (execute_1_0 ??)
2272           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2273           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2274           >H2b >(eq_instruction_to_eq … H2a)
2275           generalize in match EQ; -EQ;
2276           whd in ⊢ (???% → ?);
2277           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2278           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2279           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)))
2280           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2281           generalize in match (refl … (split bool 4 4 pc_bu))
2282           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2283           generalize in match (refl … (split bool 3 8 rest_addr))
2284           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2285           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2286           generalize in match
2287            (refl …
2288             (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc)
2289             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2290           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7   
2291           @split_eq_status try %
2292            [ change with (? = sigma ?? (address_of_word_labels ps label))
2293              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2294            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2295              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]]
2296  | (* Instruction *) -pi;  whd in ⊢ (? → ??%? → ?) *; normalize nodelta;
2297    [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1]
2298       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2299       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2300       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2301       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2302       >H2b >(eq_instruction_to_eq … H2a)
2303       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP;
2304       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1;
2305       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2306       normalize nodelta; #MAP;
2307       [1: change in ⊢ (? → %) with
2308        ((let 〈result,flags〉 ≝
2309          add_8_with_carry
2310           (get_arg_8 ? ps false ACC_A)
2311           (get_arg_8 ?
2312             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2313             false (DIRECT ARG2))
2314           ? in ?) = ?)
2315        [2,3: %]
2316        change in ⊢ (???% → ?) with
2317         (let 〈result,flags〉 ≝ add_8_with_carry ?(*(get_arg_8 ? ps false ACC_A)*) ?? in ?)
2318        >get_arg_8_set_clock
2319       [1,2: cases (addressing_mode_ok ???? ∧ addressing_mode_ok ????) in MAP ⊢ ?
2320         [2,4: #abs @⊥ normalize in abs; destruct (abs)
2321         |*:whd in ⊢ (??%? → ?) #H <(option_destruct_Some ??? H)]
2322       [ change in ⊢ (? → %) with
2323        ((let 〈result,flags〉 ≝
2324          add_8_with_carry
2325           (get_arg_8 ? ps false ACC_A)
2326           (get_arg_8 ?
2327             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2328             false (DIRECT ARG2))
2329           ? in ?) = ?)
2330          >get_arg_8_set_low_internal_ram
2331       
2332        cases (add_8_with_carry ???)
2333         
2334        [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2335       #result #flags
2336       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
2337    | (* INC *) #arg1 #H1 #H2 #EQ %[@1]
2338       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2339       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2340       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2341       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2342       >H2b >(eq_instruction_to_eq … H2a)
2343       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2344       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG]
2345       [1,2,3,4: cases (half_add ???) #carry #result
2346       | cases (half_add ???) #carry #bl normalize nodelta;
2347         cases (full_add ????) #carry' #bu normalize nodelta ]
2348        #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) -newppc';
2349        [5: %
2350        |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program
2351      (set_program_counter pseudo_assembly_program ps newppc)
2352      (\fst  (ticks_of0 〈preamble,instr_list〉
2353                   (program_counter pseudo_assembly_program ps)
2354                   (Instruction (INC Identifier (DIRECT ARG))))
2355       +clock pseudo_assembly_program
2356        (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG))
2357        [2,3: // ]
2358            <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://]
2359            whd in ⊢ (??%%)
2360            cases (split bool 4 4 ARG)
2361            #nu' #nl'
2362            normalize nodelta
2363            cases (split bool 1 3 nu')
2364            #bit_1' #ignore'
2365            normalize nodelta
2366            cases (get_index_v bool 4 nu' ? ?)
2367            [ normalize nodelta (* HERE *) whd in ⊢ (??%%) %
2368            |
2369            ]
2370cases daemon (* EASY CASES TO BE COMPLETED *)
2371qed.
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