source: src/ASM/AssemblyProof.ma @ 1333

Last change on this file since 1333 was 1333, checked in by sacerdot, 8 years ago

Avoid using the name of the construction of jmeq.

File size: 88.7 KB
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1include "ASM/Assembly.ma".
2include "ASM/Interpret.ma".
3include "ASM/StatusProofs.ma".
4
5definition bit_elim_prop: ∀P: bool → Prop. Prop ≝
6  λP.
7    P true ∧ P false.
8 
9let rec bitvector_elim_prop_internal
10  (n: nat) (P: BitVector n → Prop) (m: nat) on m: m ≤ n → BitVector (n - m) → Prop ≝
11  match m return λm. m ≤ n → BitVector (n - m) → Prop with
12  [ O    ⇒ λprf1. λprefix. P ?
13  | S n' ⇒ λprf2. λprefix. bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' ? ?)
14  ].
15  [ applyS prefix
16  | letin res ≝ (bit ::: prefix)
17    < (minus_S_S ? ?)
18    > (minus_Sn_m ? ?)
19  [ @ res
20  | @ prf2
21  ]
22  | /2/
23  ].
24qed.
25
26definition bitvector_elim_prop ≝
27  λn: nat.
28  λP: BitVector n → Prop.
29    bitvector_elim_prop_internal n P n ? ?.
30  [ @ (le_n ?)
31  | < (minus_n_n ?)
32    @ [[ ]]
33  ]
34qed.
35
36lemma bool_eq_internal_eq:
37  ∀b, c.
38    (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c.
39  #b #c
40  cases b
41  [ normalize //
42  | normalize
43    cases c
44    [ normalize //
45    | normalize //
46    ]
47  ]
48qed.
49
50definition bit_elim: ∀P: bool → bool. bool ≝
51  λP.
52    P true ∧ P false.
53
54let rec bitvector_elim_internal
55  (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝
56  match m return λm. m ≤ n → BitVector (n - m) → bool with
57  [ O    ⇒ λprf1. λprefix. P ?
58  | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?)
59  ].
60  [ applyS prefix
61  | letin res ≝ (bit ::: prefix)
62    < (minus_S_S ? ?)
63    > (minus_Sn_m ? ?)
64    [ @ res
65    | @ prf2
66    ]
67  | /2/
68  ].
69qed.
70
71definition bitvector_elim ≝
72  λn: nat.
73  λP: BitVector n → bool.
74    bitvector_elim_internal n P n ? ?.
75  [ @ (le_n ?)
76  | < (minus_n_n ?)
77    @ [[ ]]
78  ]
79qed.
80
81lemma super_rewrite2:
82 ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m.
83  ∀P: ∀m. Vector A m → Prop.
84   n=m → v1 ≃ v2 → P n v1 → P m v2.
85 #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ //
86qed.
87
88lemma vector_cons_append:
89  ∀A: Type[0].
90  ∀n: nat.
91  ∀e: A.
92  ∀v: Vector A n.
93    e ::: v = [[ e ]] @@ v.
94  # A # N # E # V
95  elim V
96  [ normalize %
97  | # NN # AA # VV # IH
98    normalize
99    %
100qed.
101
102lemma vector_cons_append2:
103  ∀A: Type[0].
104  ∀n, m: nat.
105  ∀v: Vector A n.
106  ∀q: Vector A m.
107  ∀hd: A.
108    hd:::(v@@q) = (hd:::v)@@q.
109  #A #n #m #v #q
110  elim v
111  [ #hd %
112  | #n' #hd' #tl' #ih #hd' <ih %
113  ]
114qed.
115
116lemma jmeq_cons_vector_monotone:
117  ∀A: Type[0].
118  ∀m, n: nat.
119  ∀v: Vector A m.
120  ∀q: Vector A n.
121  ∀prf: m = n.
122  ∀hd: A.
123    v ≃ q → hd:::v ≃ hd:::q.
124  #A #m #n #v #q #prf #hd #E
125  @(super_rewrite2 A … E)
126  [ assumption | % ]
127qed.
128
129lemma vector_associative_append:
130  ∀A: Type[0].
131  ∀n, m, o:  nat.
132  ∀v: Vector A n.
133  ∀q: Vector A m.
134  ∀r: Vector A o.
135    ((v @@ q) @@ r)
136    ≃
137    (v @@ (q @@ r)).
138  #A #n #m #o #v #q #r
139  elim v
140  [ %
141  | #n' #hd #tl #ih
142    <(vector_cons_append2 A … hd)
143    @jmeq_cons_vector_monotone
144    //
145  ]
146qed.
147
148lemma mem_middle_vector:
149  ∀A: Type[0].
150  ∀m, o: nat.
151  ∀eq: A → A → bool.
152  ∀reflex: ∀a. eq a a = true.
153  ∀p: Vector A m.
154  ∀a: A.
155  ∀r: Vector A o.
156    mem A eq ? (p@@(a:::r)) a = true.
157  # A # M # O # EQ # REFLEX # P # A
158  elim P
159  [ normalize
160    > (REFLEX A)
161    normalize
162    # H
163    %
164  | # NN # AA # PP # IH
165    normalize
166    cases (EQ A AA) //
167     @ IH
168  ]
169qed.
170
171lemma mem_monotonic_wrt_append:
172  ∀A: Type[0].
173  ∀m, o: nat.
174  ∀eq: A → A → bool.
175  ∀reflex: ∀a. eq a a = true.
176  ∀p: Vector A m.
177  ∀a: A.
178  ∀r: Vector A o.
179    mem A eq ? r a = true → mem A eq ? (p @@ r) a = true.
180  # A # M # O # EQ # REFLEX # P # A
181  elim P
182  [ #R #H @H
183  | #NN #AA # PP # IH #R #H
184    normalize
185    cases (EQ A AA)
186    [ normalize %
187    | @ IH @ H
188    ]
189  ]
190qed.
191
192lemma subvector_multiple_append:
193  ∀A: Type[0].
194  ∀o, n: nat.
195  ∀eq: A → A → bool.
196  ∀refl: ∀a. eq a a = true.
197  ∀h: Vector A o.
198  ∀v: Vector A n.
199  ∀m: nat.
200  ∀q: Vector A m.
201    bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)).
202  # A # O # N # EQ # REFLEX # H # V
203  elim V
204  [ normalize
205    # M # V %
206  | # NN # AA # VV # IH # MM # QQ
207    change with (bool_to_Prop (andb ??))
208    cut ((mem A EQ (O + (MM + S NN)) (H@@QQ@@AA:::VV) AA) = true)
209    [
210    | # HH > HH
211      > (vector_cons_append ? ? AA VV)
212      change with (bool_to_Prop (subvector_with ??????))
213      @(super_rewrite2 A ((MM + 1)+ NN) (MM+S NN) ??
214        (λSS.λVS.bool_to_Prop (subvector_with ?? (O+SS) ?? (H@@VS)))
215        ?
216        (vector_associative_append A ? ? ? QQ [[AA]] VV))
217      [ >associative_plus //
218      | @IH ]
219    ]
220    @(mem_monotonic_wrt_append)
221    [ @ REFLEX
222    | @(mem_monotonic_wrt_append)
223      [ @ REFLEX
224      | normalize
225        > REFLEX
226        normalize
227        %
228      ]
229    ]
230qed.
231
232lemma vector_cons_empty:
233  ∀A: Type[0].
234  ∀n: nat.
235  ∀v: Vector A n.
236    [[ ]] @@ v = v.
237  # A # N # V
238  elim V
239  [ normalize %
240  | # NN # HH # VV #H %
241  ]
242qed.
243
244corollary subvector_hd_tl:
245  ∀A: Type[0].
246  ∀o: nat.
247  ∀eq: A → A → bool.
248  ∀refl: ∀a. eq a a = true.
249  ∀h: A.
250  ∀v: Vector A o.
251    bool_to_Prop (subvector_with A ? ? eq v (h ::: v)).
252  # A # O # EQ # REFLEX # H # V
253  > (vector_cons_append A ? H V)
254  < (vector_cons_empty A ? ([[H]] @@ V))
255  @ (subvector_multiple_append A ? ? EQ REFLEX [[]] V ? [[ H ]])
256qed.
257
258lemma eq_a_reflexive:
259  ∀a. eq_a a a = true.
260  # A
261  cases A
262  %
263qed.
264
265lemma is_in_monotonic_wrt_append:
266  ∀m, n: nat.
267  ∀p: Vector addressing_mode_tag m.
268  ∀q: Vector addressing_mode_tag n.
269  ∀to_search: addressing_mode.
270    bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search).
271  # M # N # P # Q # TO_SEARCH
272  # H
273  elim Q
274  [ normalize
275    @ H
276  | # NN # PP # QQ # IH
277    normalize
278    cases (is_a PP TO_SEARCH)
279    [ normalize
280      %
281    | normalize
282      normalize in IH
283      @ IH
284    ]
285  ]
286qed.
287
288corollary is_in_hd_tl:
289  ∀to_search: addressing_mode.
290  ∀hd: addressing_mode_tag.
291  ∀n: nat.
292  ∀v: Vector addressing_mode_tag n.
293    bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search).
294  # TO_SEARCH # HD # N # V
295  elim V
296  [ # H
297    normalize in H;
298    cases H
299  | # NN # HHD # VV # IH # HH
300    > vector_cons_append
301    > (vector_cons_append ? ? HHD VV)
302    @ (is_in_monotonic_wrt_append ? 1 ([[HHD]]@@VV) [[HD]] TO_SEARCH)
303    @ HH
304  ]
305qed.
306 
307let rec list_addressing_mode_tags_elim
308  (n: nat) (l: Vector addressing_mode_tag (S n)) on l: (l → bool) → bool ≝
309  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
310   (l → bool) → bool ] with
311  [ VEmpty      ⇒  true 
312  | VCons len hd tl ⇒ λP.
313    let process_hd ≝
314      match hd return λhd. ∀P: hd:::tl → bool. bool with
315      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
316      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
317      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
318      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
319      | acc_a ⇒ λP.P ACC_A
320      | acc_b ⇒ λP.P ACC_B
321      | dptr ⇒ λP.P DPTR
322      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
323      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
324      | acc_dptr ⇒ λP.P ACC_DPTR
325      | acc_pc ⇒ λP.P ACC_PC
326      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
327      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
328      | carry ⇒ λP.P CARRY
329      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
330      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
331      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
332      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
333      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
334      ]
335    in
336      andb (process_hd P)
337       (match len return λx. x = len → bool with
338         [ O ⇒ λprf. true
339         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
340  ].
341  try %
342  [ 2: cases (sym_eq ??? prf); @tl
343  | generalize in match H; generalize in match tl; cases prf;
344    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
345    #tl
346    normalize in ⊢ (∀_: %. ?)
347    # H
348    whd
349    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
350    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
351qed.
352
353definition product_elim ≝
354  λm, n: nat.
355  λv: Vector addressing_mode_tag (S m).
356  λq: Vector addressing_mode_tag (S n).
357  λP: (v × q) → bool.
358    list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)).
359
360definition union_elim ≝
361  λA, B: Type[0].
362  λelimA: (A → bool) → bool.
363  λelimB: (B → bool) → bool.
364  λelimU: A ⊎ B → bool.
365    elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))).
366
367(*                           
368definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝
369  λP.
370    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧
371    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧
372    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧
373    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧
374    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧
375    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧
376    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧
377    list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
378    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧
379    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧
380    P (DA ? ACC_A) ∧
381    bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧
382    bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧
383    bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧
384    bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧
385    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧
386    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧
387    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧
388    list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
389    P (RL ? ACC_A) ∧
390    P (RLC ? ACC_A) ∧
391    P (RR ? ACC_A) ∧
392    P (RRC ? ACC_A) ∧
393    P (SWAP ? ACC_A) ∧
394    P (RET ?) ∧
395    P (RETI ?) ∧
396    P (NOP ?) ∧
397    bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧
398    list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧
399    bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧
400    bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧
401    union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]])
402                   (product_elim ? ? [[ registr; indirect ]] [[ data ]])
403                   (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧
404    list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧
405    union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]])
406                   (product_elim ? ? [[direct]] [[ acc_a ; data ]])
407                   (λd. P (XRL ? d)) ∧
408    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
409                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
410                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
411                   (λd. P (ANL ? d)) ∧
412    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]])
413                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
414                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
415                   (λd. P (ORL ? d)) ∧
416    union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]])
417                   (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]])
418                   (λd. P (MOVX ? d)) ∧
419    union_elim ? ? (
420      union_elim ? ? (
421        union_elim ? ? (
422          union_elim ? ? (
423            union_elim ? ?  (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
424                            (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]]))
425                            (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]]))
426                            (product_elim ? ? [[dptr]] [[data16]]))
427                            (product_elim ? ? [[carry]] [[bit_addr]]))
428                            (product_elim ? ? [[bit_addr]] [[carry]])
429                            (λd. P (MOV ? d)).
430  %
431qed.
432 
433definition instruction_elim: ∀P: instruction → bool. bool ≝
434  λP. (*
435    bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧
436    bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧
437    bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧
438    bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *)
439    bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (*  ∧
440    P (JMP INDIRECT_DPTR) ∧
441    list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧
442    preinstruction_elim (λp. P (RealInstruction p)). *)
443  %
444qed.
445
446
447axiom instruction_elim_complete:
448 ∀P. instruction_elim P = true → ∀i. P i = true.
449*)
450(*definition eq_instruction ≝
451  λi, j: instruction.
452    true.*)
453
454definition eq_addressing_mode: addressing_mode → addressing_mode → bool ≝
455  λa, b: addressing_mode.
456  match a with
457  [ DIRECT d ⇒
458    match b with
459    [ DIRECT e ⇒ eq_bv ? d e
460    | _ ⇒ false
461    ]
462  | INDIRECT b' ⇒
463    match b with
464    [ INDIRECT e ⇒ eq_b b' e
465    | _ ⇒ false
466    ]
467  | EXT_INDIRECT b' ⇒
468    match b with
469    [ EXT_INDIRECT e ⇒ eq_b b' e
470    | _ ⇒ false
471    ]
472  | REGISTER bv ⇒
473    match b with
474    [ REGISTER bv' ⇒ eq_bv ? bv bv'
475    | _ ⇒ false
476    ]
477  | ACC_A ⇒ match b with [ ACC_A ⇒ true | _ ⇒ false ]
478  | ACC_B ⇒ match b with [ ACC_B ⇒ true | _ ⇒ false ]
479  | DPTR ⇒ match b with [ DPTR ⇒ true | _ ⇒ false ]
480  | DATA b' ⇒
481    match b with
482    [ DATA e ⇒ eq_bv ? b' e
483    | _ ⇒ false
484    ]
485  | DATA16 w ⇒
486    match b with
487    [ DATA16 e ⇒ eq_bv ? w e
488    | _ ⇒ false
489    ]
490  | ACC_DPTR ⇒ match b with [ ACC_DPTR ⇒ true | _ ⇒ false ]
491  | ACC_PC ⇒ match b with [ ACC_PC ⇒ true | _ ⇒ false ]
492  | EXT_INDIRECT_DPTR ⇒ match b with [ EXT_INDIRECT_DPTR ⇒ true | _ ⇒ false ]
493  | INDIRECT_DPTR ⇒ match b with [ INDIRECT_DPTR ⇒ true | _ ⇒ false ]
494  | CARRY ⇒ match b with [ CARRY ⇒ true | _ ⇒ false ]
495  | BIT_ADDR b' ⇒
496    match b with
497    [ BIT_ADDR e ⇒ eq_bv ? b' e
498    | _ ⇒ false
499    ]
500  | N_BIT_ADDR b' ⇒
501    match b with
502    [ N_BIT_ADDR e ⇒ eq_bv ? b' e
503    | _ ⇒ false
504    ]
505  | RELATIVE n ⇒
506    match b with
507    [ RELATIVE e ⇒ eq_bv ? n e
508    | _ ⇒ false
509    ]
510  | ADDR11 w ⇒
511    match b with
512    [ ADDR11 e ⇒ eq_bv ? w e
513    | _ ⇒ false
514    ]
515  | ADDR16 w ⇒
516    match b with
517    [ ADDR16 e ⇒ eq_bv ? w e
518    | _ ⇒ false
519    ]
520  ].
521
522lemma eq_bv_refl:
523  ∀n, b.
524    eq_bv n b b = true.
525  #n #b cases b //
526qed.
527
528lemma eq_b_refl:
529  ∀b.
530    eq_b b b = true.
531  #b cases b //
532qed.
533
534lemma eq_addressing_mode_refl:
535  ∀a. eq_addressing_mode a a = true.
536  #a cases a try #arg1 try #arg2 try @eq_bv_refl try @eq_b_refl
537  try normalize %
538qed.
539 
540definition eq_sum: ∀A, B. (A → A → bool) → (B → B → bool) → (A ⊎ B) → (A ⊎ B) → bool ≝
541  λlt, rt, leq, req, left, right.
542    match left with
543    [ inl l ⇒
544      match right with
545      [ inl l' ⇒ leq l l'
546      | _ ⇒ false
547      ]
548    | inr r ⇒
549      match right with
550      [ inr r' ⇒ req r r'
551      | _ ⇒ false
552      ]
553    ].
554
555definition eq_prod: ∀A, B. (A → A → bool) → (B → B → bool) → (A × B) → (A × B) → bool ≝
556  λlt, rt, leq, req, left, right.
557    let 〈l, r〉 ≝ left in
558    let 〈l', r'〉 ≝ right in
559      leq l l' ∧ req r r'.
560
561definition eq_preinstruction: preinstruction [[relative]] → preinstruction [[relative]] → bool ≝
562  λi, j.
563  match i with
564  [ ADD arg1 arg2 ⇒
565    match j with
566    [ ADD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
567    | _ ⇒ false
568    ]
569  | ADDC arg1 arg2 ⇒
570    match j with
571    [ ADDC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
572    | _ ⇒ false
573    ]
574  | SUBB arg1 arg2 ⇒
575    match j with
576    [ SUBB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
577    | _ ⇒ false
578    ]
579  | INC arg ⇒
580    match j with
581    [ INC arg' ⇒ eq_addressing_mode arg arg'
582    | _ ⇒ false
583    ]
584  | DEC arg ⇒
585    match j with
586    [ DEC arg' ⇒ eq_addressing_mode arg arg'
587    | _ ⇒ false
588    ]
589  | MUL arg1 arg2 ⇒
590    match j with
591    [ MUL arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
592    | _ ⇒ false
593    ]
594  | DIV arg1 arg2 ⇒
595    match j with
596    [ DIV arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
597    | _ ⇒ false
598    ]
599  | DA arg ⇒
600    match j with
601    [ DA arg' ⇒ eq_addressing_mode arg arg'
602    | _ ⇒ false
603    ]
604  | JC arg ⇒
605    match j with
606    [ JC arg' ⇒ eq_addressing_mode arg arg'
607    | _ ⇒ false
608    ]
609  | JNC arg ⇒
610    match j with
611    [ JNC arg' ⇒ eq_addressing_mode arg arg'
612    | _ ⇒ false
613    ]
614  | JB arg1 arg2 ⇒
615    match j with
616    [ JB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
617    | _ ⇒ false
618    ]
619  | JNB arg1 arg2 ⇒
620    match j with
621    [ JNB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
622    | _ ⇒ false
623    ]
624  | JBC arg1 arg2 ⇒
625    match j with
626    [ JBC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
627    | _ ⇒ false
628    ]
629  | JZ arg ⇒
630    match j with
631    [ JZ arg' ⇒ eq_addressing_mode arg arg'
632    | _ ⇒ false
633    ]
634  | JNZ arg ⇒
635    match j with
636    [ JNZ arg' ⇒ eq_addressing_mode arg arg'
637    | _ ⇒ false
638    ]
639  | CJNE arg1 arg2 ⇒
640    match j with
641    [ CJNE arg1' arg2' ⇒
642      let prod_eq_left ≝ eq_prod [[acc_a]] [[direct; data]] eq_addressing_mode eq_addressing_mode in
643      let prod_eq_right ≝ eq_prod [[registr; indirect]] [[data]] eq_addressing_mode eq_addressing_mode in
644      let arg1_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
645        arg1_eq arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
646    | _ ⇒ false
647    ]
648  | DJNZ arg1 arg2 ⇒
649    match j with
650    [ DJNZ arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
651    | _ ⇒ false
652    ]
653  | CLR arg ⇒
654    match j with
655    [ CLR arg' ⇒ eq_addressing_mode arg arg'
656    | _ ⇒ false
657    ]
658  | CPL arg ⇒
659    match j with
660    [ CPL arg' ⇒ eq_addressing_mode arg arg'
661    | _ ⇒ false
662    ]
663  | RL arg ⇒
664    match j with
665    [ RL arg' ⇒ eq_addressing_mode arg arg'
666    | _ ⇒ false
667    ]
668  | RLC arg ⇒
669    match j with
670    [ RLC arg' ⇒ eq_addressing_mode arg arg'
671    | _ ⇒ false
672    ]
673  | RR arg ⇒
674    match j with
675    [ RR arg' ⇒ eq_addressing_mode arg arg'
676    | _ ⇒ false
677    ]
678  | RRC arg ⇒
679    match j with
680    [ RRC arg' ⇒ eq_addressing_mode arg arg'
681    | _ ⇒ false
682    ]
683  | SWAP arg ⇒
684    match j with
685    [ SWAP arg' ⇒ eq_addressing_mode arg arg'
686    | _ ⇒ false
687    ]
688  | SETB arg ⇒
689    match j with
690    [ SETB arg' ⇒ eq_addressing_mode arg arg'
691    | _ ⇒ false
692    ]
693  | PUSH arg ⇒
694    match j with
695    [ PUSH arg' ⇒ eq_addressing_mode arg arg'
696    | _ ⇒ false
697    ]
698  | POP arg ⇒
699    match j with
700    [ POP arg' ⇒ eq_addressing_mode arg arg'
701    | _ ⇒ false
702    ]
703  | XCH arg1 arg2 ⇒
704    match j with
705    [ XCH arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
706    | _ ⇒ false
707    ]
708  | XCHD arg1 arg2 ⇒
709    match j with
710    [ XCHD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
711    | _ ⇒ false
712    ]
713  | RET ⇒ match j with [ RET ⇒ true | _ ⇒ false ]
714  | RETI ⇒ match j with [ RETI ⇒ true | _ ⇒ false ]
715  | NOP ⇒ match j with [ NOP ⇒ true | _ ⇒ false ]
716  | MOVX arg ⇒
717    match j with
718    [ MOVX arg' ⇒
719      let prod_eq_left ≝ eq_prod [[acc_a]] [[ext_indirect; ext_indirect_dptr]] eq_addressing_mode eq_addressing_mode in
720      let prod_eq_right ≝ eq_prod [[ext_indirect; ext_indirect_dptr]] [[acc_a]] eq_addressing_mode eq_addressing_mode in
721      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
722        sum_eq arg arg'
723    | _ ⇒ false
724    ]
725  | XRL arg ⇒
726    match j with
727    [ XRL arg' ⇒
728      let prod_eq_left ≝ eq_prod [[acc_a]] [[ data ; registr ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in
729      let prod_eq_right ≝ eq_prod [[direct]] [[ acc_a ; data ]] eq_addressing_mode eq_addressing_mode in
730      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
731        sum_eq arg arg'
732    | _ ⇒ false
733    ]
734  | ORL arg ⇒
735    match j with
736    [ ORL arg' ⇒
737      let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; data ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in
738      let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in
739      let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in
740      let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in
741      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
742        sum_eq arg arg'
743    | _ ⇒ false
744    ]
745  | ANL arg ⇒
746    match j with
747    [ ANL arg' ⇒
748      let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; direct ; indirect ; data ]] eq_addressing_mode eq_addressing_mode in
749      let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in
750      let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in
751      let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in
752      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
753        sum_eq arg arg'
754    | _ ⇒ false
755    ]
756  | MOV arg ⇒
757    match j with
758    [ MOV arg' ⇒
759      let prod_eq_6 ≝ eq_prod [[acc_a]] [[registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in
760      let prod_eq_5 ≝ eq_prod [[registr; indirect]] [[acc_a; direct; data]] eq_addressing_mode eq_addressing_mode in
761      let prod_eq_4 ≝ eq_prod [[direct]] [[acc_a; registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in
762      let prod_eq_3 ≝ eq_prod [[dptr]] [[data16]] eq_addressing_mode eq_addressing_mode in
763      let prod_eq_2 ≝ eq_prod [[carry]] [[bit_addr]] eq_addressing_mode eq_addressing_mode in
764      let prod_eq_1 ≝ eq_prod [[bit_addr]] [[carry]] eq_addressing_mode eq_addressing_mode in
765      let sum_eq_1 ≝ eq_sum ? ? prod_eq_6 prod_eq_5 in
766      let sum_eq_2 ≝ eq_sum ? ? sum_eq_1 prod_eq_4 in
767      let sum_eq_3 ≝ eq_sum ? ? sum_eq_2 prod_eq_3 in
768      let sum_eq_4 ≝ eq_sum ? ? sum_eq_3 prod_eq_2 in
769      let sum_eq_5 ≝ eq_sum ? ? sum_eq_4 prod_eq_1 in
770        sum_eq_5 arg arg'
771    | _ ⇒ false
772    ]
773  ].
774
775lemma eq_sum_refl:
776  ∀A, B: Type[0].
777  ∀leq, req.
778  ∀s.
779  ∀leq_refl: (∀t. leq t t = true).
780  ∀req_refl: (∀u. req u u = true).
781    eq_sum A B leq req s s = true.
782  #A #B #leq #req #s #leq_refl #req_refl
783  cases s whd in ⊢ (? → ??%?) //
784qed.
785
786lemma eq_prod_refl:
787  ∀A, B: Type[0].
788  ∀leq, req.
789  ∀s.
790  ∀leq_refl: (∀t. leq t t = true).
791  ∀req_refl: (∀u. req u u = true).
792    eq_prod A B leq req s s = true.
793  #A #B #leq #req #s #leq_refl #req_refl
794  cases s whd in ⊢ (? → ? → ??%?) #l #r >leq_refl normalize @req_refl
795qed.
796
797lemma eq_preinstruction_refl:
798  ∀i.
799    eq_preinstruction i i = true.
800  #i cases i try #arg1 try #arg2
801  try @eq_addressing_mode_refl
802  [1,2,3,4,5,6,7,8,10,16,17,18,19,20:
803    whd in ⊢ (??%?)
804    try %
805    >eq_addressing_mode_refl
806    >eq_addressing_mode_refl %
807  |13,15:
808    whd in ⊢ (??%?)
809    cases arg1
810    [*:
811      #arg1_left normalize nodelta
812      >eq_prod_refl [*: try % #argr @eq_addressing_mode_refl]
813    ]
814  |11,12:
815    whd in ⊢ (??%?)
816    cases arg1
817    [1:
818      #arg1_left normalize nodelta
819      >(eq_sum_refl …)
820      [1: % | 2,3: #arg @eq_prod_refl ]
821      @eq_addressing_mode_refl
822    |2:
823      #arg1_left normalize nodelta
824      @eq_prod_refl [*: @eq_addressing_mode_refl ]
825    |3:
826      #arg1_left normalize nodelta
827      >(eq_sum_refl …) [1: %
828      |2,3: #arg @eq_prod_refl #arg @eq_addressing_mode_refl ]
829    |4:
830      #arg1_left normalize nodelta
831      @eq_prod_refl [*: #arg @eq_addressing_mode_refl ]
832    ]
833  |14:
834    whd in ⊢ (??%?)
835    cases arg1
836    [ #arg1_left normalize nodelta
837      @eq_sum_refl
838      [1: #arg @eq_sum_refl
839        [1: #arg @eq_sum_refl
840          [1: #arg @eq_sum_refl
841            [1: #arg @eq_prod_refl
842              [*: @eq_addressing_mode_refl ]
843            |2: #arg @eq_prod_refl
844              [*: #arg @eq_addressing_mode_refl ]
845            ]
846          |2: #arg @eq_prod_refl
847            [*: #arg @eq_addressing_mode_refl ]
848          ]
849        |2: #arg @eq_prod_refl
850          [*: #arg @eq_addressing_mode_refl ]
851        ]
852      |2: #arg @eq_prod_refl
853        [*: #arg @eq_addressing_mode_refl ]
854      ]
855    |2: #arg1_right normalize nodelta
856        @eq_prod_refl [*: #arg @eq_addressing_mode_refl ]
857    ]
858  |*:
859    whd in ⊢ (??%?)
860    cases arg1
861    [*: #arg1 >eq_sum_refl
862      [1,4: normalize @eq_addressing_mode_refl
863      |2,3,5,6: #arg @eq_prod_refl
864        [*: #arg @eq_addressing_mode_refl ]
865      ]
866    ]
867  ]
868qed.
869
870definition eq_instruction: instruction → instruction → bool ≝
871  λi, j.
872  match i with
873  [ ACALL arg ⇒
874    match j with
875    [ ACALL arg' ⇒ eq_addressing_mode arg arg'
876    | _ ⇒ false
877    ]
878  | LCALL arg ⇒
879    match j with
880    [ LCALL arg' ⇒ eq_addressing_mode arg arg'
881    | _ ⇒ false
882    ]
883  | AJMP arg ⇒
884    match j with
885    [ AJMP arg' ⇒ eq_addressing_mode arg arg'
886    | _ ⇒ false
887    ]
888  | LJMP arg ⇒
889    match j with
890    [ LJMP arg' ⇒ eq_addressing_mode arg arg'
891    | _ ⇒ false
892    ]
893  | SJMP arg ⇒
894    match j with
895    [ SJMP arg' ⇒ eq_addressing_mode arg arg'
896    | _ ⇒ false
897    ]
898  | JMP arg ⇒
899    match j with
900    [ JMP arg' ⇒ eq_addressing_mode arg arg'
901    | _ ⇒ false
902    ]
903  | MOVC arg1 arg2 ⇒
904    match j with
905    [ MOVC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
906    | _ ⇒ false
907    ]
908  | RealInstruction instr ⇒
909    match j with
910    [ RealInstruction instr' ⇒ eq_preinstruction instr instr'
911    | _ ⇒ false
912    ]
913  ].
914 
915lemma eq_instruction_refl:
916  ∀i. eq_instruction i i = true.
917  #i cases i
918  [1,2,3,4,5,6: #arg1 @eq_addressing_mode_refl
919  |7: #arg1 #arg2
920      whd in ⊢ (??%?)
921      >eq_addressing_mode_refl
922      >eq_addressing_mode_refl
923      %
924  |8: #arg @eq_preinstruction_refl
925  ]
926qed.
927
928let rec vect_member
929  (A: Type[0]) (n: nat) (eq: A → A → bool)
930  (v: Vector A n) (a: A) on v: bool ≝
931  match v with
932  [ VEmpty          ⇒ false
933  | VCons len hd tl ⇒
934    eq hd a ∨ (vect_member A ? eq tl a)
935  ].
936
937let rec list_addressing_mode_tags_elim_prop
938  (n: nat)
939  (l: Vector addressing_mode_tag (S n))
940  on l:
941  ∀P: l → Prop.
942  ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a.
943  ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a.
944  ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a.
945  ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a.
946  ∀x: l. P x ≝
947  match l return
948    λy.
949      match y with
950      [ O    ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True
951      | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop.
952               ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True.
953               ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True.
954               ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True.
955               ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True.
956               ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True.
957               ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True.
958               ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True.
959               ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True.
960               ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True.
961               ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True.
962               ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True.
963               ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True.
964               ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True.
965               ∀carry_a: if vect_member … eq_a l carry then P CARRY else True.
966               ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True.
967               ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True.
968               ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True.
969               ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True.
970               ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True.
971               ∀x:l. P x
972      ] with
973  [ VEmpty          ⇒ λAbsurd. ⊥
974  | VCons len hd tl ⇒ λProof. ?
975  ] (refl ? (S n)). cases daemon. qed. (*
976  [ destruct(Absurd)
977  | # A1 # A2 # A3 # A4 # A5 # A6 # A7
978    # A8 # A9 # A10 # A11 # A12 # A13 # A14
979    # A15 # A16 # A17 # A18 # A19 # X
980    cases X
981    # SUB cases daemon ] qed.
982    cases SUB
983    [ # BYTE
984    normalize
985  ].
986 
987 
988(*    let prepare_hd ≝
989      match hd with
990      [ direct ⇒ λdirect_prf. ?
991      | indirect ⇒ λindirect_prf. ?
992      | ext_indirect ⇒ λext_indirect_prf. ?
993      | registr ⇒ λregistr_prf. ?
994      | acc_a ⇒ λacc_a_prf. ?
995      | acc_b ⇒ λacc_b_prf. ?
996      | dptr ⇒ λdptr_prf. ?
997      | data ⇒ λdata_prf. ?
998      | data16 ⇒ λdata16_prf. ?
999      | acc_dptr ⇒ λacc_dptr_prf. ?
1000      | acc_pc ⇒ λacc_pc_prf. ?
1001      | ext_indirect_dptr ⇒ λext_indirect_prf. ?
1002      | indirect_dptr ⇒ λindirect_prf. ?
1003      | carry ⇒ λcarry_prf. ?
1004      | bit_addr ⇒ λbit_addr_prf. ?
1005      | n_bit_addr ⇒ λn_bit_addr_prf. ?
1006      | relative ⇒ λrelative_prf. ?
1007      | addr11 ⇒ λaddr11_prf. ?
1008      | addr16 ⇒ λaddr16_prf. ?
1009      ]
1010    in ? *)
1011  ].
1012  [ 1: destruct(absd)
1013  | 2: # A1 # A2 # A3 # A4 # A5 # A6
1014       # A7 # A8 # A9 # A10 # A11 # A12
1015       # A13 # A14 # A15 # A16 # A17 # A18
1016       # A19 *
1017  ].
1018
1019
1020  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
1021   (l → bool) → bool ] with
1022  [ VEmpty      ⇒  true 
1023  | VCons len hd tl ⇒ λP.
1024    let process_hd ≝
1025      match hd return λhd. ∀P: hd:::tl → bool. bool with
1026      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
1027      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
1028      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
1029      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
1030      | acc_a ⇒ λP.P ACC_A
1031      | acc_b ⇒ λP.P ACC_B
1032      | dptr ⇒ λP.P DPTR
1033      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
1034      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
1035      | acc_dptr ⇒ λP.P ACC_DPTR
1036      | acc_pc ⇒ λP.P ACC_PC
1037      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
1038      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
1039      | carry ⇒ λP.P CARRY
1040      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
1041      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
1042      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
1043      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
1044      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
1045      ]
1046    in
1047      andb (process_hd P)
1048       (match len return λx. x = len → bool with
1049         [ O ⇒ λprf. true
1050         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
1051  ].
1052  try %
1053  [ 2: cases (sym_eq ??? prf); @tl
1054  | generalize in match H; generalize in match tl; cases prf;
1055    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
1056    #tl
1057    normalize in ⊢ (∀_: %. ?)
1058    # H
1059    whd
1060    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1061    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
1062qed.
1063*)
1064
1065definition load_code_memory_aux ≝
1066 fold_left_i_aux … (
1067   λi, mem, v.
1068     insert … (bitvector_of_nat … i) v mem) (Stub Byte 16).
1069
1070lemma split_zero:
1071  ∀A,m.
1072  ∀v: Vector A m.
1073    〈[[]], v〉 = split A 0 m v.
1074  #A #m #v
1075  elim v
1076  [ %
1077  | #n #hd #tl #ih
1078    normalize in ⊢ (???%) %
1079  ]
1080qed.
1081
1082lemma Vector_O: ∀A. ∀v: Vector A 0. v ≃ VEmpty A.
1083 #A #v generalize in match (refl … 0) cases v in ⊢ (??%? → ?%%??) //
1084 #n #hd #tl #abs @⊥ destruct (abs)
1085qed.
1086
1087lemma Vector_Sn: ∀A. ∀n.∀v:Vector A (S n).
1088 ∃hd.∃tl.v ≃ VCons A n hd tl.
1089 #A #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??)))
1090 [ #abs @⊥ destruct (abs)
1091 | #m #hd #tl #EQ <(injective_S … EQ) %[@hd] %[@tl] // ]
1092qed.
1093
1094lemma vector_append_zero:
1095  ∀A,m.
1096  ∀v: Vector A m.
1097  ∀q: Vector A 0.
1098    v = q@@v.
1099  #A #m #v #q
1100  >(Vector_O A q) %
1101qed.
1102
1103lemma prod_eq_left:
1104  ∀A: Type[0].
1105  ∀p, q, r: A.
1106    p = q → 〈p, r〉 = 〈q, r〉.
1107  #A #p #q #r #hyp
1108  >hyp %
1109qed.
1110
1111lemma prod_eq_right:
1112  ∀A: Type[0].
1113  ∀p, q, r: A.
1114    p = q → 〈r, p〉 = 〈r, q〉.
1115  #A #p #q #r #hyp
1116  >hyp %
1117qed.
1118
1119corollary prod_vector_zero_eq_left:
1120  ∀A, n.
1121  ∀q: Vector A O.
1122  ∀r: Vector A n.
1123    〈q, r〉 = 〈[[ ]], r〉.
1124  #A #n #q #r
1125  generalize in match (Vector_O A q …)
1126  #hyp
1127  >hyp in ⊢ (??%?)
1128  %
1129qed.
1130
1131axiom split_succ:
1132  ∀A, m, n.
1133  ∀l: Vector A m.
1134  ∀r: Vector A n.
1135  ∀v: Vector A (m + n).
1136  ∀hd: A.
1137    v = l@@r → (〈l, r〉 = split A m n v → 〈hd:::l, r〉 = split A (S m) n (hd:::v)).
1138(*
1139  #A #m
1140  elim m
1141  [ #n #l #r #v #hd #equal #hyp
1142    normalize >(Vector_O A l) >equal
1143    >(Vector_O A l) %
1144  | #n' #ih #n #l #r #v #hd
1145    #equal #hyp
1146    cases(Vector_Sn A n' l)
1147    #hd' #exists
1148    cases exists #tl #jmeq
1149    >jmeq *)
1150
1151lemma split_prod:
1152  ∀A,m,n.
1153  ∀p: Vector A (m + n).
1154  ∀v: Vector A m.
1155  ∀q: Vector A n.
1156    p = v@@q → 〈v, q〉 = split A m n p.
1157  #A #m
1158  elim m
1159  [ #n #p #v #q #hyp
1160    >hyp <(vector_append_zero A n q v)
1161    >(prod_vector_zero_eq_left A …)
1162    @split_zero
1163  | #r #ih #n #p #v #q #hyp
1164    >hyp
1165    cases (Vector_Sn A r v)
1166    #hd #exists
1167    cases exists
1168    #tl #jmeq >jmeq
1169    @split_succ [1: % |2: @ih % ]
1170  ]
1171qed.
1172
1173lemma split_prod_exists:
1174  ∀A, m, n.
1175  ∀p: Vector A (m + n).
1176  ∃v: Vector A m.
1177  ∃q: Vector A n.
1178    〈v, q〉 = split A m n p.
1179  #A #m #n #p
1180  elim m
1181  @ex_intro
1182  [1:
1183  |2: @ex_intro
1184      [1:
1185      |2:
1186      ]
1187  ]
1188
1189lemma split_elim:
1190 ∀A,l,m,v.∀P: (Vector A l) × (Vector A m) → Prop.
1191  (∀vl,vm. v = vl@@vm → P 〈vl,vm〉) → P (split A l m v).
1192  #A #l #m #v #P #hyp
1193  normalize
1194  <(split_prod A l m v ? ? ?)
1195
1196example half_add_SO:
1197 ∀pc.
1198 \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc).
1199 cases daemon.
1200qed.
1201
1202(*
1203axiom not_eqvb_S:
1204 ∀pc.
1205 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))).
1206
1207axiom not_eqvb_SS:
1208 ∀pc.
1209 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))).
1210 
1211axiom bitvector_elim_complete:
1212 ∀n,P. bitvector_elim n P = true → ∀bv. P bv.
1213
1214lemma bitvector_elim_complete':
1215 ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true.
1216 #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv
1217 generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ //
1218qed.
1219*)
1220
1221(*
1222lemma andb_elim':
1223 ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true.
1224 #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize //
1225qed.
1226*)
1227
1228let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word)
1229                       (encoding: list Byte) on encoding: Prop ≝
1230  match encoding with
1231  [ nil ⇒ final_pc = pc
1232  | cons hd tl ⇒
1233    let 〈new_pc, byte〉 ≝ next code_memory pc in
1234      hd = byte ∧ encoding_check code_memory new_c final_pc tl
1235  ].
1236
1237lemma encoding_check_append: ∀code_memory,final_pc,l1,pc,l2.
1238 encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … final_pc) (l1@l2) →
1239  let intermediate_pc ≝ pc + length … l1 in
1240   encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … intermediate_pc) l1 ∧
1241    encoding_check code_memory (bitvector_of_nat … intermediate_pc) (bitvector_of_nat … final_pc) l2.
1242 #code_memory #final_pc #l1 elim l1
1243  [ #pc #l2 whd in ⊢ (????% → ?) #H <plus_n_O whd whd in ⊢ (?%?) /2/
1244  | #hd #tl #IH #pc #l2 * #H1 #H2 >half_add_SO in H2; #H2 cases (IH … H2) <plus_n_Sm
1245    #K1 #K2 % [2:@K2] whd % // >half_add_SO @K1 ]
1246qed.
1247
1248axiom bitvector_3_elim_prop:
1249 ∀P: BitVector 3 → Prop.
1250  P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] →
1251  P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] →
1252  P [[true;true;false]] → P [[true;true;true]] → ∀v. P v.
1253
1254definition ticks_of_instruction ≝
1255 λi.
1256  let trivial_code_memory ≝ assembly1 i in
1257  let trivial_status ≝ load_code_memory trivial_code_memory in
1258   \snd (fetch trivial_status (zero ?)).
1259
1260lemma fetch_assembly:
1261  ∀pc,i,code_memory,assembled.
1262    assembled = assembly1 i →
1263      let len ≝ length … assembled in
1264      encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
1265      let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in
1266      let 〈instr_pc, ticks〉 ≝ fetched in
1267      let 〈instr,pc'〉 ≝ instr_pc in
1268       (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' (bitvector_of_nat … (pc + len))) = true.
1269 #pc #i #code_memory #assembled cases i [8: *]
1270 [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]]
1271 [47,48,49:
1272 |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg
1273  [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,
1274   59,60,63,64,65,66,67: #ARG]]
1275 [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,
1276  56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2
1277  [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,
1278   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,
1279   68,69,70,71: #ARG2]]
1280 [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3]
1281 normalize in ⊢ (???% → ?)
1282 [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)]
1283  normalize in ⊢ (???% → ?)]
1284 #H >H * #H1 try (whd in ⊢ (% → ?) * #H2)
1285 try (whd in ⊢ (% → ?) * #H3) whd in ⊢ (% → ?) #H4
1286 change in ⊢ (let fetched ≝ % in ?) with (fetch0 ??)
1287 whd in ⊢ (let fetched ≝ ??% in ?) <H1 whd in ⊢ (let fetched ≝ % in ?)
1288 [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3]
1289 [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,
1290  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,
1291  69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2]
1292 whd >eq_instruction_refl >H4 @eq_bv_refl
1293qed.
1294
1295let rec fetch_many code_memory final_pc pc expected on expected: Prop ≝
1296 match expected with
1297  [ nil ⇒ eq_bv … pc final_pc = true
1298  | cons i tl ⇒
1299     let fetched ≝ fetch code_memory pc in
1300     let 〈instr_pc, ticks〉 ≝ fetched in
1301     let 〈instr,pc'〉 ≝ instr_pc in
1302      eq_instruction instr i = true ∧
1303      ticks = (ticks_of_instruction i) ∧
1304      fetch_many code_memory final_pc pc' tl].
1305
1306lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b.
1307 #A #a #b #EQ destruct //
1308qed.
1309
1310axiom eq_instruction_to_eq:
1311  ∀i1,i2. eq_instruction i1 i2 = true → i1 = i2.
1312               
1313
1314notation < "❰ x ❱" with precedence 90 for @{'dp $x $y }.
1315interpretation "dp" 'dp x y = (dp x y).
1316
1317lemma fetch_assembly_pseudo':
1318 ∀program.∀pol:policy program.∀ppc,lookup_labels,lookup_datalabels.
1319  ∀pi,code_memory,len,assembled,instructions,pc.
1320   let expansion ≝ pol ppc in
1321   Some ? instructions = expand_pseudo_instruction_safe lookup_labels lookup_datalabels (bitvector_of_nat ? pc) expansion pi →
1322    Some … 〈len,assembled〉 = assembly_1_pseudoinstruction_safe program pol ppc (bitvector_of_nat ? pc) lookup_labels lookup_datalabels pi →
1323     encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
1324      fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions.
1325 #program #pol #ppc #lookup_labels #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc
1326 #EQ1 whd in ⊢ (???% → ?) <EQ1 whd in ⊢ (???% → ?) #EQ2
1327 cases (pair_destruct ?????? (option_destruct_Some … EQ2)) -EQ2; #EQ2a #EQ2b
1328 >EQ2a >EQ2b -EQ2a EQ2b;
1329  generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc
1330  generalize in match pc elim instructions
1331  [ #pc whd in ⊢ (% → %) #H >H @eq_bv_refl
1332  | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd
1333    generalize in match (fetch_assembly pc i code_memory … (refl …) H1)
1334    cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %)
1335    cases newi_pc #newi #newpc whd in ⊢ (% → %) #K cases (conjunction_true … K) -K; #K1
1336    cases (conjunction_true … K1) -K1; #K1 #K2 #K3 % try %
1337    [ @K1 | @eqb_true_to_eq <(eq_instruction_to_eq … K1) @K2 | >(eq_bv_eq … K3) @IH @H2 ]
1338qed.
1339
1340axiom bitvector_of_nat_nat_of_bitvector:
1341  ∀n,v.
1342    bitvector_of_nat n (nat_of_bitvector n v) = v.
1343
1344(* CSC: soo long next one; can we merge with previous one now? *)
1345lemma fetch_assembly_pseudo:
1346 ∀program.∀pol:policy program.∀ppc.
1347  ∀code_memory.
1348   let lookup_labels ≝ ? in
1349   let lookup_datalabels ≝ ? in
1350   let pc ≝ ? in
1351   let pi ≝  \fst  (fetch_pseudo_instruction (\snd  program) ppc) in
1352   let instructions ≝ expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels pc (refl …) (refl …) (refl …) in
1353   let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi (refl …) (refl …) (refl …) in
1354     encoding_check code_memory pc (bitvector_of_nat … (nat_of_bitvector ? pc + len)) assembled →
1355      fetch_many code_memory (bitvector_of_nat … (nat_of_bitvector ? pc + len)) pc instructions.
1356 #program #pol #ppc #code_memory
1357 letin lookup_labels ≝ (λx:Identifier
1358    .sigma program pol (address_of_word_labels_code_mem (\snd  program) x))
1359 letin lookup_datalabels ≝
1360  (λx:BitVector 16
1361    .lookup Identifier 16 x (construct_datalabels (\fst  program)) (zero 16))
1362 letin pc ≝ (sigma program pol ppc)
1363 letin pi ≝ (\fst  (fetch_pseudo_instruction (\snd  program) ppc))
1364 letin instructions ≝ (expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels pc (refl …) (refl …) (refl …))
1365 @pair_elim' #len #assembled #EQ1 #H
1366 generalize in match
1367  (fetch_assembly_pseudo' program pol ppc lookup_labels lookup_datalabels pi
1368  code_memory len assembled instructions (nat_of_bitvector ? pc) ???) in ⊢ ?;
1369 [ >bitvector_of_nat_nat_of_bitvector //
1370 | >bitvector_of_nat_nat_of_bitvector normalize nodelta >(sig2 ?? (expand_pseudo_instruction …)) %
1371 | >bitvector_of_nat_nat_of_bitvector <EQ1 @assembly_1_pseudoinstruction_ok1
1372 | //]
1373qed.
1374
1375(* This is a trivial consequence of fetch_assembly + the proof that the
1376   function that load the code in memory is correct. The latter is based
1377   on missing properties from the standard library on the BitVectorTrie
1378   data structrure. *)
1379axiom assembly_ok:
1380 ∀program,pol,assembled,costs'.
1381  let 〈labels,costs〉 ≝ build_maps program pol in
1382  〈assembled,costs'〉 = assembly program pol →
1383  costs = costs' ∧
1384  let code_memory ≝ load_code_memory assembled in
1385  let preamble ≝ \fst program in
1386  let datalabels ≝ construct_datalabels preamble in
1387  let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in
1388  let lookup_datalabels ≝ λx. lookup ?? x datalabels (zero ?) in
1389  ∀ppc.
1390  let pi_newppc ≝ fetch_pseudo_instruction (\snd program) ppc in
1391   ∀len,assembledi.
1392     〈len,assembledi〉 = assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels (\fst pi_newppc) (refl …) (refl …) (refl …) →
1393      encoding_check code_memory (sigma program pol ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len)) assembledi ∧
1394       sigma program pol (\snd pi_newppc) = bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len).
1395
1396lemma fetch_assembly_pseudo2:
1397 ∀program,pol,ppc.
1398(*  let 〈labels,costs〉 ≝ build_maps program pol in *)
1399  let assembled ≝ \fst (assembly program pol) in
1400  let code_memory ≝ load_code_memory assembled in
1401  let data_labels ≝ construct_datalabels (\fst program) in
1402  let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in
1403  let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in
1404  let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1405  let instructions ≝ expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels (sigma program pol ppc) (refl …) (refl …) (refl …) in
1406   fetch_many code_memory (sigma program pol newppc) (sigma program pol ppc) instructions.
1407 * #preamble #instr_list #pol #ppc
1408 letin assembled ≝ (\fst (assembly 〈preamble,instr_list〉 pol))
1409 letin code_memory ≝ (load_code_memory assembled)
1410 letin data_labels ≝ (construct_datalabels preamble)
1411 letin lookup_labels ≝ (λx. sigma … pol (address_of_word_labels_code_mem instr_list x))
1412 letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?))
1413 @pair_elim' #pi #newppc #EQ1 change in EQ1 with (fetch_pseudo_instruction instr_list ? = ?)
1414 generalize in match (assembly_ok ? pol assembled (\snd (assembly 〈preamble,instr_list〉 pol)))
1415 @pair_elim' #labels #costs #EQ2 <eq_pair_fst_snd
1416 #H cases (H (refl \ldots)) -H; #EQ3
1417 generalize in match (refl … (assembly_1_pseudoinstruction … pol ppc lookup_labels lookup_datalabels ? (refl …) (refl …) (refl …)))
1418 cases (assembly_1_pseudoinstruction ?????????) in ⊢ (???% → ?) #len #assembledi #EQ4
1419 #H cases (H ppc len assembledi ?) [2: <EQ4 %] -H; #H1 #H2
1420 generalize in match (fetch_assembly_pseudo … pol ppc (load_code_memory assembled)) in ⊢ ?
1421 >EQ4 #H generalize in match (H H1) in ⊢ ? -H; >(pair_destruct_2 ????? (sym_eq … EQ1))
1422 >H2 #K @K
1423qed.
1424
1425(* OLD?
1426definition assembly_specification:
1427  ∀assembly_program: pseudo_assembly_program.
1428  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1429  λpseudo_assembly_program.
1430  λcode_mem.
1431    ∀pc: Word.
1432      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1433      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1434      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1435      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1436      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1437       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1438      match pre_assembled with
1439       [ None ⇒ True
1440       | Some pc_code ⇒
1441          let 〈new_pc,code〉 ≝ pc_code in
1442           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1443
1444axiom assembly_meets_specification:
1445  ∀pseudo_assembly_program.
1446    match assembly pseudo_assembly_program with
1447    [ None ⇒ True
1448    | Some code_mem_cost ⇒
1449      let 〈code_mem, cost〉 ≝ code_mem_cost in
1450        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1451    ].
1452(*
1453  # PROGRAM
1454  [ cases PROGRAM
1455    # PREAMBLE
1456    # INSTR_LIST
1457    elim INSTR_LIST
1458    [ whd
1459      whd in ⊢ (∀_. %)
1460      # PC
1461      whd
1462    | # INSTR
1463      # INSTR_LIST_TL
1464      # H
1465      whd
1466      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1467    ]
1468  | cases not_implemented
1469  ] *)
1470*)
1471
1472definition internal_pseudo_address_map ≝ list (BitVector 8).
1473
1474axiom low_internal_ram_of_pseudo_low_internal_ram:
1475 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1476
1477axiom high_internal_ram_of_pseudo_high_internal_ram:
1478 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1479
1480axiom low_internal_ram_of_pseudo_internal_ram_hit:
1481 ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀pol:policy (code_memory … s).∀addr:BitVector 7.
1482  member ? (eq_bv 8) (false:::addr) M = true →
1483   let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1484   let pbl ≝ lookup ? 7 addr (low_internal_ram ? s) (zero 8) in
1485   let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram ? s) (zero 8) in
1486   let bl ≝ lookup ? 7 addr ram (zero 8) in
1487   let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in
1488    bu@@bl = sigma (code_memory … s) pol (pbu@@pbl).
1489
1490(* changed from add to sub *)
1491axiom low_internal_ram_of_pseudo_internal_ram_miss:
1492 ∀T.∀M:internal_pseudo_address_map.∀s:PreStatus T.∀addr:BitVector 7.
1493  let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1494  let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in
1495  let carr ≝ get_index_v ? ? flags 1 ? in
1496  carr = false →
1497  member ? (eq_bv 8) (false:::Saddr) M = false →
1498   member ? (eq_bv 8) (false:::addr) M = false →
1499    lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?).
1500  //
1501qed.
1502
1503definition addressing_mode_ok ≝
1504 λT.λM:internal_pseudo_address_map.λs:PreStatus T.
1505  λaddr:addressing_mode.
1506   match addr with
1507    [ DIRECT d ⇒
1508       ¬(member ? (eq_bv 8) d M) ∧
1509       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1510    | INDIRECT i ⇒
1511       let d ≝ get_register ? s [[false;false;i]] in
1512       ¬(member ? (eq_bv 8) d M) ∧
1513       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1514    | EXT_INDIRECT _ ⇒ true
1515    | REGISTER _ ⇒ true
1516    | ACC_A ⇒ true
1517    | ACC_B ⇒ true
1518    | DPTR ⇒ true
1519    | DATA _ ⇒ true
1520    | DATA16 _ ⇒ true
1521    | ACC_DPTR ⇒ true
1522    | ACC_PC ⇒ true
1523    | EXT_INDIRECT_DPTR ⇒ true
1524    | INDIRECT_DPTR ⇒ true
1525    | CARRY ⇒ true
1526    | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1527    | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1528    | RELATIVE _ ⇒ true
1529    | ADDR11 _ ⇒ true
1530    | ADDR16 _ ⇒ true ].
1531   
1532definition next_internal_pseudo_address_map0 ≝
1533  λT.
1534  λfetched.
1535  λM: internal_pseudo_address_map.
1536  λs: PreStatus T.
1537   match fetched with
1538    [ Comment _ ⇒ Some ? M
1539    | Cost _ ⇒ Some … M
1540    | Jmp _ ⇒ Some … M
1541    | Call _ ⇒
1542       Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M)
1543    | Mov _ _ ⇒ Some … M
1544    | Instruction instr ⇒
1545       match instr with
1546        [ ADD addr1 addr2 ⇒
1547           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1548            Some ? M
1549           else
1550            None ?
1551        | ADDC addr1 addr2 ⇒
1552           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1553            Some ? M
1554           else
1555            None ?
1556        | SUBB addr1 addr2 ⇒
1557           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1558            Some ? M
1559           else
1560            None ?       
1561        | _ ⇒ (* TO BE COMPLETED *) Some ? M ]].
1562 
1563
1564definition next_internal_pseudo_address_map ≝
1565 λM:internal_pseudo_address_map.
1566  λs:PseudoStatus.
1567    next_internal_pseudo_address_map0 ?
1568     (\fst (fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s))) M s.
1569   
1570definition status_of_pseudo_status:
1571 internal_pseudo_address_map → ∀ps:PseudoStatus. policy (code_memory … ps) → Status ≝
1572 λM,ps,pol.
1573  let pap ≝ code_memory … ps in
1574  let p ≝ assembly pap pol in
1575  let cm ≝ load_code_memory (\fst p) in
1576  let pc ≝ sigma pap pol (program_counter ? ps) in
1577  let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in
1578  let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in
1579     mk_PreStatus (BitVectorTrie Byte 16)
1580      cm
1581      lir
1582      hir
1583      (external_ram … ps)
1584      pc
1585      (special_function_registers_8051 … ps)
1586      (special_function_registers_8052 … ps)
1587      (p1_latch … ps)
1588      (p3_latch … ps)
1589      (clock … ps).
1590
1591(*
1592definition write_at_stack_pointer':
1593 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1594  λM: Type[0].
1595  λs: PreStatus M.
1596  λv: Byte.
1597    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1598    let bit_zero ≝ get_index_v… nu O ? in
1599    let bit_1 ≝ get_index_v… nu 1 ? in
1600    let bit_2 ≝ get_index_v… nu 2 ? in
1601    let bit_3 ≝ get_index_v… nu 3 ? in
1602      if bit_zero then
1603        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1604                              v (low_internal_ram ? s) in
1605          set_low_internal_ram ? s memory
1606      else
1607        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1608                              v (high_internal_ram ? s) in
1609          set_high_internal_ram ? s memory.
1610  [ cases l0 %
1611  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1612qed.
1613
1614definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1615 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1616
1617  λticks_of.
1618  λs.
1619  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1620  let ticks ≝ ticks_of (program_counter ? s) in
1621  let s ≝ set_clock ? s (clock ? s + ticks) in
1622  let s ≝ set_program_counter ? s pc in
1623    match instr with
1624    [ Instruction instr ⇒
1625       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1626    | Comment cmt ⇒ s
1627    | Cost cst ⇒ s
1628    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1629    | Call call ⇒
1630      let a ≝ address_of_word_labels s call in
1631      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1632      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1633      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1634      let s ≝ write_at_stack_pointer' ? s pc_bl 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 s ≝ write_at_stack_pointer' ? s pc_bu in
1638        set_program_counter ? s a
1639    | Mov dptr ident ⇒
1640       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1641    ].
1642 [
1643 |2,3,4: %
1644 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1645 |
1646 | %
1647 ]
1648 cases not_implemented
1649qed.
1650*)
1651
1652axiom execute_1_pseudo_instruction_preserves_code_memory:
1653 ∀ticks_of,ps.
1654  code_memory … (execute_1_pseudo_instruction ticks_of ps) = code_memory … ps.
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 → Word → ? → nat × nat ≝
1709  λprogram: pseudo_assembly_program.λpol.
1710  λppc: Word.
1711  λfetched.
1712    match fetched with
1713    [ Instruction instr ⇒
1714      match instr with
1715      [ JC lbl ⇒
1716        match pol ppc with
1717        [ short_jump ⇒ 〈2, 2〉
1718        | medium_jump ⇒ ?
1719        | long_jump ⇒ 〈4, 4〉
1720        ]
1721      | JNC lbl ⇒
1722        match pol ppc with
1723        [ short_jump ⇒ 〈2, 2〉
1724        | medium_jump ⇒ ?
1725        | long_jump ⇒ 〈4, 4〉
1726        ]
1727      | JB bit lbl ⇒
1728        match pol ppc with
1729        [ short_jump ⇒ 〈2, 2〉
1730        | medium_jump ⇒ ?
1731        | long_jump ⇒ 〈4, 4〉
1732        ]
1733      | JNB bit lbl ⇒
1734        match pol ppc with
1735        [ short_jump ⇒ 〈2, 2〉
1736        | medium_jump ⇒ ?
1737        | long_jump ⇒ 〈4, 4〉
1738        ]
1739      | JBC bit lbl ⇒
1740        match pol ppc with
1741        [ short_jump ⇒ 〈2, 2〉
1742        | medium_jump ⇒ ?
1743        | long_jump ⇒ 〈4, 4〉
1744        ]
1745      | JZ lbl ⇒
1746        match pol ppc with
1747        [ short_jump ⇒ 〈2, 2〉
1748        | medium_jump ⇒ ?
1749        | long_jump ⇒ 〈4, 4〉
1750        ]
1751      | JNZ lbl ⇒
1752        match pol ppc with
1753        [ short_jump ⇒ 〈2, 2〉
1754        | medium_jump ⇒ ?
1755        | long_jump ⇒ 〈4, 4〉
1756        ]
1757      | CJNE arg lbl ⇒
1758        match pol ppc with
1759        [ short_jump ⇒ 〈2, 2〉
1760        | medium_jump ⇒ ?
1761        | long_jump ⇒ 〈4, 4〉
1762        ]
1763      | DJNZ arg lbl ⇒
1764        match pol 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 → Word → nat × nat ≝
1864  λprogram: pseudo_assembly_program.λpol.
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 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,s1,s2,s3.∀pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
1979  get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP →
1980  low_internal_ram ? s2 = low_internal_ram T2 s3 →
1981  sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) →
1982  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
1983  bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) →
1984   low_internal_ram T1
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,s1,s2,s3,pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2005  get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP →
2006  high_internal_ram ? s2 = high_internal_ram T2 s3 →
2007  sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) →
2008  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
2009  bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) →
2010   high_internal_ram T1
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
2034theorem main_thm:
2035 ∀M,M',ps,pol.
2036  next_internal_pseudo_address_map M ps = Some … M' →
2037   ∃n.
2038      execute n (status_of_pseudo_status M ps pol)
2039    = status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps) ?.
2040 [2: >execute_1_pseudo_instruction_preserves_code_memory @pol]
2041 #M #M' #ps #pol #SAFE
2042 cut
2043  (∀ps'.
2044    ∀prf:ps'=execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps.
2045    ∃n. execute n (status_of_pseudo_status M ps pol) = status_of_pseudo_status M' ps' ?)
2046 [ >prf >execute_1_pseudo_instruction_preserves_code_memory @pol |3: #K @(K ? (refl …))]
2047 #ps' #EQ
2048 whd in ⊢ (??(λ_.??(??%)?))
2049 change with
2050  (∃n.
2051    execute n
2052     (set_low_internal_ram ?
2053       (set_high_internal_ram ?
2054         (set_program_counter ?
2055           (set_code_memory ?? ps (load_code_memory ?))
2056          (sigma ? pol (program_counter ? ps)))
2057        (high_internal_ram_of_pseudo_high_internal_ram M ?))
2058      (low_internal_ram_of_pseudo_low_internal_ram M ?))
2059   = set_low_internal_ram ?
2060      (set_high_internal_ram ?
2061        (set_program_counter ?
2062          (set_code_memory ?? ? (load_code_memory ?))
2063         (sigma ???)) ?) ?)
2064 >EQ whd in match eq_rect_Type0_r normalize nodelta
2065 >execute_1_pseudo_instruction_preserves_code_memory normalize nodelta
2066 generalize in match EQ -EQ;
2067 generalize in match (refl … (code_memory pseudo_assembly_program ps))
2068 generalize in match pol -pol; generalize in ⊢ (∀_.??%? → ?)
2069 * #preamble #instr_list #pol #EQ1 generalize in match pol -pol <EQ1 #pol #EQps' <EQps'
2070 (* Dependent types madness ends here *)
2071 letin ppc ≝ (program_counter … ps)
2072 generalize in match (fetch_assembly_pseudo2 ? pol ppc) in ⊢ ?
2073 letin assembled ≝ (\fst (assembly ? pol))
2074 letin lookup_labels ≝ (λx. sigma ? pol (address_of_word_labels_code_mem instr_list x))
2075 letin lookup_datalabels ≝ (λx. lookup … x (construct_datalabels preamble) (zero 16))
2076 @pair_elim' #pi #newppc #EQ2
2077 letin instructions ≝
2078  (expand_pseudo_instruction ? pol ppc lookup_labels lookup_datalabels
2079    (sigma ? pol ppc) (refl …) (refl …) (refl …))
2080 change with (fetch_many ???? → ?) #H1
2081 change in EQ2 with (fetch_pseudo_instruction instr_list ppc = ?)
2082 change in SAFE with (next_internal_pseudo_address_map0 ???? = ?) <EQ1 in SAFE;
2083 >EQ2 whd in ⊢ (??(??%??)? → ?) #SAFE
2084 whd in EQps':(???%); <EQ1 in EQps'; >EQ2 normalize nodelta
2085 generalize in match H1; -H1; generalize in match instructions -instructions
2086 * #instructions >EQ2 change in match (\fst 〈pi,newppc〉) with pi
2087 whd in match ticks_of normalize nodelta <EQ1 >EQ2
2088 cases pi in SAFE
2089  [2,3: (* Comment, Cost *) #ARG whd in ⊢ (??%? → ???% → ?)
2090   @Some_Some_elim #MAP #EQ3 @(Some_Some_elim ????? EQ3) #EQ3'
2091   whd in match eject normalize nodelta >EQ3' in ⊢ (% → ?) whd in ⊢ (% → ?)
2092   #H2 #EQ %[1,3:@0]
2093   <MAP >(eq_bv_eq … H2) >EQ %
2094  |4: (* Jmp *) #label whd in ⊢ (??%? → ???% → ?)
2095   @Some_Some_elim #MAP cases (pol ?) normalize nodelta
2096       [3: (* long *) #EQ3 @(Some_Some_elim ????? EQ3) #EQ3'
2097         whd in match eject normalize nodelta >EQ3' in ⊢ (% → ?) whd in ⊢ (% → ?)
2098         @pair_elim' * #instr #newppc' #ticks #EQ4       
2099         * * #H2a #H2b whd in ⊢ (% → ?) #H2
2100         >H2b >(eq_instruction_to_eq … H2a)
2101         #EQ %[@1]
2102         <MAP >(eq_bv_eq … H2) >EQ
2103         whd in ⊢ (??%?) >EQ4 whd in ⊢ (??%?)
2104         cases ps in EQ4; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXX >XXX %
2105         whd in ⊢ (??%?)
2106         whd in ⊢ (??(match ?%? with [_ ⇒ ?])?)
2107         cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
2108  |6: (* Mov *) #arg1 #arg2
2109       #H1 #H2 #EQ %[@1]
2110       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2111       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2112       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2113       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2114       >H2b >(eq_instruction_to_eq … H2a)
2115       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2116       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?)
2117       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2118       normalize nodelta;
2119       [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2120       #result #flags
2121       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
2122  |5: (* Call *) #label #MAP
2123      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP;
2124      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2125       [ (* short *) #abs @⊥ destruct (abs)
2126       |3: (* long *) #H1 #H2 #EQ %[@1]
2127           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2128           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2129           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2130           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2131           >H2b >(eq_instruction_to_eq … H2a)
2132           generalize in match EQ; -EQ;
2133           whd in ⊢ (???% → ??%?);
2134           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;
2135           >(eq_bv_eq … H2c)
2136           change with
2137            ((?=let 〈ppc_bu,ppc_bl〉 ≝ split bool 8 8 newppc in ?) →
2138                (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2139           generalize in match (refl … (split … 8 8 newppc)) cases (split bool 8 8 newppc) in ⊢ (??%? → %) #ppc_bu #ppc_bl #EQppc
2140           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;
2141           >get_8051_sfr_write_at_stack_pointer >get_8051_sfr_write_at_stack_pointer
2142           >get_8051_sfr_set_8051_sfr >get_8051_sfr_set_8051_sfr
2143           generalize in match (refl … (half_add ? new_sp (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry' #new_sp' #EQ2 normalize nodelta;
2144           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2145           @split_eq_status;
2146            [ >code_memory_write_at_stack_pointer whd in ⊢ (??%?)
2147              >code_memory_write_at_stack_pointer %
2148            | >set_program_counter_set_low_internal_ram
2149              >set_clock_set_low_internal_ram
2150              @low_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            | >set_low_internal_ram_set_high_internal_ram
2162              >set_program_counter_set_high_internal_ram
2163              >set_clock_set_high_internal_ram
2164              @high_internal_ram_write_at_stack_pointer
2165               [ >EQ0 @pol | % | %
2166               | @(pair_destruct_2 … EQ1)
2167               | @(pair_destruct_2 … EQ2)
2168               | >(pair_destruct_1 ????? EQpc)
2169                 >(pair_destruct_2 ????? EQpc)
2170                 @split_elim #x #y #H <H -x y H;
2171                 >(pair_destruct_1 ????? EQppc)
2172                 >(pair_destruct_2 ????? EQppc)
2173                 @split_elim #x #y #H <H -x y H;
2174                 >EQ0 % ]           
2175            | >external_ram_write_at_stack_pointer whd in ⊢ (??%?)
2176              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
2177              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
2178              >external_ram_write_at_stack_pointer %
2179            | change with (? = sigma ?? (address_of_word_labels_code_mem (\snd (code_memory ? ps)) ?))
2180              >EQ0 %
2181            | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (??%?)
2182              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
2183              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
2184              >special_function_registers_8051_write_at_stack_pointer %
2185            | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (??%?)
2186              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
2187              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
2188              >special_function_registers_8052_write_at_stack_pointer %
2189            | >p1_latch_write_at_stack_pointer whd in ⊢ (??%?)
2190              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
2191              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
2192              >p1_latch_write_at_stack_pointer %
2193            | >p3_latch_write_at_stack_pointer whd in ⊢ (??%?)
2194              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
2195              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
2196              >p3_latch_write_at_stack_pointer %
2197            | >clock_write_at_stack_pointer whd in ⊢ (??%?)
2198              >clock_write_at_stack_pointer whd in ⊢ (???%)
2199              >clock_write_at_stack_pointer whd in ⊢ (???%)
2200              >clock_write_at_stack_pointer %]
2201       (*| (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2202         @pair_elim' #fst_5_addr #rest_addr #EQ1
2203         @pair_elim' #fst_5_pc #rest_pc #EQ2
2204         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2205         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2206         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2207         change in ⊢ (? →??%?) with (execute_1_0 ??)
2208         @pair_elim' * #instr #newppc' #ticks #EQn
2209          * * #H2a #H2b whd in ⊢ (% → ?) #H2c >H2b >(eq_instruction_to_eq … H2a) whd in ⊢ (??%?)
2210          generalize in match EQ; -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2211          @pair_elim' #carry #new_sp change with (half_add ? (get_8051_sfr ? ps ?) ? = ? → ?) #EQ4
2212          @split_elim' #pc_bu #pc_bl >program_counter_set_8051_sfr XXX change with (newppc = ?) #EQ5
2213          @pair_elim' #carry' #new_sp' #EQ6 normalize nodelta; #EQx >EQx -EQx;
2214          change in ⊢ (??(match ????% with [_ ⇒ ?])?) with (sigma … newppc)
2215          @split_elim' #pc_bu' #pc_bl' #EQ7 change with (newppc' = ? → ?)
2216          >get_8051_sfr_set_8051_sfr
2217         
2218          whd in EQ:(???%) ⊢ ? >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2219           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?)
2220           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc)))
2221           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2222           generalize in match (refl … (split bool 4 4 pc_bu))
2223           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2224           generalize in match (refl … (split bool 3 8 rest_addr))
2225           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2226           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2227           generalize in match
2228            (refl …
2229             (half_add 16 (sigma 〈preamble,instr_list〉 newppc)
2230             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2231           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7
2232           @split_eq_status try %
2233            [ change with (? = sigma ? (address_of_word_labels ps label))
2234              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2235            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2236              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] *)]
2237  |4: (* Jmp *) #label #MAP
2238      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP >MAP -MAP;
2239      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2240       [3: (* long *) #H1 #H2 #EQ %[@1]
2241           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2242           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2243           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2244           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2245           >H2b >(eq_instruction_to_eq … H2a)
2246           generalize in match EQ; -EQ;
2247           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2248           cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
2249       |1: (* short *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2250           generalize in match
2251            (refl ?
2252             (sub_16_with_carry
2253              (sigma 〈preamble,instr_list〉 pol (program_counter … ps))
2254              (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))
2255              false))
2256           cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta;
2257           generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta;
2258           generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta;
2259           #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)]
2260           generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2261           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2262           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2263           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2264           >H2b >(eq_instruction_to_eq … H2a)
2265           generalize in match EQ; -EQ;
2266           whd in ⊢ (???% → ?);
2267           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2268           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ???) ? in ?) = ?)
2269           generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) (sign_extension lower)))
2270           cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4
2271           @split_eq_status try % change with (newpc = sigma ?? (address_of_word_labels ps label))
2272           (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2273       | (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2274         generalize in match
2275          (refl …
2276            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))))
2277         cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1
2278         generalize in match
2279          (refl …
2280            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (program_counter … ps))))
2281         cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2
2282         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2283         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2284         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2285         change in ⊢ (? →??%?) with (execute_1_0 ??)
2286           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2287           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2288           >H2b >(eq_instruction_to_eq … H2a)
2289           generalize in match EQ; -EQ;
2290           whd in ⊢ (???% → ?);
2291           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2292           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2293           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)))
2294           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2295           generalize in match (refl … (split bool 4 4 pc_bu))
2296           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2297           generalize in match (refl … (split bool 3 8 rest_addr))
2298           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2299           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2300           generalize in match
2301            (refl …
2302             (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc)
2303             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2304           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7   
2305           @split_eq_status try %
2306            [ change with (? = sigma ?? (address_of_word_labels ps label))
2307              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2308            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2309              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]]
2310  | (* Instruction *) -pi;  whd in ⊢ (? → ??%? → ?) *; normalize nodelta;
2311    [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1]
2312       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2313       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2314       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2315       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2316       >H2b >(eq_instruction_to_eq … H2a)
2317       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP;
2318       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1;
2319       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2320       normalize nodelta; #MAP;
2321       [1: change in ⊢ (? → %) with
2322        ((let 〈result,flags〉 ≝
2323          add_8_with_carry
2324           (get_arg_8 ? ps false ACC_A)
2325           (get_arg_8 ?
2326             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2327             false (DIRECT ARG2))
2328           ? in ?) = ?)
2329        [2,3: %]
2330        change in ⊢ (???% → ?) with
2331         (let 〈result,flags〉 ≝ add_8_with_carry ?(*(get_arg_8 ? ps false ACC_A)*) ?? in ?)
2332        >get_arg_8_set_clock
2333       [1,2: cases (addressing_mode_ok ???? ∧ addressing_mode_ok ????) in MAP ⊢ ?
2334         [2,4: #abs @⊥ normalize in abs; destruct (abs)
2335         |*:whd in ⊢ (??%? → ?) #H <(option_destruct_Some ??? H)]
2336       [ change in ⊢ (? → %) with
2337        ((let 〈result,flags〉 ≝
2338          add_8_with_carry
2339           (get_arg_8 ? ps false ACC_A)
2340           (get_arg_8 ?
2341             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2342             false (DIRECT ARG2))
2343           ? in ?) = ?)
2344          >get_arg_8_set_low_internal_ram
2345       
2346        cases (add_8_with_carry ???)
2347         
2348        [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2349       #result #flags
2350       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
2351    | (* INC *) #arg1 #H1 #H2 #EQ %[@1]
2352       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2353       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2354       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2355       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2356       >H2b >(eq_instruction_to_eq … H2a)
2357       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2358       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG]
2359       [1,2,3,4: cases (half_add ???) #carry #result
2360       | cases (half_add ???) #carry #bl normalize nodelta;
2361         cases (full_add ????) #carry' #bu normalize nodelta ]
2362        #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) -newppc';
2363        [5: %
2364        |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program
2365      (set_program_counter pseudo_assembly_program ps newppc)
2366      (\fst  (ticks_of0 〈preamble,instr_list〉
2367                   (program_counter pseudo_assembly_program ps)
2368                   (Instruction (INC Identifier (DIRECT ARG))))
2369       +clock pseudo_assembly_program
2370        (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG))
2371        [2,3: // ]
2372            <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://]
2373            whd in ⊢ (??%%)
2374            cases (split bool 4 4 ARG)
2375            #nu' #nl'
2376            normalize nodelta
2377            cases (split bool 1 3 nu')
2378            #bit_1' #ignore'
2379            normalize nodelta
2380            cases (get_index_v bool 4 nu' ? ?)
2381            [ normalize nodelta (* HERE *) whd in ⊢ (??%%) %
2382            |
2383            ]
2384cases daemon (* EASY CASES TO BE COMPLETED *)
2385qed.
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