source: src/ASM/AssemblyProof.ma @ 1607

Last change on this file since 1607 was 1607, checked in by sacerdot, 8 years ago

Porting to new library.

File size: 88.6 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
1173(*
1174lemma split_prod_exists:
1175  ∀A, m, n.
1176  ∀p: Vector A (m + n).
1177  ∃v: Vector A m.
1178  ∃q: Vector A n.
1179    〈v, q〉 = split A m n p.
1180  #A #m #n #p
1181  elim m
1182  @ex_intro
1183  [1:
1184  |2: @ex_intro
1185      [1:
1186      |2:
1187      ]
1188  ]
1189*)
1190
1191axiom split_elim:
1192 ∀A,l,m,v.∀P: (Vector A l) × (Vector A m) → Prop.
1193  (∀vl,vm. v = vl@@vm → P 〈vl,vm〉) → P (split A l m v).
1194(*  #A #l #m #v #P #hyp
1195  normalize
1196  <(split_prod A l m v ? ? ?)
1197*)
1198
1199example half_add_SO:
1200 ∀pc.
1201 \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc).
1202 cases daemon.
1203qed.
1204
1205(*
1206axiom not_eqvb_S:
1207 ∀pc.
1208 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))).
1209
1210axiom not_eqvb_SS:
1211 ∀pc.
1212 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))).
1213 
1214axiom bitvector_elim_complete:
1215 ∀n,P. bitvector_elim n P = true → ∀bv. P bv.
1216
1217lemma bitvector_elim_complete':
1218 ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true.
1219 #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv
1220 generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ //
1221qed.
1222*)
1223
1224(*
1225lemma andb_elim':
1226 ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true.
1227 #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize //
1228qed.
1229*)
1230
1231let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word)
1232                       (encoding: list Byte) on encoding: Prop ≝
1233  match encoding with
1234  [ nil ⇒ final_pc = pc
1235  | cons hd tl ⇒
1236    let 〈new_pc, byte〉 ≝ next code_memory pc in
1237      hd = byte ∧ encoding_check code_memory new_pc final_pc tl
1238  ].
1239
1240lemma encoding_check_append: ∀code_memory,final_pc,l1,pc,l2.
1241 encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … final_pc) (l1@l2) →
1242  let intermediate_pc ≝ pc + length … l1 in
1243   encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … intermediate_pc) l1 ∧
1244    encoding_check code_memory (bitvector_of_nat … intermediate_pc) (bitvector_of_nat … final_pc) l2.
1245 #code_memory #final_pc #l1 elim l1
1246  [ #pc #l2 whd in ⊢ (????% → ?) #H <plus_n_O whd whd in ⊢ (?%?) /2/
1247  | #hd #tl #IH #pc #l2 * #H1 #H2 >half_add_SO in H2; #H2 cases (IH … H2) <plus_n_Sm
1248    #K1 #K2 % [2:@K2] whd % // >half_add_SO @K1 ]
1249qed.
1250
1251axiom bitvector_3_elim_prop:
1252 ∀P: BitVector 3 → Prop.
1253  P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] →
1254  P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] →
1255  P [[true;true;false]] → P [[true;true;true]] → ∀v. P v.
1256
1257definition ticks_of_instruction ≝
1258 λi.
1259  let trivial_code_memory ≝ assembly1 i in
1260  let trivial_status ≝ load_code_memory trivial_code_memory in
1261   \snd (fetch trivial_status (zero ?)).
1262
1263lemma fetch_assembly:
1264  ∀pc,i,code_memory,assembled.
1265    assembled = assembly1 i →
1266      let len ≝ length … assembled in
1267      encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
1268      let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in
1269      let 〈instr_pc, ticks〉 ≝ fetched in
1270      let 〈instr,pc'〉 ≝ instr_pc in
1271       (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' (bitvector_of_nat … (pc + len))) = true.
1272 #pc #i #code_memory #assembled cases i [8: *]
1273 [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]]
1274 [47,48,49:
1275 |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg
1276  [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,
1277   59,60,63,64,65,66,67: #ARG]]
1278 [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,
1279  56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2
1280  [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,
1281   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,
1282   68,69,70,71: #ARG2]]
1283 [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3]
1284 normalize in ⊢ (???% → ?)
1285 [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)]
1286  normalize in ⊢ (???% → ?)]
1287 #H >H * #H1 try (whd in ⊢ (% → ?) * #H2)
1288 try (whd in ⊢ (% → ?) * #H3) whd in ⊢ (% → ?) #H4
1289 change in ⊢ (let fetched ≝ % in ?) with (fetch0 ??)
1290 whd in ⊢ (let fetched ≝ ??% in ?) <H1 whd in ⊢ (let fetched ≝ % in ?)
1291 [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3]
1292 [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,
1293  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,
1294  69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2]
1295 whd >eq_instruction_refl >H4 @eq_bv_refl
1296qed.
1297
1298let rec fetch_many code_memory final_pc pc expected on expected: Prop ≝
1299 match expected with
1300  [ nil ⇒ eq_bv … pc final_pc = true
1301  | cons i tl ⇒
1302     let fetched ≝ fetch code_memory pc in
1303     let 〈instr_pc, ticks〉 ≝ fetched in
1304     let 〈instr,pc'〉 ≝ instr_pc in
1305      eq_instruction instr i = true ∧
1306      ticks = (ticks_of_instruction i) ∧
1307      fetch_many code_memory final_pc pc' tl].
1308
1309lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b.
1310 #A #a #b #EQ destruct //
1311qed.
1312
1313axiom eq_instruction_to_eq:
1314  ∀i1,i2. eq_instruction i1 i2 = true → i1 = i2.
1315               
1316lemma fetch_assembly_pseudo':
1317 ∀program.∀pol:policy program.∀ppc,lookup_labels,lookup_datalabels.
1318  ∀pi,code_memory,len,assembled,instructions,pc.
1319   let expansion ≝ pol ppc in
1320   Some ? instructions = expand_pseudo_instruction_safe lookup_labels lookup_datalabels (bitvector_of_nat ? pc) expansion pi →
1321    Some … 〈len,assembled〉 = assembly_1_pseudoinstruction_safe program pol ppc (bitvector_of_nat ? pc) lookup_labels lookup_datalabels pi →
1322     encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
1323      fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions.
1324 #program #pol #ppc #lookup_labels #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc
1325 #EQ1 whd in ⊢ (???% → ?) <EQ1 whd in ⊢ (???% → ?) #EQ2
1326 cases (pair_destruct ?????? (option_destruct_Some … EQ2)) -EQ2; #EQ2a #EQ2b
1327 >EQ2a >EQ2b -EQ2a EQ2b;
1328  generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc
1329  generalize in match pc elim instructions
1330  [ #pc whd in ⊢ (% → %) #H >H @eq_bv_refl
1331  | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd
1332    generalize in match (fetch_assembly pc i code_memory … (refl …) H1)
1333    cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %)
1334    cases newi_pc #newi #newpc whd in ⊢ (% → %) #K cases (conjunction_true … K) -K; #K1
1335    cases (conjunction_true … K1) -K1; #K1 #K2 #K3 % try %
1336    [ @K1 | @eqb_true_to_eq <(eq_instruction_to_eq … K1) @K2 | >(eq_bv_eq … K3) @IH @H2 ]
1337qed.
1338
1339axiom bitvector_of_nat_nat_of_bitvector:
1340  ∀n,v.
1341    bitvector_of_nat n (nat_of_bitvector n v) = v.
1342
1343(* CSC: soo long next one; can we merge with previous one now? *)
1344lemma fetch_assembly_pseudo:
1345 ∀program.∀pol:policy program.∀ppc.
1346  ∀code_memory.
1347   let lookup_labels ≝ ? in
1348   let lookup_datalabels ≝ ? in
1349   let pc ≝ ? in
1350   let pi ≝  \fst  (fetch_pseudo_instruction (\snd  program) ppc) in
1351   let instructions ≝ expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels pc (refl …) (refl …) (refl …) in
1352   let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi (refl …) (refl …) (refl …) in
1353     encoding_check code_memory pc (bitvector_of_nat … (nat_of_bitvector ? pc + len)) assembled →
1354      fetch_many code_memory (bitvector_of_nat … (nat_of_bitvector ? pc + len)) pc instructions.
1355 #program #pol #ppc #code_memory
1356 letin lookup_labels ≝ (λx:Identifier
1357    .sigma program pol (address_of_word_labels_code_mem (\snd  program) x))
1358 letin lookup_datalabels ≝
1359  (λx:BitVector 16
1360    .lookup Identifier 16 x (construct_datalabels (\fst  program)) (zero 16))
1361 letin pc ≝ (sigma program pol ppc)
1362 letin pi ≝ (\fst  (fetch_pseudo_instruction (\snd  program) ppc))
1363 letin instructions ≝ (expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels pc (refl …) (refl …) (refl …))
1364 @pair_elim' #len #assembled #EQ1 #H
1365 generalize in match
1366  (fetch_assembly_pseudo' program pol ppc lookup_labels lookup_datalabels pi
1367  code_memory len assembled instructions (nat_of_bitvector ? pc) ???) in ⊢ ?;
1368 [ >bitvector_of_nat_nat_of_bitvector //
1369 | >bitvector_of_nat_nat_of_bitvector normalize nodelta >(sig2 ?? (expand_pseudo_instruction …)) %
1370 | >bitvector_of_nat_nat_of_bitvector <EQ1 @assembly_1_pseudoinstruction_ok1
1371 | //]
1372qed.
1373
1374(* This is a trivial consequence of fetch_assembly + the proof that the
1375   function that load the code in memory is correct. The latter is based
1376   on missing properties from the standard library on the BitVectorTrie
1377   data structrure. *)
1378axiom assembly_ok:
1379 ∀program,pol,assembled,costs'.
1380  let 〈labels,costs〉 ≝ build_maps program pol in
1381  〈assembled,costs'〉 = assembly program pol →
1382  costs = costs' ∧
1383  let code_memory ≝ load_code_memory assembled in
1384  let preamble ≝ \fst program in
1385  let datalabels ≝ construct_datalabels preamble in
1386  let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in
1387  let lookup_datalabels ≝ λx. lookup ?? x datalabels (zero ?) in
1388  ∀ppc.
1389  let pi_newppc ≝ fetch_pseudo_instruction (\snd program) ppc in
1390   ∀len,assembledi.
1391     〈len,assembledi〉 = assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels (\fst pi_newppc) (refl …) (refl …) (refl …) →
1392      encoding_check code_memory (sigma program pol ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len)) assembledi ∧
1393       sigma program pol (\snd pi_newppc) = bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len).
1394
1395lemma fetch_assembly_pseudo2:
1396 ∀program,pol,ppc.
1397(*  let 〈labels,costs〉 ≝ build_maps program pol in *)
1398  let assembled ≝ \fst (assembly program pol) in
1399  let code_memory ≝ load_code_memory assembled in
1400  let data_labels ≝ construct_datalabels (\fst program) in
1401  let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in
1402  let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in
1403  let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1404  let instructions ≝ expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels (sigma program pol ppc) (refl …) (refl …) (refl …) in
1405   fetch_many code_memory (sigma program pol newppc) (sigma program pol ppc) instructions.
1406 * #preamble #instr_list #pol #ppc
1407 letin assembled ≝ (\fst (assembly 〈preamble,instr_list〉 pol))
1408 letin code_memory ≝ (load_code_memory assembled)
1409 letin data_labels ≝ (construct_datalabels preamble)
1410 letin lookup_labels ≝ (λx. sigma … pol (address_of_word_labels_code_mem instr_list x))
1411 letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?))
1412 @pair_elim' #pi #newppc #EQ1 change in EQ1 with (fetch_pseudo_instruction instr_list ? = ?)
1413 generalize in match (assembly_ok ? pol assembled (\snd (assembly 〈preamble,instr_list〉 pol)))
1414 @pair_elim' #labels #costs #EQ2 <eq_pair_fst_snd
1415 #H cases (H (refl \ldots)) -H; #EQ3
1416 generalize in match (refl … (assembly_1_pseudoinstruction … pol ppc lookup_labels lookup_datalabels ? (refl …) (refl …) (refl …)))
1417 cases (assembly_1_pseudoinstruction ?????????) in ⊢ (???% → ?) #len #assembledi #EQ4
1418 #H cases (H ppc len assembledi ?) [2: <EQ4 %] -H; #H1 #H2
1419 generalize in match (fetch_assembly_pseudo … pol ppc (load_code_memory assembled)) in ⊢ ?
1420 >EQ4 #H generalize in match (H H1) in ⊢ ? -H; >(pair_destruct_2 ????? (sym_eq … EQ1))
1421 >H2 #K @K
1422qed.
1423
1424(* OLD?
1425definition assembly_specification:
1426  ∀assembly_program: pseudo_assembly_program.
1427  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1428  λpseudo_assembly_program.
1429  λcode_mem.
1430    ∀pc: Word.
1431      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1432      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1433      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1434      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1435      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1436       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1437      match pre_assembled with
1438       [ None ⇒ True
1439       | Some pc_code ⇒
1440          let 〈new_pc,code〉 ≝ pc_code in
1441           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1442
1443axiom assembly_meets_specification:
1444  ∀pseudo_assembly_program.
1445    match assembly pseudo_assembly_program with
1446    [ None ⇒ True
1447    | Some code_mem_cost ⇒
1448      let 〈code_mem, cost〉 ≝ code_mem_cost in
1449        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1450    ].
1451(*
1452  # PROGRAM
1453  [ cases PROGRAM
1454    # PREAMBLE
1455    # INSTR_LIST
1456    elim INSTR_LIST
1457    [ whd
1458      whd in ⊢ (∀_. %)
1459      # PC
1460      whd
1461    | # INSTR
1462      # INSTR_LIST_TL
1463      # H
1464      whd
1465      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1466    ]
1467  | cases not_implemented
1468  ] *)
1469*)
1470
1471definition internal_pseudo_address_map ≝ list (BitVector 8).
1472
1473axiom low_internal_ram_of_pseudo_low_internal_ram:
1474 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1475
1476axiom high_internal_ram_of_pseudo_high_internal_ram:
1477 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1478
1479axiom low_internal_ram_of_pseudo_internal_ram_hit:
1480 ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀pol:policy (code_memory … s).∀addr:BitVector 7.
1481  member ? (eq_bv 8) (false:::addr) M = true →
1482   let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1483   let pbl ≝ lookup ? 7 addr (low_internal_ram ? s) (zero 8) in
1484   let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram ? s) (zero 8) in
1485   let bl ≝ lookup ? 7 addr ram (zero 8) in
1486   let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in
1487    bu@@bl = sigma (code_memory … s) pol (pbu@@pbl).
1488
1489(* changed from add to sub *)
1490axiom low_internal_ram_of_pseudo_internal_ram_miss:
1491 ∀T.∀M:internal_pseudo_address_map.∀s:PreStatus T.∀addr:BitVector 7.
1492  let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1493  let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in
1494  let carr ≝ get_index_v ? ? flags 1 ? in
1495  carr = false →
1496  member ? (eq_bv 8) (false:::Saddr) M = false →
1497   member ? (eq_bv 8) (false:::addr) M = false →
1498    lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?).
1499  //
1500qed.
1501
1502definition addressing_mode_ok ≝
1503 λT.λM:internal_pseudo_address_map.λs:PreStatus T.
1504  λaddr:addressing_mode.
1505   match addr with
1506    [ DIRECT d ⇒
1507       ¬(member ? (eq_bv 8) d M) ∧
1508       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1509    | INDIRECT i ⇒
1510       let d ≝ get_register ? s [[false;false;i]] in
1511       ¬(member ? (eq_bv 8) d M) ∧
1512       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1513    | EXT_INDIRECT _ ⇒ true
1514    | REGISTER _ ⇒ true
1515    | ACC_A ⇒ true
1516    | ACC_B ⇒ true
1517    | DPTR ⇒ true
1518    | DATA _ ⇒ true
1519    | DATA16 _ ⇒ true
1520    | ACC_DPTR ⇒ true
1521    | ACC_PC ⇒ true
1522    | EXT_INDIRECT_DPTR ⇒ true
1523    | INDIRECT_DPTR ⇒ true
1524    | CARRY ⇒ true
1525    | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1526    | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1527    | RELATIVE _ ⇒ true
1528    | ADDR11 _ ⇒ true
1529    | ADDR16 _ ⇒ true ].
1530   
1531definition next_internal_pseudo_address_map0 ≝
1532  λT.
1533  λfetched.
1534  λM: internal_pseudo_address_map.
1535  λs: PreStatus T.
1536   match fetched with
1537    [ Comment _ ⇒ Some ? M
1538    | Cost _ ⇒ Some … M
1539    | Jmp _ ⇒ Some … M
1540    | Call _ ⇒
1541       Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M)
1542    | Mov _ _ ⇒ Some … M
1543    | Instruction instr ⇒
1544       match instr with
1545        [ ADD addr1 addr2 ⇒
1546           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1547            Some ? M
1548           else
1549            None ?
1550        | ADDC addr1 addr2 ⇒
1551           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1552            Some ? M
1553           else
1554            None ?
1555        | SUBB addr1 addr2 ⇒
1556           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1557            Some ? M
1558           else
1559            None ?       
1560        | _ ⇒ (* TO BE COMPLETED *) Some ? M ]].
1561 
1562
1563definition next_internal_pseudo_address_map ≝
1564 λM:internal_pseudo_address_map.
1565  λs:PseudoStatus.
1566    next_internal_pseudo_address_map0 ?
1567     (\fst (fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s))) M s.
1568   
1569definition status_of_pseudo_status:
1570 internal_pseudo_address_map → ∀ps:PseudoStatus. policy (code_memory … ps) → Status ≝
1571 λM,ps,pol.
1572  let pap ≝ code_memory … ps in
1573  let p ≝ assembly pap pol in
1574  let cm ≝ load_code_memory (\fst p) in
1575  let pc ≝ sigma pap pol (program_counter ? ps) in
1576  let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in
1577  let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in
1578     mk_PreStatus (BitVectorTrie Byte 16)
1579      cm
1580      lir
1581      hir
1582      (external_ram … ps)
1583      pc
1584      (special_function_registers_8051 … ps)
1585      (special_function_registers_8052 … ps)
1586      (p1_latch … ps)
1587      (p3_latch … ps)
1588      (clock … ps).
1589
1590(*
1591definition write_at_stack_pointer':
1592 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1593  λM: Type[0].
1594  λs: PreStatus M.
1595  λv: Byte.
1596    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1597    let bit_zero ≝ get_index_v… nu O ? in
1598    let bit_1 ≝ get_index_v… nu 1 ? in
1599    let bit_2 ≝ get_index_v… nu 2 ? in
1600    let bit_3 ≝ get_index_v… nu 3 ? in
1601      if bit_zero then
1602        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1603                              v (low_internal_ram ? s) in
1604          set_low_internal_ram ? s memory
1605      else
1606        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1607                              v (high_internal_ram ? s) in
1608          set_high_internal_ram ? s memory.
1609  [ cases l0 %
1610  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1611qed.
1612
1613definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1614 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1615
1616  λticks_of.
1617  λs.
1618  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1619  let ticks ≝ ticks_of (program_counter ? s) in
1620  let s ≝ set_clock ? s (clock ? s + ticks) in
1621  let s ≝ set_program_counter ? s pc in
1622    match instr with
1623    [ Instruction instr ⇒
1624       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1625    | Comment cmt ⇒ s
1626    | Cost cst ⇒ s
1627    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1628    | Call call ⇒
1629      let a ≝ address_of_word_labels s call in
1630      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1631      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1632      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1633      let s ≝ write_at_stack_pointer' ? s pc_bl in
1634      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1635      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1636      let s ≝ write_at_stack_pointer' ? s pc_bu in
1637        set_program_counter ? s a
1638    | Mov dptr ident ⇒
1639       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1640    ].
1641 [
1642 |2,3,4: %
1643 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1644 |
1645 | %
1646 ]
1647 cases not_implemented
1648qed.
1649*)
1650
1651axiom execute_1_pseudo_instruction_preserves_code_memory:
1652 ∀ticks_of,ps.
1653  code_memory … (execute_1_pseudo_instruction ticks_of ps) = code_memory … ps.
1654
1655(*
1656lemma execute_code_memory_unchanged:
1657 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1658 #ticks #ps whd in ⊢ (??? (??%))
1659 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1660  (program_counter pseudo_assembly_program ps)) #instr #pc
1661 whd in ⊢ (??? (??%)) cases instr
1662  [ #pre cases pre
1663     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1664       cases (split ????) #z1 #z2 %
1665     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1666       cases (split ????) #z1 #z2 %
1667     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1668       cases (split ????) #z1 #z2 %
1669     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1670       [ #x1 whd in ⊢ (??? (??%))
1671     | *: cases not_implemented
1672     ]
1673  | #comment %
1674  | #cost %
1675  | #label %
1676  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1677    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1678    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1679    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1680    (* CSC: ??? *)
1681  | #dptr #label (* CSC: ??? *)
1682  ]
1683  cases not_implemented
1684qed.
1685*)
1686
1687(* DEAD CODE?
1688lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1689 ∀M:internal_pseudo_address_map.
1690 ∀ps,ps': PseudoStatus.
1691 ∀pol.
1692  ∀prf:code_memory … ps = code_memory … ps'.
1693   let pol' ≝ ? in
1694   match status_of_pseudo_status M ps pol with
1695    [ None ⇒ status_of_pseudo_status M ps' pol' = None …
1696    | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w
1697    ].
1698 [2: <prf @pol]
1699 #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1700 generalize in match (refl … (assembly (code_memory … ps) pol))
1701 cases (assembly ??) in ⊢ (???% → %)
1702  [ #K whd whd in ⊢ (??%?) <H >K %
1703  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1704qed.
1705*)
1706
1707definition ticks_of0: ∀p:pseudo_assembly_program. policy p → Word → ? → nat × nat ≝
1708  λprogram: pseudo_assembly_program.λpol.
1709  λppc: Word.
1710  λfetched.
1711    match fetched with
1712    [ Instruction instr ⇒
1713      match instr with
1714      [ JC lbl ⇒
1715        match pol ppc with
1716        [ short_jump ⇒ 〈2, 2〉
1717        | medium_jump ⇒ ?
1718        | long_jump ⇒ 〈4, 4〉
1719        ]
1720      | JNC lbl ⇒
1721        match pol ppc with
1722        [ short_jump ⇒ 〈2, 2〉
1723        | medium_jump ⇒ ?
1724        | long_jump ⇒ 〈4, 4〉
1725        ]
1726      | JB bit lbl ⇒
1727        match pol ppc with
1728        [ short_jump ⇒ 〈2, 2〉
1729        | medium_jump ⇒ ?
1730        | long_jump ⇒ 〈4, 4〉
1731        ]
1732      | JNB bit lbl ⇒
1733        match pol ppc with
1734        [ short_jump ⇒ 〈2, 2〉
1735        | medium_jump ⇒ ?
1736        | long_jump ⇒ 〈4, 4〉
1737        ]
1738      | JBC bit lbl ⇒
1739        match pol ppc with
1740        [ short_jump ⇒ 〈2, 2〉
1741        | medium_jump ⇒ ?
1742        | long_jump ⇒ 〈4, 4〉
1743        ]
1744      | JZ lbl ⇒
1745        match pol ppc with
1746        [ short_jump ⇒ 〈2, 2〉
1747        | medium_jump ⇒ ?
1748        | long_jump ⇒ 〈4, 4〉
1749        ]
1750      | JNZ lbl ⇒
1751        match pol ppc with
1752        [ short_jump ⇒ 〈2, 2〉
1753        | medium_jump ⇒ ?
1754        | long_jump ⇒ 〈4, 4〉
1755        ]
1756      | CJNE arg lbl ⇒
1757        match pol ppc with
1758        [ short_jump ⇒ 〈2, 2〉
1759        | medium_jump ⇒ ?
1760        | long_jump ⇒ 〈4, 4〉
1761        ]
1762      | DJNZ arg lbl ⇒
1763        match pol ppc with
1764        [ short_jump ⇒ 〈2, 2〉
1765        | medium_jump ⇒ ?
1766        | long_jump ⇒ 〈4, 4〉
1767        ]
1768      | ADD arg1 arg2 ⇒
1769        let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in
1770         〈ticks, ticks〉
1771      | ADDC arg1 arg2 ⇒
1772        let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in
1773         〈ticks, ticks〉
1774      | SUBB arg1 arg2 ⇒
1775        let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in
1776         〈ticks, ticks〉
1777      | INC arg ⇒
1778        let ticks ≝ ticks_of_instruction (INC ? arg) in
1779         〈ticks, ticks〉
1780      | DEC arg ⇒
1781        let ticks ≝ ticks_of_instruction (DEC ? arg) in
1782         〈ticks, ticks〉
1783      | MUL arg1 arg2 ⇒
1784        let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in
1785         〈ticks, ticks〉
1786      | DIV arg1 arg2 ⇒
1787        let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in
1788         〈ticks, ticks〉
1789      | DA arg ⇒
1790        let ticks ≝ ticks_of_instruction (DA ? arg) in
1791         〈ticks, ticks〉
1792      | ANL arg ⇒
1793        let ticks ≝ ticks_of_instruction (ANL ? arg) in
1794         〈ticks, ticks〉
1795      | ORL arg ⇒
1796        let ticks ≝ ticks_of_instruction (ORL ? arg) in
1797         〈ticks, ticks〉
1798      | XRL arg ⇒
1799        let ticks ≝ ticks_of_instruction (XRL ? arg) in
1800         〈ticks, ticks〉
1801      | CLR arg ⇒
1802        let ticks ≝ ticks_of_instruction (CLR ? arg) in
1803         〈ticks, ticks〉
1804      | CPL arg ⇒
1805        let ticks ≝ ticks_of_instruction (CPL ? arg) in
1806         〈ticks, ticks〉
1807      | RL arg ⇒
1808        let ticks ≝ ticks_of_instruction (RL ? arg) in
1809         〈ticks, ticks〉
1810      | RLC arg ⇒
1811        let ticks ≝ ticks_of_instruction (RLC ? arg) in
1812         〈ticks, ticks〉
1813      | RR arg ⇒
1814        let ticks ≝ ticks_of_instruction (RR ? arg) in
1815         〈ticks, ticks〉
1816      | RRC arg ⇒
1817        let ticks ≝ ticks_of_instruction (RRC ? arg) in
1818         〈ticks, ticks〉
1819      | SWAP arg ⇒
1820        let ticks ≝ ticks_of_instruction (SWAP ? arg) in
1821         〈ticks, ticks〉
1822      | MOV arg ⇒
1823        let ticks ≝ ticks_of_instruction (MOV ? arg) in
1824         〈ticks, ticks〉
1825      | MOVX arg ⇒
1826        let ticks ≝ ticks_of_instruction (MOVX ? arg) in
1827         〈ticks, ticks〉
1828      | SETB arg ⇒
1829        let ticks ≝ ticks_of_instruction (SETB ? arg) in
1830         〈ticks, ticks〉
1831      | PUSH arg ⇒
1832        let ticks ≝ ticks_of_instruction (PUSH ? arg) in
1833         〈ticks, ticks〉
1834      | POP arg ⇒
1835        let ticks ≝ ticks_of_instruction (POP ? arg) in
1836         〈ticks, ticks〉
1837      | XCH arg1 arg2 ⇒
1838        let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in
1839         〈ticks, ticks〉
1840      | XCHD arg1 arg2 ⇒
1841        let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in
1842         〈ticks, ticks〉
1843      | RET ⇒
1844        let ticks ≝ ticks_of_instruction (RET ?) in
1845         〈ticks, ticks〉
1846      | RETI ⇒
1847        let ticks ≝ ticks_of_instruction (RETI ?) in
1848         〈ticks, ticks〉
1849      | NOP ⇒
1850        let ticks ≝ ticks_of_instruction (NOP ?) in
1851         〈ticks, ticks〉
1852      ]
1853    | Comment comment ⇒ 〈0, 0〉
1854    | Cost cost ⇒ 〈0, 0〉
1855    | Jmp jmp ⇒ 〈2, 2〉
1856    | Call call ⇒ 〈2, 2〉
1857    | Mov dptr tgt ⇒ 〈2, 2〉
1858    ].
1859  cases not_implemented (* policy returned medium_jump for conditional jumping = impossible *)
1860qed.
1861
1862definition ticks_of: ∀p:pseudo_assembly_program. policy p → Word → nat × nat ≝
1863  λprogram: pseudo_assembly_program.λpol.
1864  λppc: Word.
1865    let 〈preamble, pseudo〉 ≝ program in
1866    let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in
1867     ticks_of0 program pol ppc fetched.
1868
1869lemma eq_rect_Type1_r:
1870  ∀A: Type[1].
1871  ∀a:A.
1872  ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p.
1873  #A #a #P #H #x #p
1874  generalize in match H
1875  generalize in match P
1876  cases p
1877  //
1878qed.
1879
1880axiom split_append:
1881  ∀A: Type[0].
1882  ∀m, n: nat.
1883  ∀v, v': Vector A m.
1884  ∀q, q': Vector A n.
1885    let 〈v', q'〉 ≝ split A m n (v@@q) in
1886      v = v' ∧ q = q'.
1887
1888axiom split_vector_singleton:
1889  ∀A: Type[0].
1890  ∀n: nat.
1891  ∀v: Vector A (S n).
1892  ∀rest: Vector A n.
1893  ∀s: Vector A 1.
1894  ∀prf.
1895    v = s @@ rest →
1896    ((get_index_v A ? v 0 prf) ::: rest) = v.
1897
1898example sub_minus_one_seven_eight:
1899  ∀v: BitVector 7.
1900  false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) =
1901  \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false).
1902 cases daemon.
1903qed.
1904
1905(*
1906lemma blah:
1907  ∀m: internal_pseudo_address_map.
1908  ∀s: PseudoStatus.
1909  ∀arg: Byte.
1910  ∀b: bool.
1911    addressing_mode_ok m s (DIRECT arg) = true →
1912      get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) =
1913      get_arg_8 ? s b (DIRECT arg).
1914  [2, 3: normalize % ]
1915  #m #s #arg #b #hyp
1916  whd in ⊢ (??%%)
1917  @split_elim''
1918  #nu' #nl' #arg_nu_nl_eq
1919  normalize nodelta
1920  generalize in match (refl ? (get_index_v bool 4 nu' ? ?))
1921  cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %)
1922  #get_index_v_eq
1923  normalize nodelta
1924  [2:
1925    normalize nodelta
1926    @split_elim''
1927    #bit_one' #three_bits' #bit_one_three_bit_eq
1928    generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl'))
1929    normalize nodelta
1930    generalize in match (refl ? (sub_7_with_carry ? ? ?))
1931    cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %)
1932    #Saddr #carr' #Saddr_carr_eq
1933    normalize nodelta
1934    #carr_hyp'
1935    @carr_hyp'
1936    [1:
1937    |2: whd in hyp:(??%?); generalize in match hyp; -hyp;
1938        generalize in match (refl ? (¬(member (BitVector 8) ? arg m)))
1939        cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %)
1940        #member_eq
1941        normalize nodelta
1942        [2: #destr destruct(destr)
1943        |1: -carr_hyp';
1944            >arg_nu_nl_eq
1945            <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq)
1946            [1: >get_index_v_eq in ⊢ (??%? → ?)
1947            |2: @le_S @le_S @le_S @le_n
1948            ]
1949            cases (member (BitVector 8) ? (\fst ?) ?)
1950            [1: #destr normalize in destr; destruct(destr)
1951            |2:
1952            ]
1953        ]
1954    |3: >get_index_v_eq in ⊢ (??%?)
1955        change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl')
1956        >(split_vector_singleton … bit_one_three_bit_eq)
1957        <arg_nu_nl_eq
1958        whd in hyp:(??%?)
1959        cases (member (BitVector 8) (eq_bv 8) arg m) in hyp
1960        normalize nodelta [*: #ignore @sym_eq ]
1961    ]
1962  |
1963  ].
1964*)
1965(*
1966map_address0 ... (DIRECT arg) = Some .. →
1967  get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) =
1968  get_arg_8 (internal_ram ...) (DIRECT arg)
1969
1970((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) 
1971                     then Some internal_pseudo_address_map M 
1972                     else None internal_pseudo_address_map )
1973                    =Some internal_pseudo_address_map M')
1974*)
1975
1976axiom low_internal_ram_write_at_stack_pointer:
1977 ∀T1,T2,M,s1,s2,s3.∀pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
1978  get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP →
1979  low_internal_ram ? s2 = low_internal_ram T2 s3 →
1980  sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) →
1981  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
1982  bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) →
1983   low_internal_ram T1
1984     (write_at_stack_pointer ?
1985       (set_8051_sfr ?
1986         (write_at_stack_pointer ?
1987           (set_8051_sfr ?
1988             (set_low_internal_ram ? s1
1989               (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram ? s2)))
1990             SFR_SP sp1)
1991          bl)
1992        SFR_SP sp2)
1993      bu)
1994   = low_internal_ram_of_pseudo_low_internal_ram (sp1::M)
1995      (low_internal_ram ?
1996       (write_at_stack_pointer ?
1997         (set_8051_sfr ?
1998           (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl)
1999          SFR_SP sp2)
2000        pbu)).
2001
2002axiom high_internal_ram_write_at_stack_pointer:
2003 ∀T1,T2,M,s1,s2,s3,pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2004  get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP →
2005  high_internal_ram ? s2 = high_internal_ram T2 s3 →
2006  sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) →
2007  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
2008  bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) →
2009   high_internal_ram T1
2010     (write_at_stack_pointer ?
2011       (set_8051_sfr ?
2012         (write_at_stack_pointer ?
2013           (set_8051_sfr ?
2014             (set_high_internal_ram ? s1
2015               (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram ? s2)))
2016             SFR_SP sp1)
2017          bl)
2018        SFR_SP sp2)
2019      bu)
2020   = high_internal_ram_of_pseudo_high_internal_ram (sp1::M)
2021      (high_internal_ram ?
2022       (write_at_stack_pointer ?
2023         (set_8051_sfr ?
2024           (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl)
2025          SFR_SP sp2)
2026        pbu)).
2027
2028lemma Some_Some_elim:
2029 ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P.
2030 #T #x #y #P #H #K @H @option_destruct_Some //
2031qed.
2032
2033theorem main_thm:
2034 ∀M,M',ps,pol.
2035  next_internal_pseudo_address_map M ps = Some … M' →
2036   ∃n.
2037      execute n (status_of_pseudo_status M ps pol)
2038    = status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps) ?.
2039 [2: >execute_1_pseudo_instruction_preserves_code_memory @pol]
2040 #M #M' #ps #pol #SAFE
2041 cut
2042  (∀ps'.
2043    ∀prf:ps'=execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps.
2044    ∃n. execute n (status_of_pseudo_status M ps pol) = status_of_pseudo_status M' ps' ?)
2045 [ >prf >execute_1_pseudo_instruction_preserves_code_memory @pol |3: #K @(K ? (refl …))]
2046 #ps' #EQ
2047 whd in ⊢ (??(λ_.??(??%)?))
2048 change with
2049  (∃n.
2050    execute n
2051     (set_low_internal_ram ?
2052       (set_high_internal_ram ?
2053         (set_program_counter ?
2054           (set_code_memory ?? ps (load_code_memory ?))
2055          (sigma ? pol (program_counter ? ps)))
2056        (high_internal_ram_of_pseudo_high_internal_ram M ?))
2057      (low_internal_ram_of_pseudo_low_internal_ram M ?))
2058   = set_low_internal_ram ?
2059      (set_high_internal_ram ?
2060        (set_program_counter ?
2061          (set_code_memory ?? ? (load_code_memory ?))
2062         (sigma ???)) ?) ?)
2063 >EQ whd in match eq_rect_Type0_r normalize nodelta
2064 >execute_1_pseudo_instruction_preserves_code_memory normalize nodelta
2065 generalize in match EQ -EQ;
2066 generalize in match (refl … (code_memory pseudo_assembly_program ps))
2067 generalize in match pol -pol; generalize in ⊢ (∀_.??%? → ?)
2068 * #preamble #instr_list #pol #EQ1 generalize in match pol -pol <EQ1 #pol #EQps' <EQps'
2069 (* Dependent types madness ends here *)
2070 letin ppc ≝ (program_counter … ps)
2071 generalize in match (fetch_assembly_pseudo2 ? pol ppc) in ⊢ ?
2072 letin assembled ≝ (\fst (assembly ? pol))
2073 letin lookup_labels ≝ (λx. sigma ? pol (address_of_word_labels_code_mem instr_list x))
2074 letin lookup_datalabels ≝ (λx. lookup … x (construct_datalabels preamble) (zero 16))
2075 @pair_elim' #pi #newppc #EQ2
2076 letin instructions ≝
2077  (expand_pseudo_instruction ? pol ppc lookup_labels lookup_datalabels
2078    (sigma ? pol ppc) (refl …) (refl …) (refl …))
2079 change with (fetch_many ???? → ?) #H1
2080 change in EQ2 with (fetch_pseudo_instruction instr_list ppc = ?)
2081 change in SAFE with (next_internal_pseudo_address_map0 ???? = ?) <EQ1 in SAFE;
2082 >EQ2 whd in ⊢ (??(??%??)? → ?) #SAFE
2083 whd in EQps':(???%); <EQ1 in EQps'; >EQ2 normalize nodelta
2084 generalize in match H1; -H1; generalize in match instructions -instructions
2085 * #instructions >EQ2 change in match (\fst 〈pi,newppc〉) with pi
2086 whd in match ticks_of normalize nodelta <EQ1 >EQ2
2087 cases pi in SAFE
2088  [2,3: (* Comment, Cost *) #ARG whd in ⊢ (??%? → ???% → ?)
2089   @Some_Some_elim #MAP #EQ3 @(Some_Some_elim ????? EQ3) #EQ3'
2090   whd in match eject normalize nodelta >EQ3' in ⊢ (% → ?) whd in ⊢ (% → ?)
2091   #H2 #EQ %[1,3:@0]
2092   <MAP >(eq_bv_eq … H2) >EQ %
2093  |4: (* Jmp *) #label whd in ⊢ (??%? → ???% → ?)
2094   @Some_Some_elim #MAP cases (pol ?) normalize nodelta
2095       [3: (* long *) #EQ3 @(Some_Some_elim ????? EQ3) #EQ3'
2096         whd in match eject normalize nodelta >EQ3' in ⊢ (% → ?) whd in ⊢ (% → ?)
2097         @pair_elim' * #instr #newppc' #ticks #EQ4       
2098         * * #H2a #H2b whd in ⊢ (% → ?) #H2
2099         >H2b >(eq_instruction_to_eq … H2a)
2100         #EQ %[@1]
2101         <MAP >(eq_bv_eq … H2) >EQ
2102         whd in ⊢ (??%?) >EQ4 whd in ⊢ (??%?)
2103         cases ps in EQ4; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXX >XXX %
2104         whd in ⊢ (??%?)
2105         whd in ⊢ (??(match ?%? with [_ ⇒ ?])?)
2106         cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
2107  |6: (* Mov *) #arg1 #arg2
2108       #H1 #H2 #EQ %[@1]
2109       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2110       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2111       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2112       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2113       >H2b >(eq_instruction_to_eq … H2a)
2114       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2115       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?)
2116       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2117       normalize nodelta;
2118       [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2119       #result #flags
2120       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
2121  |5: (* Call *) #label #MAP
2122      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP;
2123      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2124       [ (* short *) #abs @⊥ destruct (abs)
2125       |3: (* long *) #H1 #H2 #EQ %[@1]
2126           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2127           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2128           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2129           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2130           >H2b >(eq_instruction_to_eq … H2a)
2131           generalize in match EQ; -EQ;
2132           whd in ⊢ (???% → ??%?);
2133           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;
2134           >(eq_bv_eq … H2c)
2135           change with
2136            ((?=let 〈ppc_bu,ppc_bl〉 ≝ split bool 8 8 newppc in ?) →
2137                (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2138           generalize in match (refl … (split … 8 8 newppc)) cases (split bool 8 8 newppc) in ⊢ (??%? → %) #ppc_bu #ppc_bl #EQppc
2139           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;
2140           >get_8051_sfr_write_at_stack_pointer >get_8051_sfr_write_at_stack_pointer
2141           >get_8051_sfr_set_8051_sfr >get_8051_sfr_set_8051_sfr
2142           generalize in match (refl … (half_add ? new_sp (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry' #new_sp' #EQ2 normalize nodelta;
2143           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2144           @split_eq_status;
2145            [ >code_memory_write_at_stack_pointer whd in ⊢ (??%?)
2146              >code_memory_write_at_stack_pointer %
2147            | >set_program_counter_set_low_internal_ram
2148              >set_clock_set_low_internal_ram
2149              @low_internal_ram_write_at_stack_pointer
2150               [ >EQ0 @pol | % | %
2151               | @(pair_destruct_2 … EQ1)
2152               | @(pair_destruct_2 … EQ2)
2153               | >(pair_destruct_1 ????? EQpc)
2154                 >(pair_destruct_2 ????? EQpc)
2155                 @split_elim #x #y #H <H -x y H;
2156                 >(pair_destruct_1 ????? EQppc)
2157                 >(pair_destruct_2 ????? EQppc)
2158                 @split_elim #x #y #H <H -x y H;
2159                 >EQ0 % ]
2160            | >set_low_internal_ram_set_high_internal_ram
2161              >set_program_counter_set_high_internal_ram
2162              >set_clock_set_high_internal_ram
2163              @high_internal_ram_write_at_stack_pointer
2164               [ >EQ0 @pol | % | %
2165               | @(pair_destruct_2 … EQ1)
2166               | @(pair_destruct_2 … EQ2)
2167               | >(pair_destruct_1 ????? EQpc)
2168                 >(pair_destruct_2 ????? EQpc)
2169                 @split_elim #x #y #H <H -x y H;
2170                 >(pair_destruct_1 ????? EQppc)
2171                 >(pair_destruct_2 ????? EQppc)
2172                 @split_elim #x #y #H <H -x y H;
2173                 >EQ0 % ]           
2174            | >external_ram_write_at_stack_pointer whd in ⊢ (??%?)
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 %
2178            | change with (? = sigma ?? (address_of_word_labels_code_mem (\snd (code_memory ? ps)) ?))
2179              >EQ0 %
2180            | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (??%?)
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 %
2184            | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (??%?)
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 %
2188            | >p1_latch_write_at_stack_pointer whd in ⊢ (??%?)
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 %
2192            | >p3_latch_write_at_stack_pointer whd in ⊢ (??%?)
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 %
2196            | >clock_write_at_stack_pointer whd in ⊢ (??%?)
2197              >clock_write_at_stack_pointer whd in ⊢ (???%)
2198              >clock_write_at_stack_pointer whd in ⊢ (???%)
2199              >clock_write_at_stack_pointer %]
2200       (*| (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2201         @pair_elim' #fst_5_addr #rest_addr #EQ1
2202         @pair_elim' #fst_5_pc #rest_pc #EQ2
2203         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2204         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2205         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2206         change in ⊢ (? →??%?) with (execute_1_0 ??)
2207         @pair_elim' * #instr #newppc' #ticks #EQn
2208          * * #H2a #H2b whd in ⊢ (% → ?) #H2c >H2b >(eq_instruction_to_eq … H2a) whd in ⊢ (??%?)
2209          generalize in match EQ; -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2210          @pair_elim' #carry #new_sp change with (half_add ? (get_8051_sfr ? ps ?) ? = ? → ?) #EQ4
2211          @split_elim' #pc_bu #pc_bl >program_counter_set_8051_sfr XXX change with (newppc = ?) #EQ5
2212          @pair_elim' #carry' #new_sp' #EQ6 normalize nodelta; #EQx >EQx -EQx;
2213          change in ⊢ (??(match ????% with [_ ⇒ ?])?) with (sigma … newppc)
2214          @split_elim' #pc_bu' #pc_bl' #EQ7 change with (newppc' = ? → ?)
2215          >get_8051_sfr_set_8051_sfr
2216         
2217          whd in EQ:(???%) ⊢ ? >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2218           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?)
2219           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc)))
2220           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2221           generalize in match (refl … (split bool 4 4 pc_bu))
2222           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2223           generalize in match (refl … (split bool 3 8 rest_addr))
2224           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2225           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2226           generalize in match
2227            (refl …
2228             (half_add 16 (sigma 〈preamble,instr_list〉 newppc)
2229             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2230           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7
2231           @split_eq_status try %
2232            [ change with (? = sigma ? (address_of_word_labels ps label))
2233              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2234            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2235              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] *)]
2236  |4: (* Jmp *) #label #MAP
2237      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP >MAP -MAP;
2238      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2239       [3: (* long *) #H1 #H2 #EQ %[@1]
2240           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2241           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2242           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2243           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2244           >H2b >(eq_instruction_to_eq … H2a)
2245           generalize in match EQ; -EQ;
2246           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2247           cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
2248       |1: (* short *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2249           generalize in match
2250            (refl ?
2251             (sub_16_with_carry
2252              (sigma 〈preamble,instr_list〉 pol (program_counter … ps))
2253              (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))
2254              false))
2255           cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta;
2256           generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta;
2257           generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta;
2258           #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)]
2259           generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2260           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2261           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2262           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2263           >H2b >(eq_instruction_to_eq … H2a)
2264           generalize in match EQ; -EQ;
2265           whd in ⊢ (???% → ?);
2266           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2267           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ???) ? in ?) = ?)
2268           generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) (sign_extension lower)))
2269           cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4
2270           @split_eq_status try % change with (newpc = sigma ?? (address_of_word_labels ps label))
2271           (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2272       | (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2273         generalize in match
2274          (refl …
2275            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))))
2276         cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1
2277         generalize in match
2278          (refl …
2279            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (program_counter … ps))))
2280         cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2
2281         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2282         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2283         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2284         change in ⊢ (? →??%?) with (execute_1_0 ??)
2285           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2286           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2287           >H2b >(eq_instruction_to_eq … H2a)
2288           generalize in match EQ; -EQ;
2289           whd in ⊢ (???% → ?);
2290           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2291           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2292           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)))
2293           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2294           generalize in match (refl … (split bool 4 4 pc_bu))
2295           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2296           generalize in match (refl … (split bool 3 8 rest_addr))
2297           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2298           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2299           generalize in match
2300            (refl …
2301             (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc)
2302             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2303           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7   
2304           @split_eq_status try %
2305            [ change with (? = sigma ?? (address_of_word_labels ps label))
2306              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2307            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2308              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]]
2309  | (* Instruction *) -pi;  whd in ⊢ (? → ??%? → ?) *; normalize nodelta;
2310    [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1]
2311       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2312       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2313       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2314       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2315       >H2b >(eq_instruction_to_eq … H2a)
2316       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP;
2317       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1;
2318       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2319       normalize nodelta; #MAP;
2320       [1: change in ⊢ (? → %) with
2321        ((let 〈result,flags〉 ≝
2322          add_8_with_carry
2323           (get_arg_8 ? ps false ACC_A)
2324           (get_arg_8 ?
2325             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2326             false (DIRECT ARG2))
2327           ? in ?) = ?)
2328        [2,3: %]
2329        change in ⊢ (???% → ?) with
2330         (let 〈result,flags〉 ≝ add_8_with_carry ?(*(get_arg_8 ? ps false ACC_A)*) ?? in ?)
2331        >get_arg_8_set_clock
2332       [1,2: cases (addressing_mode_ok ???? ∧ addressing_mode_ok ????) in MAP ⊢ ?
2333         [2,4: #abs @⊥ normalize in abs; destruct (abs)
2334         |*:whd in ⊢ (??%? → ?) #H <(option_destruct_Some ??? H)]
2335       [ change in ⊢ (? → %) with
2336        ((let 〈result,flags〉 ≝
2337          add_8_with_carry
2338           (get_arg_8 ? ps false ACC_A)
2339           (get_arg_8 ?
2340             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2341             false (DIRECT ARG2))
2342           ? in ?) = ?)
2343          >get_arg_8_set_low_internal_ram
2344       
2345        cases (add_8_with_carry ???)
2346         
2347        [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2348       #result #flags
2349       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
2350    | (* INC *) #arg1 #H1 #H2 #EQ %[@1]
2351       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2352       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2353       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2354       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2355       >H2b >(eq_instruction_to_eq … H2a)
2356       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2357       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG]
2358       [1,2,3,4: cases (half_add ???) #carry #result
2359       | cases (half_add ???) #carry #bl normalize nodelta;
2360         cases (full_add ????) #carry' #bu normalize nodelta ]
2361        #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) -newppc';
2362        [5: %
2363        |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program
2364      (set_program_counter pseudo_assembly_program ps newppc)
2365      (\fst  (ticks_of0 〈preamble,instr_list〉
2366                   (program_counter pseudo_assembly_program ps)
2367                   (Instruction (INC Identifier (DIRECT ARG))))
2368       +clock pseudo_assembly_program
2369        (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG))
2370        [2,3: // ]
2371            <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://]
2372            whd in ⊢ (??%%)
2373            cases (split bool 4 4 ARG)
2374            #nu' #nl'
2375            normalize nodelta
2376            cases (split bool 1 3 nu')
2377            #bit_1' #ignore'
2378            normalize nodelta
2379            cases (get_index_v bool 4 nu' ? ?)
2380            [ normalize nodelta (* HERE *) whd in ⊢ (??%%) %
2381            |
2382            ]
2383cases daemon (* EASY CASES TO BE COMPLETED *)
2384qed.
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