source: src/ASM/AssemblyProof.ma @ 993

Last change on this file since 993 was 993, checked in by sacerdot, 9 years ago

More Russell everywhere; getting closer to the goal.

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1include "ASM/Assembly.ma".
2include "ASM/Interpret.ma".
3
4definition bit_elim_prop: ∀P: bool → Prop. Prop ≝
5  λP.
6    P true ∧ P false.
7 
8let rec bitvector_elim_prop_internal
9  (n: nat) (P: BitVector n → Prop) (m: nat) on m: m ≤ n → BitVector (n - m) → Prop ≝
10  match m return λm. m ≤ n → BitVector (n - m) → Prop with
11  [ O    ⇒ λprf1. λprefix. P ?
12  | S n' ⇒ λprf2. λprefix. bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' ? ?)
13  ].
14  [ applyS prefix
15  | letin res ≝ (bit ::: prefix)
16    < (minus_S_S ? ?)
17    > (minus_Sn_m ? ?)
18  [ @ res
19  | @ prf2
20  ]
21  | /2/
22  ].
23qed.
24
25definition bitvector_elim_prop ≝
26  λn: nat.
27  λP: BitVector n → Prop.
28    bitvector_elim_prop_internal n P n ? ?.
29  [ @ (le_n ?)
30  | < (minus_n_n ?)
31    @ [[ ]]
32  ]
33qed.
34
35lemma bool_eq_internal_eq:
36  ∀b, c.
37    (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c.
38  #b #c
39  cases b
40  [ normalize //
41  | normalize
42    cases c
43    [ normalize //
44    | normalize //
45    ]
46  ]
47qed.
48
49definition bit_elim: ∀P: bool → bool. bool ≝
50  λP.
51    P true ∧ P false.
52
53let rec bitvector_elim_internal
54  (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝
55  match m return λm. m ≤ n → BitVector (n - m) → bool with
56  [ O    ⇒ λprf1. λprefix. P ?
57  | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?)
58  ].
59  [ applyS prefix
60  | letin res ≝ (bit ::: prefix)
61    < (minus_S_S ? ?)
62    > (minus_Sn_m ? ?)
63    [ @ res
64    | @ prf2
65    ]
66  | /2/
67  ].
68qed.
69
70definition bitvector_elim ≝
71  λn: nat.
72  λP: BitVector n → bool.
73    bitvector_elim_internal n P n ? ?.
74  [ @ (le_n ?)
75  | < (minus_n_n ?)
76    @ [[ ]]
77  ]
78qed.
79
80lemma super_rewrite2:
81 ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m.
82  ∀P: ∀m. Vector A m → Prop.
83   n=m → v1 ≃ v2 → P n v1 → P m v2.
84 #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ //
85qed.
86
87lemma vector_cons_append:
88  ∀A: Type[0].
89  ∀n: nat.
90  ∀e: A.
91  ∀v: Vector A n.
92    e ::: v = [[ e ]] @@ v.
93  # A # N # E # V
94  elim V
95  [ normalize %
96  | # NN # AA # VV # IH
97    normalize
98    %
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 | @jmrefl ]
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
1070axiom split_elim:
1071 ∀A,l,m,v.∀P: (Vector A l) × (Vector A m) → Prop.
1072  (∀vl,vm. v = vl@@vm → P 〈vl,vm〉) → P (split A l m v).
1073
1074example half_add_SO:
1075 ∀pc.
1076 \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc).
1077 cases daemon.
1078qed.
1079
1080(*
1081axiom not_eqvb_S:
1082 ∀pc.
1083 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))).
1084
1085axiom not_eqvb_SS:
1086 ∀pc.
1087 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))).
1088 
1089axiom bitvector_elim_complete:
1090 ∀n,P. bitvector_elim n P = true → ∀bv. P bv.
1091
1092lemma bitvector_elim_complete':
1093 ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true.
1094 #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv
1095 generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ //
1096qed.
1097*)
1098
1099(*
1100lemma andb_elim':
1101 ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true.
1102 #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize //
1103qed.
1104*)
1105
1106let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word)
1107                       (encoding: list Byte) on encoding: Prop ≝
1108  match encoding with
1109  [ nil ⇒ final_pc = pc
1110  | cons hd tl ⇒
1111    let 〈new_pc, byte〉 ≝ next code_memory pc in
1112      hd = byte ∧ encoding_check code_memory new_pc final_pc tl
1113  ].
1114
1115lemma encoding_check_append: ∀code_memory,final_pc,l1,pc,l2.
1116 encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … final_pc) (l1@l2) →
1117  let intermediate_pc ≝ pc + length … l1 in
1118   encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … intermediate_pc) l1 ∧
1119    encoding_check code_memory (bitvector_of_nat … intermediate_pc) (bitvector_of_nat … final_pc) l2.
1120 #code_memory #final_pc #l1 elim l1
1121  [ #pc #l2 whd in ⊢ (????% → ?) #H <plus_n_O whd whd in ⊢ (?%?) /2/
1122  | #hd #tl #IH #pc #l2 * #H1 #H2 >half_add_SO in H2; #H2 cases (IH … H2) <plus_n_Sm
1123    #K1 #K2 % [2:@K2] whd % // >half_add_SO @K1 ]
1124qed.
1125
1126axiom bitvector_3_elim_prop:
1127 ∀P: BitVector 3 → Prop.
1128  P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] →
1129  P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] →
1130  P [[true;true;false]] → P [[true;true;true]] → ∀v. P v.
1131
1132definition ticks_of_instruction ≝
1133 λi.
1134  let trivial_code_memory ≝ assembly1 i in
1135  let trivial_status ≝ load_code_memory trivial_code_memory in
1136   \snd (fetch trivial_status (zero ?)).
1137
1138axiom fetch_assembly:
1139  ∀pc,i,code_memory,assembled.
1140    assembled = assembly1 i →
1141      let len ≝ length … assembled in
1142      encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
1143      let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in
1144      let 〈instr_pc, ticks〉 ≝ fetched in
1145      let 〈instr,pc'〉 ≝ instr_pc in
1146       (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' (bitvector_of_nat … (pc + len))) = true.
1147(* #pc #i #code_memory #assembled cases i [8: *]
1148 [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]]
1149 [47,48,49:
1150 |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg
1151  [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,
1152   59,60,63,64,65,66,67: #ARG]]
1153 [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,
1154  56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2
1155  [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,
1156   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,
1157   68,69,70,71: #ARG2]]
1158 [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3]
1159 normalize in ⊢ (???% → ?)
1160 [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)]
1161  normalize in ⊢ (???% → ?)]
1162 #H >H * #H1 try (change in ⊢ (% → ?) with (? ∧ ?) * #H2)
1163 try (change in ⊢ (% → ?) with (? ∧ ?) * #H3) whd in ⊢ (% → ?) #H4
1164 change in ⊢ (let fetched ≝ % in ?) with (fetch0 ??)
1165 whd in ⊢ (let fetched ≝ ??% in ?) <H1 whd in ⊢ (let fetched ≝ % in ?)
1166 [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3]
1167 [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,
1168  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,
1169  69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2]
1170 whd >eq_instruction_refl >H4 @eq_bv_refl
1171qed. *)
1172
1173let rec fetch_many code_memory final_pc pc expected on expected: Prop ≝
1174 match expected with
1175  [ nil ⇒ eq_bv … pc final_pc = true
1176  | cons i tl ⇒
1177     let fetched ≝ fetch code_memory pc in
1178     let 〈instr_pc, ticks〉 ≝ fetched in
1179     let 〈instr,pc'〉 ≝ instr_pc in
1180      eq_instruction instr i = true ∧
1181      ticks = (ticks_of_instruction i) ∧
1182      fetch_many code_memory final_pc pc' tl].
1183
1184lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b.
1185 #A #a #b #EQ destruct //
1186qed.
1187
1188axiom eq_instruction_to_eq: ∀i1,i2. eq_instruction i1 i2 = true → i1 = i2.
1189
1190lemma fetch_assembly_pseudo:
1191 ∀program.∀pol:policy program.∀ppc,lookup_labels,lookup_datalabels.
1192  ∀pi,code_memory,len,assembled,instructions,pc.
1193   let expansion ≝ pol ppc in
1194   Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (bitvector_of_nat ? pc) expansion pi →
1195    Some … 〈len,assembled〉 = assembly_1_pseudoinstruction program pol ppc (bitvector_of_nat ? pc) lookup_labels lookup_datalabels pi →
1196     encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
1197      fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions.
1198 #program #pol #ppc #lookup_labels #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc
1199 #EQ1 whd in ⊢ (???% → ?) <EQ1 whd in ⊢ (???% → ?) #EQ2
1200 cases (pair_destruct ?????? (option_destruct_Some … EQ2)) -EQ2; #EQ2a #EQ2b
1201 >EQ2a >EQ2b -EQ2a EQ2b;
1202  generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc
1203  generalize in match pc elim instructions
1204  [ #pc whd in ⊢ (% → %) #H >H @eq_bv_refl
1205  | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd
1206    generalize in match (fetch_assembly pc i code_memory … (refl …) H1)
1207    cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %)
1208    cases newi_pc #newi #newpc whd in ⊢ (% → %) #K cases (conjunction_true … K) -K; #K1
1209    cases (conjunction_true … K1) -K1; #K1 #K2 #K3 % try %
1210    [ @K1 | @eqb_true_to_eq <(eq_instruction_to_eq … K1) @K2 | >(eq_bv_eq … K3) @IH @H2 ]
1211qed.
1212
1213(*
1214lemma rev_preserves_length:
1215 ∀A.∀l. length … (rev A l) = length … l.
1216  #A #l elim l
1217   [ %
1218   | #hd #tl #IH normalize >length_append normalize /2/ ]
1219qed.
1220
1221lemma rev_append:
1222 ∀A.∀l1,l2.
1223  rev A (l1@l2) = rev A l2 @ rev A l1.
1224 #A #l1 elim l1 normalize //
1225qed.
1226 
1227lemma rev_rev: ∀A.∀l. rev … (rev A l) = l.
1228 #A #l elim l
1229  [ //
1230  | #hd #tl #IH normalize >rev_append normalize // ]
1231qed.
1232
1233lemma split_len_Sn:
1234 ∀A:Type[0].∀l:list A.∀len.
1235  length … l = S len →
1236   Σl'.Σa. l = l'@[a] ∧ length … l' = len.
1237 #A #l elim l
1238  [ normalize #len #abs destruct
1239  | #hd #tl #IH #len
1240    generalize in match (rev_rev … tl)
1241    cases (rev A tl) in ⊢ (??%? → ?)
1242     [ #H <H normalize #EQ % [@[ ]] % [@hd] normalize /2/ 
1243     | #a #l' #H <H normalize #EQ
1244      %[@(hd::rev … l')] %[@a] % //
1245      >length_append in EQ #EQ normalize in EQ; normalize;
1246      generalize in match (injective_S … EQ) #EQ2 /2/ ]]
1247qed.
1248
1249lemma list_elim_rev:
1250 ∀A:Type[0].∀P:list A → Type[0].
1251  P [ ] → (∀l,a. P l → P (l@[a])) →
1252   ∀l. P l.
1253 #A #P #H1 #H2 #l
1254 generalize in match (refl … (length … l))
1255 generalize in ⊢ (???% → ?) #n generalize in match l
1256 elim n
1257  [ #L cases L [ // | #x #w #abs (normalize in abs) @⊥ // ]
1258  | #m #IH #L #EQ
1259    cases (split_len_Sn … EQ) #l' * #a * /3/ ]
1260qed.
1261
1262axiom is_prefix: ∀A:Type[0]. list A → list A → Prop.
1263axiom prefix_of_append:
1264 ∀A:Type[0].∀l,l1,l2:list A.
1265  is_prefix … l l1 → is_prefix … l (l1@l2).
1266axiom prefix_reflexive: ∀A,l. is_prefix A l l.
1267axiom nil_prefix: ∀A,l. is_prefix A [ ] l.
1268
1269record Propify (A:Type[0]) : Type[0] (*Prop*) ≝ { in_propify: A }.
1270
1271definition Propify_elim: ∀A. ∀P:Prop. (A → P) → (Propify A → P) ≝
1272 λA,P,H,x. match x with [ mk_Propify p ⇒ H p ].
1273
1274definition app ≝
1275 λA:Type[0].λl1:Propify (list A).λl2:list A.
1276  match l1 with
1277   [ mk_Propify l1 ⇒ mk_Propify … (l1@l2) ].
1278
1279lemma app_nil: ∀A,l1. app A l1 [ ] = l1.
1280 #A * /3/
1281qed.
1282
1283lemma app_assoc: ∀A,l1,l2,l3. app A (app A l1 l2) l3 = app A l1 (l2@l3).
1284 #A * #l1 normalize //
1285qed.
1286*)
1287
1288axiom assembly_ok:
1289 ∀program,pol,assembled.
1290  let 〈labels,costs〉 ≝ build_maps program pol in
1291  Some … 〈assembled,costs〉 = assembly program pol →
1292  let code_memory ≝ load_code_memory assembled in
1293  let preamble ≝ \fst program in
1294  let datalabels ≝ construct_datalabels preamble in
1295  let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in
1296  let lookup_datalabels ≝ λx. lookup ?? x datalabels (zero ?) in
1297   ∀ppc,len,assembledi.
1298    let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1299     Some … 〈len,assembledi〉 = assembly_1_pseudoinstruction program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi →
1300      encoding_check code_memory (sigma program pol ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len)) assembledi ∧
1301       sigma program pol newppc = bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len).
1302
1303axiom bitvector_of_nat_nat_of_bitvector:
1304  ∀n,v.
1305    bitvector_of_nat n (nat_of_bitvector n v) = v.
1306
1307axiom assembly_ok_to_expand_pseudo_instruction_ok:
1308 ∀program,pol,assembled,costs.
1309  Some … 〈assembled,costs〉 = assembly program pol →
1310   ∀ppc.
1311    let code_memory ≝ load_code_memory assembled in
1312    let preamble ≝ \fst program in   
1313    let data_labels ≝ construct_datalabels preamble in
1314    let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in
1315    let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in
1316    let expansion ≝ pol ppc in
1317    let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1318     ∃instructions.
1319      Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program pol ppc) expansion pi.
1320
1321axiom split_elim':
1322  ∀A: Type[0].
1323  ∀B: Type[1].
1324  ∀l, m, v.
1325  ∀T: Vector A l → Vector A m → B.
1326  ∀P: B → Prop.
1327    (∀lft, rgt. v = lft @@ rgt → P (T lft rgt)) →
1328      P (let 〈lft, rgt〉 ≝ split A l m v in T lft rgt).
1329
1330axiom split_elim'':
1331  ∀A: Type[0].
1332  ∀B,B': Type[1].
1333  ∀l, m, v.
1334  ∀T: Vector A l → Vector A m → B.
1335  ∀T': Vector A l → Vector A m → B'.
1336  ∀P: B → B' → Prop.
1337    (∀lft, rgt. v = lft @@ rgt → P (T lft rgt) (T' lft rgt)) →
1338      P (let 〈lft, rgt〉 ≝ split A l m v in T lft rgt)
1339        (let 〈lft, rgt〉 ≝ split A l m v in T' lft rgt).
1340
1341lemma fetch_assembly_pseudo2:
1342 ∀program,pol,assembled.
1343  let 〈labels,costs〉 ≝ build_maps program pol in
1344  Some … 〈assembled,costs〉 = assembly program pol →
1345   ∀ppc.
1346    let code_memory ≝ load_code_memory assembled in
1347    let preamble ≝ \fst program in
1348    let data_labels ≝ construct_datalabels preamble in
1349    let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in
1350    let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in
1351    let expansion ≝ pol ppc in
1352    let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1353     ∃instructions.
1354      Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program pol ppc) expansion pi ∧
1355       fetch_many code_memory (sigma program pol newppc) (sigma program pol ppc) instructions.
1356 #program #pol #assembled @pair_elim' #labels #costs #BUILD_MAPS #ASSEMBLY #ppc
1357 generalize in match (assembly_ok_to_expand_pseudo_instruction_ok program pol assembled costs ASSEMBLY ppc)
1358 letin code_memory ≝ (load_code_memory assembled)
1359 letin preamble ≝ (\fst program)
1360 letin data_labels ≝ (construct_datalabels preamble)
1361 letin lookup_labels ≝ (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x))
1362 letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?))
1363 whd in ⊢ (% → %) generalize in match (assembly_ok program pol assembled) >BUILD_MAPS #XX generalize in match (XX ASSEMBLY ppc) -XX;
1364 cases (fetch_pseudo_instruction (\snd program) ppc) #pi #newppc
1365 generalize in match (fetch_assembly_pseudo program pol ppc
1366  (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) (λx. lookup ?? x data_labels (zero ?)) pi
1367  (load_code_memory assembled));
1368 whd in ⊢ ((∀_.∀_.∀_.∀_.%) → (∀_.∀_.%) → % → %) #H1 #H2 * #instructions #EXPAND
1369 whd in H1:(∀_.∀_.∀_.∀_.? → ???% → ?) H2:(∀_.∀_.???% → ?);
1370 normalize nodelta in EXPAND; (* HERE *)
1371 generalize in match (λlen, assembled.H1 len assembled instructions (nat_of_bitvector … (sigma program pol ppc))) -H1; #H1
1372 >bitvector_of_nat_nat_of_bitvector in H1; #H1
1373 <EXPAND in H1 H2; whd in ⊢ ((∀_.∀_.? → ???% → ?) → (∀_.∀_.???% → ?) → ?)
1374 #H1 #H2
1375 cases (H2 ?? (refl …)) -H2; #K1 #K2 >K2
1376 generalize in match (H1 ?? (refl …) (refl …) ?) -H1;
1377  [ #K3 % [2: % [% | @K3]] | @K1 ]
1378qed.
1379
1380(* OLD?
1381definition assembly_specification:
1382  ∀assembly_program: pseudo_assembly_program.
1383  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1384  λpseudo_assembly_program.
1385  λcode_mem.
1386    ∀pc: Word.
1387      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1388      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1389      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1390      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1391      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1392       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1393      match pre_assembled with
1394       [ None ⇒ True
1395       | Some pc_code ⇒
1396          let 〈new_pc,code〉 ≝ pc_code in
1397           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1398
1399axiom assembly_meets_specification:
1400  ∀pseudo_assembly_program.
1401    match assembly pseudo_assembly_program with
1402    [ None ⇒ True
1403    | Some code_mem_cost ⇒
1404      let 〈code_mem, cost〉 ≝ code_mem_cost in
1405        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1406    ].
1407(*
1408  # PROGRAM
1409  [ cases PROGRAM
1410    # PREAMBLE
1411    # INSTR_LIST
1412    elim INSTR_LIST
1413    [ whd
1414      whd in ⊢ (∀_. %)
1415      # PC
1416      whd
1417    | # INSTR
1418      # INSTR_LIST_TL
1419      # H
1420      whd
1421      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1422    ]
1423  | cases not_implemented
1424  ] *)
1425*)
1426
1427definition internal_pseudo_address_map ≝ list (BitVector 8).
1428
1429axiom low_internal_ram_of_pseudo_low_internal_ram:
1430 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1431
1432axiom high_internal_ram_of_pseudo_high_internal_ram:
1433 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1434
1435axiom low_internal_ram_of_pseudo_internal_ram_hit:
1436 ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀pol:policy (code_memory … s).∀addr:BitVector 7.
1437  member ? (eq_bv 8) (false:::addr) M = true →
1438   let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1439   let pbl ≝ lookup ? 7 addr (low_internal_ram ? s) (zero 8) in
1440   let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram ? s) (zero 8) in
1441   let bl ≝ lookup ? 7 addr ram (zero 8) in
1442   let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in
1443    bu@@bl = sigma (code_memory … s) pol (pbu@@pbl).
1444
1445(* changed from add to sub *)
1446axiom low_internal_ram_of_pseudo_internal_ram_miss:
1447 ∀T.∀M:internal_pseudo_address_map.∀s:PreStatus T.∀addr:BitVector 7.
1448  let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1449  let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in
1450  let carr ≝ get_index_v ? ? flags 1 ? in
1451  carr = false →
1452  member ? (eq_bv 8) (false:::Saddr) M = false →
1453   member ? (eq_bv 8) (false:::addr) M = false →
1454    lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?).
1455  //
1456qed.
1457
1458definition addressing_mode_ok ≝
1459 λT.λM:internal_pseudo_address_map.λs:PreStatus T.
1460  λaddr:addressing_mode.
1461   match addr with
1462    [ DIRECT d ⇒
1463       ¬(member ? (eq_bv 8) d M) ∧
1464       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1465    | INDIRECT i ⇒
1466       let d ≝ get_register ? s [[false;false;i]] in
1467       ¬(member ? (eq_bv 8) d M) ∧
1468       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1469    | EXT_INDIRECT _ ⇒ true
1470    | REGISTER _ ⇒ true
1471    | ACC_A ⇒ true
1472    | ACC_B ⇒ true
1473    | DPTR ⇒ true
1474    | DATA _ ⇒ true
1475    | DATA16 _ ⇒ true
1476    | ACC_DPTR ⇒ true
1477    | ACC_PC ⇒ true
1478    | EXT_INDIRECT_DPTR ⇒ true
1479    | INDIRECT_DPTR ⇒ true
1480    | CARRY ⇒ true
1481    | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1482    | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1483    | RELATIVE _ ⇒ true
1484    | ADDR11 _ ⇒ true
1485    | ADDR16 _ ⇒ true ].
1486   
1487definition next_internal_pseudo_address_map0 ≝
1488  λT.
1489  λfetched.
1490  λM: internal_pseudo_address_map.
1491  λs: PreStatus T.
1492   match fetched with
1493    [ Comment _ ⇒ Some ? M
1494    | Cost _ ⇒ Some … M
1495    | Jmp _ ⇒ Some … M
1496    | Call _ ⇒
1497       Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M)
1498    | Mov _ _ ⇒ Some … M
1499    | Instruction instr ⇒
1500       match instr with
1501        [ ADD addr1 addr2 ⇒
1502           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1503            Some ? M
1504           else
1505            None ?
1506        | ADDC addr1 addr2 ⇒
1507           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1508            Some ? M
1509           else
1510            None ?
1511        | SUBB addr1 addr2 ⇒
1512           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1513            Some ? M
1514           else
1515            None ?       
1516        | _ ⇒ (* TO BE COMPLETED *) Some ? M ]].
1517 
1518
1519definition next_internal_pseudo_address_map ≝
1520 λM:internal_pseudo_address_map.
1521  λs:PseudoStatus.
1522    next_internal_pseudo_address_map0 ?
1523     (\fst (fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s))) M s.
1524   
1525definition status_of_pseudo_status:
1526 internal_pseudo_address_map → ∀ps:PseudoStatus. policy (code_memory … ps) → option Status ≝
1527 λM,ps,pol.
1528  let pap ≝ code_memory … ps in
1529   match assembly pap pol with
1530    [ None ⇒ None …
1531    | Some p ⇒
1532       let cm ≝ load_code_memory (\fst p) in
1533       let pc ≝ sigma pap pol (program_counter ? ps) in
1534       let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in
1535       let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in
1536        Some …
1537         (mk_PreStatus (BitVectorTrie Byte 16)
1538           cm
1539           lir
1540           hir
1541           (external_ram … ps)
1542           pc
1543           (special_function_registers_8051 … ps)
1544           (special_function_registers_8052 … ps)
1545           (p1_latch … ps)
1546           (p3_latch … ps)
1547           (clock … ps)) ].
1548
1549(*
1550definition write_at_stack_pointer':
1551 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1552  λM: Type[0].
1553  λs: PreStatus M.
1554  λv: Byte.
1555    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1556    let bit_zero ≝ get_index_v… nu O ? in
1557    let bit_1 ≝ get_index_v… nu 1 ? in
1558    let bit_2 ≝ get_index_v… nu 2 ? in
1559    let bit_3 ≝ get_index_v… nu 3 ? in
1560      if bit_zero then
1561        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1562                              v (low_internal_ram ? s) in
1563          set_low_internal_ram ? s memory
1564      else
1565        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1566                              v (high_internal_ram ? s) in
1567          set_high_internal_ram ? s memory.
1568  [ cases l0 %
1569  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1570qed.
1571
1572definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1573 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1574
1575  λticks_of.
1576  λs.
1577  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1578  let ticks ≝ ticks_of (program_counter ? s) in
1579  let s ≝ set_clock ? s (clock ? s + ticks) in
1580  let s ≝ set_program_counter ? s pc in
1581    match instr with
1582    [ Instruction instr ⇒
1583       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1584    | Comment cmt ⇒ s
1585    | Cost cst ⇒ s
1586    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1587    | Call call ⇒
1588      let a ≝ address_of_word_labels s call in
1589      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1590      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1591      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1592      let s ≝ write_at_stack_pointer' ? s pc_bl in
1593      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1594      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1595      let s ≝ write_at_stack_pointer' ? s pc_bu in
1596        set_program_counter ? s a
1597    | Mov dptr ident ⇒
1598       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1599    ].
1600 [
1601 |2,3,4: %
1602 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1603 |
1604 | %
1605 ]
1606 cases not_implemented
1607qed.
1608*)
1609
1610axiom execute_1_pseudo_instruction_preserves_code_memory:
1611 ∀ticks_of,ps.
1612  code_memory … (execute_1_pseudo_instruction ticks_of ps) = code_memory … ps.
1613
1614(*
1615lemma execute_code_memory_unchanged:
1616 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1617 #ticks #ps whd in ⊢ (??? (??%))
1618 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1619  (program_counter pseudo_assembly_program ps)) #instr #pc
1620 whd in ⊢ (??? (??%)) cases instr
1621  [ #pre cases pre
1622     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1623       cases (split ????) #z1 #z2 %
1624     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1625       cases (split ????) #z1 #z2 %
1626     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1627       cases (split ????) #z1 #z2 %
1628     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1629       [ #x1 whd in ⊢ (??? (??%))
1630     | *: cases not_implemented
1631     ]
1632  | #comment %
1633  | #cost %
1634  | #label %
1635  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1636    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1637    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1638    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1639    (* CSC: ??? *)
1640  | #dptr #label (* CSC: ??? *)
1641  ]
1642  cases not_implemented
1643qed.
1644*)
1645
1646lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1647 ∀M:internal_pseudo_address_map.
1648 ∀ps,ps': PseudoStatus.
1649 ∀pol.
1650  ∀prf:code_memory … ps = code_memory … ps'.
1651   let pol' ≝ ? in
1652   match status_of_pseudo_status M ps pol with
1653    [ None ⇒ status_of_pseudo_status M ps' pol' = None …
1654    | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w
1655    ].
1656 [2: <prf @pol]
1657 #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1658 generalize in match (refl … (assembly (code_memory … ps) pol))
1659 cases (assembly ??) in ⊢ (???% → %)
1660  [ #K whd whd in ⊢ (??%?) <H >K %
1661  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1662qed.
1663
1664definition ticks_of0: ∀p:pseudo_assembly_program. policy p → Word → ? → nat × nat ≝
1665  λprogram: pseudo_assembly_program.λpol.
1666  λppc: Word.
1667  λfetched.
1668    match fetched with
1669    [ Instruction instr ⇒
1670      match instr with
1671      [ JC lbl ⇒
1672        match pol ppc with
1673        [ short_jump ⇒ 〈2, 2〉
1674        | medium_jump ⇒ ?
1675        | long_jump ⇒ 〈4, 4〉
1676        ]
1677      | JNC lbl ⇒
1678        match pol ppc with
1679        [ short_jump ⇒ 〈2, 2〉
1680        | medium_jump ⇒ ?
1681        | long_jump ⇒ 〈4, 4〉
1682        ]
1683      | JB bit lbl ⇒
1684        match pol ppc with
1685        [ short_jump ⇒ 〈2, 2〉
1686        | medium_jump ⇒ ?
1687        | long_jump ⇒ 〈4, 4〉
1688        ]
1689      | JNB bit lbl ⇒
1690        match pol ppc with
1691        [ short_jump ⇒ 〈2, 2〉
1692        | medium_jump ⇒ ?
1693        | long_jump ⇒ 〈4, 4〉
1694        ]
1695      | JBC bit lbl ⇒
1696        match pol ppc with
1697        [ short_jump ⇒ 〈2, 2〉
1698        | medium_jump ⇒ ?
1699        | long_jump ⇒ 〈4, 4〉
1700        ]
1701      | JZ lbl ⇒
1702        match pol ppc with
1703        [ short_jump ⇒ 〈2, 2〉
1704        | medium_jump ⇒ ?
1705        | long_jump ⇒ 〈4, 4〉
1706        ]
1707      | JNZ lbl ⇒
1708        match pol ppc with
1709        [ short_jump ⇒ 〈2, 2〉
1710        | medium_jump ⇒ ?
1711        | long_jump ⇒ 〈4, 4〉
1712        ]
1713      | CJNE arg lbl ⇒
1714        match pol ppc with
1715        [ short_jump ⇒ 〈2, 2〉
1716        | medium_jump ⇒ ?
1717        | long_jump ⇒ 〈4, 4〉
1718        ]
1719      | DJNZ arg lbl ⇒
1720        match pol ppc with
1721        [ short_jump ⇒ 〈2, 2〉
1722        | medium_jump ⇒ ?
1723        | long_jump ⇒ 〈4, 4〉
1724        ]
1725      | ADD arg1 arg2 ⇒
1726        let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in
1727         〈ticks, ticks〉
1728      | ADDC arg1 arg2 ⇒
1729        let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in
1730         〈ticks, ticks〉
1731      | SUBB arg1 arg2 ⇒
1732        let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in
1733         〈ticks, ticks〉
1734      | INC arg ⇒
1735        let ticks ≝ ticks_of_instruction (INC ? arg) in
1736         〈ticks, ticks〉
1737      | DEC arg ⇒
1738        let ticks ≝ ticks_of_instruction (DEC ? arg) in
1739         〈ticks, ticks〉
1740      | MUL arg1 arg2 ⇒
1741        let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in
1742         〈ticks, ticks〉
1743      | DIV arg1 arg2 ⇒
1744        let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in
1745         〈ticks, ticks〉
1746      | DA arg ⇒
1747        let ticks ≝ ticks_of_instruction (DA ? arg) in
1748         〈ticks, ticks〉
1749      | ANL arg ⇒
1750        let ticks ≝ ticks_of_instruction (ANL ? arg) in
1751         〈ticks, ticks〉
1752      | ORL arg ⇒
1753        let ticks ≝ ticks_of_instruction (ORL ? arg) in
1754         〈ticks, ticks〉
1755      | XRL arg ⇒
1756        let ticks ≝ ticks_of_instruction (XRL ? arg) in
1757         〈ticks, ticks〉
1758      | CLR arg ⇒
1759        let ticks ≝ ticks_of_instruction (CLR ? arg) in
1760         〈ticks, ticks〉
1761      | CPL arg ⇒
1762        let ticks ≝ ticks_of_instruction (CPL ? arg) in
1763         〈ticks, ticks〉
1764      | RL arg ⇒
1765        let ticks ≝ ticks_of_instruction (RL ? arg) in
1766         〈ticks, ticks〉
1767      | RLC arg ⇒
1768        let ticks ≝ ticks_of_instruction (RLC ? arg) in
1769         〈ticks, ticks〉
1770      | RR arg ⇒
1771        let ticks ≝ ticks_of_instruction (RR ? arg) in
1772         〈ticks, ticks〉
1773      | RRC arg ⇒
1774        let ticks ≝ ticks_of_instruction (RRC ? arg) in
1775         〈ticks, ticks〉
1776      | SWAP arg ⇒
1777        let ticks ≝ ticks_of_instruction (SWAP ? arg) in
1778         〈ticks, ticks〉
1779      | MOV arg ⇒
1780        let ticks ≝ ticks_of_instruction (MOV ? arg) in
1781         〈ticks, ticks〉
1782      | MOVX arg ⇒
1783        let ticks ≝ ticks_of_instruction (MOVX ? arg) in
1784         〈ticks, ticks〉
1785      | SETB arg ⇒
1786        let ticks ≝ ticks_of_instruction (SETB ? arg) in
1787         〈ticks, ticks〉
1788      | PUSH arg ⇒
1789        let ticks ≝ ticks_of_instruction (PUSH ? arg) in
1790         〈ticks, ticks〉
1791      | POP arg ⇒
1792        let ticks ≝ ticks_of_instruction (POP ? arg) in
1793         〈ticks, ticks〉
1794      | XCH arg1 arg2 ⇒
1795        let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in
1796         〈ticks, ticks〉
1797      | XCHD arg1 arg2 ⇒
1798        let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in
1799         〈ticks, ticks〉
1800      | RET ⇒
1801        let ticks ≝ ticks_of_instruction (RET ?) in
1802         〈ticks, ticks〉
1803      | RETI ⇒
1804        let ticks ≝ ticks_of_instruction (RETI ?) in
1805         〈ticks, ticks〉
1806      | NOP ⇒
1807        let ticks ≝ ticks_of_instruction (NOP ?) in
1808         〈ticks, ticks〉
1809      ]
1810    | Comment comment ⇒ 〈0, 0〉
1811    | Cost cost ⇒ 〈0, 0〉
1812    | Jmp jmp ⇒ 〈2, 2〉
1813    | Call call ⇒ 〈2, 2〉
1814    | Mov dptr tgt ⇒ 〈2, 2〉
1815    ].
1816  cases not_implemented (* policy returned medium_jump for conditional jumping = impossible *)
1817qed.
1818
1819definition ticks_of: ∀p:pseudo_assembly_program. policy p → Word → nat × nat ≝
1820  λprogram: pseudo_assembly_program.λpol.
1821  λppc: Word.
1822    let 〈preamble, pseudo〉 ≝ program in
1823    let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in
1824     ticks_of0 program pol ppc fetched.
1825
1826lemma get_register_set_program_counter:
1827 ∀T,s,pc. get_register T (set_program_counter … s pc) = get_register … s.
1828 #T #s #pc %
1829qed.
1830
1831lemma get_8051_sfr_set_program_counter:
1832 ∀T,s,pc. get_8051_sfr T (set_program_counter … s pc) = get_8051_sfr … s.
1833 #T #s #pc %
1834qed.
1835
1836lemma get_bit_addressable_sfr_set_program_counter:
1837 ∀T,s,pc. get_bit_addressable_sfr T (set_program_counter … s pc) = get_bit_addressable_sfr … s.
1838 #T #s #pc %
1839qed.
1840
1841lemma low_internal_ram_set_program_counter:
1842 ∀T,s,pc. low_internal_ram T (set_program_counter … s pc) = low_internal_ram … s.
1843 #T #s #pc %
1844qed.
1845
1846lemma get_arg_8_set_program_counter:
1847 ∀n.∀l:Vector addressing_mode_tag (S n). ¬(is_in ? l ACC_PC) →
1848  ∀T,s,pc,b.∀arg:l.
1849   get_arg_8 T (set_program_counter … s pc) b arg = get_arg_8 … s b arg.
1850 [2,3: cases arg; *; normalize //]
1851 #n #l #H #T #s #pc #b * *; [11: #NH @⊥ //] #X try #Y %
1852qed.
1853
1854lemma set_bit_addressable_sfr_set_code_memory:
1855  ∀T, U: Type[0].
1856  ∀ps: PreStatus ?.
1857  ∀code_mem.
1858  ∀x.
1859  ∀val.
1860  set_bit_addressable_sfr ? (set_code_memory T U ps code_mem) x val =
1861  set_code_memory T U (set_bit_addressable_sfr ? ps x val) code_mem.
1862  #T #U #ps #code_mem #x #val
1863  whd in ⊢ (??%?)
1864  whd in ⊢ (???(???%?))
1865  cases (eqb ? 128) [ normalize nodelta cases not_implemented
1866  | normalize nodelta
1867  cases (eqb (nat_of_bitvector 8 x) ?) [ normalize nodelta cases not_implemented
1868  | normalize nodelta
1869  cases (eqb (nat_of_bitvector 8 x) ?) [ normalize nodelta cases not_implemented
1870  | normalize nodelta
1871  cases (eqb ? 176) [ normalize nodelta %
1872  | normalize nodelta
1873  cases (eqb ? 153) [ normalize nodelta %
1874  | normalize nodelta
1875  cases (eqb ? 138) [ normalize nodelta %
1876  | normalize nodelta
1877  cases (eqb ? 139) [ normalize nodelta %
1878  | normalize nodelta
1879  cases (eqb ? 140) [ normalize nodelta %
1880  | normalize nodelta
1881  cases (eqb ? 141) [ normalize nodelta %
1882  | normalize nodelta
1883  cases (eqb ? 200) [ normalize nodelta %
1884  | normalize nodelta
1885  cases (eqb ? 202) [ normalize nodelta %
1886  | normalize nodelta
1887  cases (eqb ? 203) [ normalize nodelta %
1888  | normalize nodelta
1889  cases (eqb ? 204) [ normalize nodelta %
1890  | normalize nodelta
1891  cases (eqb ? 205) [ normalize nodelta %
1892  | normalize nodelta
1893  cases (eqb ? 135) [ normalize nodelta %
1894  | normalize nodelta
1895  cases (eqb ? 136) [ normalize nodelta %
1896  | normalize nodelta
1897  cases (eqb ? 137) [ normalize nodelta %
1898  | normalize nodelta
1899  cases (eqb ? 152) [ normalize nodelta %
1900  | normalize nodelta
1901  cases (eqb ? 168) [ normalize nodelta %
1902  | normalize nodelta
1903  cases (eqb ? 184) [ normalize nodelta %
1904  | normalize nodelta
1905  cases (eqb ? 129) [ normalize nodelta %
1906  | normalize nodelta
1907  cases (eqb ? 130) [ normalize nodelta %
1908  | normalize nodelta
1909  cases (eqb ? 131) [ normalize nodelta %
1910  | normalize nodelta
1911  cases (eqb ? 208) [ normalize nodelta %
1912  | normalize nodelta
1913  cases (eqb ? 224) [ normalize nodelta %
1914  | normalize nodelta
1915  cases (eqb ? 240) [ normalize nodelta %
1916  | normalize nodelta
1917    cases not_implemented
1918  ]]]]]]]]]]]]]]]]]]]]]]]]]]
1919qed.
1920
1921lemma set_arg_8_set_code_memory:
1922  ∀n:nat.
1923  ∀l:Vector addressing_mode_tag (S n).
1924    ¬(is_in ? l ACC_PC) →
1925    ∀T: Type[0].
1926    ∀U: Type[0].
1927    ∀ps: PreStatus ?.
1928    ∀code_mem.
1929    ∀val.
1930    ∀b: l.
1931  set_arg_8 ? (set_code_memory T U ps code_mem) b val =
1932  set_code_memory T U (set_arg_8 ? ps b val) code_mem.
1933  [2,3: cases b; *; normalize //]
1934  #n #l #prf #T #U #ps #code_mem #val * *;
1935  [*:
1936    [1,2,3,4,8,9,15,16,17,18,19: #x]
1937    try #y whd in ⊢ (??(%)?)
1938    whd in ⊢ (???(???(%)?))
1939    [1,2:
1940      cases (split bool 4 4 ?)
1941      #nu' #nl'
1942      normalize nodelta
1943      cases (split bool 1 3 nu')
1944      #bit1' #ignore'
1945      normalize nodelta
1946      cases (get_index_v bool 4 nu' 0 ?)
1947      [1,2,3,4:
1948        normalize nodelta
1949        try %
1950        try (@(set_bit_addressable_sfr_set_code_memory T U ps code_mem x val))
1951        try (normalize in ⊢ (???(???%?)))
1952      ]
1953    |3,4: %
1954    |*:
1955       try normalize nodelta
1956       normalize cases (not_implemented)
1957    ]
1958 ]
1959 
1960
1961qed.
1962
1963axiom set_arg_8_set_program_counter:
1964  ∀n:nat.
1965  ∀l:Vector addressing_mode_tag (S n).
1966    ¬(is_in ? l ACC_PC) →
1967    ∀T: Type[0].
1968    ∀ps: PreStatus ?.
1969    ∀pc.
1970    ∀val.
1971    ∀b: l.
1972  set_arg_8 ? (set_program_counter T ps pc) b val =
1973  set_program_counter T (set_arg_8 ? ps b val) pc.
1974  [1,2: cases b; *; normalize //]
1975qed.
1976 
1977
1978lemma get_arg_8_set_code_memory:
1979 ∀T1,T2,s,code_mem,b,arg.
1980   get_arg_8 T1 (set_code_memory T2 T1 s code_mem) b arg = get_arg_8 … s b arg.
1981 #T1 #T2 #s #code_mem #b #arg %
1982qed.
1983
1984lemma set_code_memory_set_flags:
1985 ∀T1,T2,s,f1,f2,f3,code_mem.
1986  set_code_memory T1 T2 (set_flags T1 s f1 f2 f3) code_mem =
1987   set_flags … (set_code_memory … s code_mem) f1 f2 f3.
1988 #T1 #T2 #s #f1 #f2 #f3 #code_mem %
1989qed.
1990
1991lemma set_program_counter_set_flags:
1992 ∀T1,s,f1,f2,f3,pc.
1993  set_program_counter T1 (set_flags T1 s f1 f2 f3) pc =
1994   set_flags … (set_program_counter … s pc) f1 f2 f3.
1995 #T1 #s #f1 #f2 #f3 #pc  %
1996qed.
1997
1998lemma program_counter_set_flags:
1999 ∀T1,s,f1,f2,f3.
2000  program_counter T1 (set_flags T1 s f1 f2 f3) = program_counter … s.
2001 #T1 #s #f1 #f2 #f3 %
2002qed.
2003
2004lemma special_function_registers_8051_write_at_stack_pointer:
2005 ∀T,s,x.
2006    special_function_registers_8051 T (write_at_stack_pointer T s x)
2007  = special_function_registers_8051 T s.
2008 #T #s #x whd in ⊢ (??(??%)?)
2009 cases (split ????) #nu #nl normalize nodelta;
2010 cases (get_index_v bool ????) %
2011qed.
2012
2013lemma get_8051_sfr_write_at_stack_pointer:
2014 ∀T,s,x,y. get_8051_sfr T (write_at_stack_pointer T s x) y = get_8051_sfr T s y.
2015 #T #s #x #y whd in ⊢ (??%%) //
2016qed.
2017
2018lemma code_memory_write_at_stack_pointer:
2019 ∀T,s,x. code_memory T (write_at_stack_pointer T s x) = code_memory T s.
2020 #T #s #x whd in ⊢ (??(??%)?)
2021 cases (split ????) #nu #nl normalize nodelta;
2022 cases (get_index_v bool ????) %
2023qed.
2024
2025axiom low_internal_ram_write_at_stack_pointer:
2026 ∀T1,T2,M,s1,s2,s3.∀pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2027  get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP →
2028  low_internal_ram ? s2 = low_internal_ram T2 s3 →
2029  sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) →
2030  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
2031  bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) →
2032   low_internal_ram T1
2033     (write_at_stack_pointer ?
2034       (set_8051_sfr ?
2035         (write_at_stack_pointer ?
2036           (set_8051_sfr ?
2037             (set_low_internal_ram ? s1
2038               (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram ? s2)))
2039             SFR_SP sp1)
2040          bl)
2041        SFR_SP sp2)
2042      bu)
2043   = low_internal_ram_of_pseudo_low_internal_ram (sp1::M)
2044      (low_internal_ram ?
2045       (write_at_stack_pointer ?
2046         (set_8051_sfr ?
2047           (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl)
2048          SFR_SP sp2)
2049        pbu)).
2050       
2051axiom high_internal_ram_write_at_stack_pointer:
2052 ∀T1,T2,M,s1,s2,s3,pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2053  get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP →
2054  high_internal_ram ? s2 = high_internal_ram T2 s3 →
2055  sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) →
2056  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
2057  bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) →
2058   high_internal_ram T1
2059     (write_at_stack_pointer ?
2060       (set_8051_sfr ?
2061         (write_at_stack_pointer ?
2062           (set_8051_sfr ?
2063             (set_high_internal_ram ? s1
2064               (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram ? s2)))
2065             SFR_SP sp1)
2066          bl)
2067        SFR_SP sp2)
2068      bu)
2069   = high_internal_ram_of_pseudo_high_internal_ram (sp1::M)
2070      (high_internal_ram ?
2071       (write_at_stack_pointer ?
2072         (set_8051_sfr ?
2073           (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl)
2074          SFR_SP sp2)
2075        pbu)).
2076
2077lemma set_program_counter_set_low_internal_ram:
2078 ∀T,s,x,y.
2079  set_program_counter T (set_low_internal_ram … s x) y =
2080   set_low_internal_ram … (set_program_counter … s y) x.
2081 //
2082qed.
2083
2084lemma set_clock_set_low_internal_ram:
2085 ∀T,s,x,y.
2086  set_clock T (set_low_internal_ram … s x) y =
2087   set_low_internal_ram … (set_clock … s y) x.
2088 //
2089qed.
2090
2091lemma set_program_counter_set_high_internal_ram:
2092 ∀T,s,x,y.
2093  set_program_counter T (set_high_internal_ram … s x) y =
2094   set_high_internal_ram … (set_program_counter … s y) x.
2095 //
2096qed.
2097
2098lemma set_clock_set_high_internal_ram:
2099 ∀T,s,x,y.
2100  set_clock T (set_high_internal_ram … s x) y =
2101   set_high_internal_ram … (set_clock … s y) x.
2102 //
2103qed.
2104
2105lemma set_low_internal_ram_set_high_internal_ram:
2106 ∀T,s,x,y.
2107  set_low_internal_ram T (set_high_internal_ram … s x) y =
2108   set_high_internal_ram … (set_low_internal_ram … s y) x.
2109 //
2110qed.
2111
2112lemma external_ram_write_at_stack_pointer:
2113 ∀T,s,x. external_ram T (write_at_stack_pointer T s x) = external_ram T s.
2114 #T #s #x whd in ⊢ (??(??%)?)
2115 cases (split ????) #nu #nl normalize nodelta;
2116 cases (get_index_v bool ????) %
2117qed.
2118
2119lemma special_function_registers_8052_write_at_stack_pointer:
2120 ∀T,s,x.
2121    special_function_registers_8052 T (write_at_stack_pointer T s x)
2122  = special_function_registers_8052 T s.
2123 #T #s #x whd in ⊢ (??(??%)?)
2124 cases (split ????) #nu #nl normalize nodelta;
2125 cases (get_index_v bool ????) %
2126qed.
2127
2128lemma p1_latch_write_at_stack_pointer:
2129 ∀T,s,x. p1_latch T (write_at_stack_pointer T s x) = p1_latch T s.
2130 #T #s #x whd in ⊢ (??(??%)?)
2131 cases (split ????) #nu #nl normalize nodelta;
2132 cases (get_index_v bool ????) %
2133qed.
2134
2135lemma p3_latch_write_at_stack_pointer:
2136 ∀T,s,x. p3_latch T (write_at_stack_pointer T s x) = p3_latch T s.
2137 #T #s #x whd in ⊢ (??(??%)?)
2138 cases (split ????) #nu #nl normalize nodelta;
2139 cases (get_index_v bool ????) %
2140qed.
2141
2142lemma clock_write_at_stack_pointer:
2143 ∀T,s,x. clock T (write_at_stack_pointer T s x) = clock T s.
2144 #T #s #x whd in ⊢ (??(??%)?)
2145 cases (split ????) #nu #nl normalize nodelta;
2146 cases (get_index_v bool ????) %
2147qed.
2148
2149axiom get_index_v_set_index:
2150 ∀T,n,x,m,p,y. get_index_v T n (set_index T n x m y p) m p = y.
2151
2152lemma get_8051_sfr_set_8051_sfr:
2153 ∀T,s,x,y. get_8051_sfr T (set_8051_sfr ? s x y) x = y.
2154 #T *; #A #B #C #D #E #F #G #H #I #J *; #y whd in ⊢ (??(??%?)?)
2155 whd in match get_8051_sfr; normalize nodelta @get_index_v_set_index
2156qed.
2157
2158lemma program_counter_set_8051_sfr:
2159 ∀T,s,x,y. program_counter T (set_8051_sfr ? s x y) = program_counter ? s.
2160 //
2161qed.
2162
2163axiom get_arg_8_set_low_internal_ram:
2164 ∀M,s,x,b,z. get_arg_8 M (set_low_internal_ram ? s x) b z = get_arg_8 ? s b z.
2165
2166lemma eq_rect_Type1_r:
2167  ∀A: Type[1].
2168  ∀a:A.
2169  ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p.
2170  #A #a #P #H #x #p
2171  generalize in match H
2172  generalize in match P
2173  cases p
2174  //
2175qed.
2176
2177lemma split_eq_status:
2178 ∀T.
2179 ∀A1,A2,A3,A4,A5,A6,A7,A8,A9,A10.
2180 ∀B1,B2,B3,B4,B5,B6,B7,B8,B9,B10.
2181  A1=B1 → A2=B2 → A3=B3 → A4=B4 → A5=B5 → A6=B6 → A7=B7 → A8=B8 → A9=B9 → A10=B10 →
2182   mk_PreStatus T A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 =
2183   mk_PreStatus T B1 B2 B3 B4 B5 B6 B7 B8 B9 B10.
2184 //
2185qed.
2186
2187axiom split_append:
2188  ∀A: Type[0].
2189  ∀m, n: nat.
2190  ∀v, v': Vector A m.
2191  ∀q, q': Vector A n.
2192    let 〈v', q'〉 ≝ split A m n (v@@q) in
2193      v = v' ∧ q = q'.
2194
2195axiom split_vector_singleton:
2196  ∀A: Type[0].
2197  ∀n: nat.
2198  ∀v: Vector A (S n).
2199  ∀rest: Vector A n.
2200  ∀s: Vector A 1.
2201  ∀prf.
2202    v = s @@ rest →
2203    ((get_index_v A ? v 0 prf) ::: rest) = v.
2204
2205example sub_minus_one_seven_eight:
2206  ∀v: BitVector 7.
2207  false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) =
2208  \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false).
2209 cases daemon.
2210qed.
2211
2212(*
2213lemma blah:
2214  ∀m: internal_pseudo_address_map.
2215  ∀s: PseudoStatus.
2216  ∀arg: Byte.
2217  ∀b: bool.
2218    addressing_mode_ok m s (DIRECT arg) = true →
2219      get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) =
2220      get_arg_8 ? s b (DIRECT arg).
2221  [2, 3: normalize % ]
2222  #m #s #arg #b #hyp
2223  whd in ⊢ (??%%)
2224  @split_elim''
2225  #nu' #nl' #arg_nu_nl_eq
2226  normalize nodelta
2227  generalize in match (refl ? (get_index_v bool 4 nu' ? ?))
2228  cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %)
2229  #get_index_v_eq
2230  normalize nodelta
2231  [2:
2232    normalize nodelta
2233    @split_elim''
2234    #bit_one' #three_bits' #bit_one_three_bit_eq
2235    generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl'))
2236    normalize nodelta
2237    generalize in match (refl ? (sub_7_with_carry ? ? ?))
2238    cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %)
2239    #Saddr #carr' #Saddr_carr_eq
2240    normalize nodelta
2241    #carr_hyp'
2242    @carr_hyp'
2243    [1:
2244    |2: whd in hyp:(??%?); generalize in match hyp; -hyp;
2245        generalize in match (refl ? (¬(member (BitVector 8) ? arg m)))
2246        cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %)
2247        #member_eq
2248        normalize nodelta
2249        [2: #destr destruct(destr)
2250        |1: -carr_hyp';
2251            >arg_nu_nl_eq
2252            <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq)
2253            [1: >get_index_v_eq in ⊢ (??%? → ?)
2254            |2: @le_S @le_S @le_S @le_n
2255            ]
2256            cases (member (BitVector 8) ? (\fst ?) ?)
2257            [1: #destr normalize in destr; destruct(destr)
2258            |2:
2259            ]
2260        ]
2261    |3: >get_index_v_eq in ⊢ (??%?)
2262        change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl')
2263        >(split_vector_singleton … bit_one_three_bit_eq)
2264        <arg_nu_nl_eq
2265        whd in hyp:(??%?)
2266        cases (member (BitVector 8) (eq_bv 8) arg m) in hyp
2267        normalize nodelta [*: #ignore @sym_eq ]
2268    ]
2269  |
2270  ].
2271*)
2272(*
2273map_address0 ... (DIRECT arg) = Some .. →
2274  get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) =
2275  get_arg_8 (internal_ram ...) (DIRECT arg)
2276
2277((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) 
2278                     then Some internal_pseudo_address_map M 
2279                     else None internal_pseudo_address_map )
2280                    =Some internal_pseudo_address_map M')
2281*)
2282
2283lemma main_thm:
2284 ∀M,M',ps,s,s'',pol.
2285  next_internal_pseudo_address_map M ps = Some … M' →
2286  status_of_pseudo_status M ps pol = Some … s →
2287  status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps) ? = Some … s'' →
2288   ∃n. execute n s = s''.
2289 [2: >execute_1_pseudo_instruction_preserves_code_memory @pol]
2290 #M #M' #ps #s #s'' #pol
2291 generalize in match (fetch_assembly_pseudo2 (code_memory … ps) pol)
2292 whd in ⊢ (? → ? → ??%? → ??%? → ?)
2293 >execute_1_pseudo_instruction_preserves_code_memory
2294 generalize in match (refl … (assembly (code_memory … ps) pol))
2295 generalize in match (assembly (code_memory … ps) pol) in ⊢ (??%? → %) #ASS whd in ⊢ (???% → ?)
2296 cases (build_maps (code_memory … ps) pol)
2297 #labels #costs change in ⊢ (? → ? → ??%? → ?) with (next_internal_pseudo_address_map0 ? ? ? ?)
2298 generalize in match pol; -pol;
2299 @(match code_memory … ps return λx. ∀e:code_memory … ps = x.? with [pair preamble instr_list ⇒ ?]) [%]
2300 #EQ0 #pol normalize nodelta;
2301 generalize in ⊢ (???(match % with [_ ⇒ ? | _ ⇒ ?]) → ?) *; normalize nodelta;
2302  [ #EQ >EQ #_ #_ #abs @⊥ normalize in abs; destruct (abs) ]
2303 * #final_ppc * #final_pc #assembled #EQ >EQ -EQ ASS; normalize nodelta;
2304 #H generalize in match (H ? (refl …)) -H; #H;
2305 #MAP
2306 #H1 generalize in match (option_destruct_Some ??? H1) -H1; #H1 <H1 -H1;
2307 #H2 generalize in match (option_destruct_Some ??? H2) -H2; #H2 <H2 -H2;
2308 change with
2309  (∃n.
2310    execute n
2311     (set_low_internal_ram ?
2312       (set_high_internal_ram ?
2313         (set_program_counter ?
2314           (set_code_memory ?? ps (load_code_memory assembled))
2315          (sigma 〈preamble,instr_list〉 pol (program_counter ? ps)))
2316        (high_internal_ram_of_pseudo_high_internal_ram M ?))
2317      (low_internal_ram_of_pseudo_low_internal_ram M ?))
2318   = set_low_internal_ram ?
2319      (set_high_internal_ram ?
2320        (set_program_counter ?
2321          (set_code_memory ?? (execute_1_pseudo_instruction ? ps) (load_code_memory assembled))
2322         (sigma ???))
2323       ?)
2324     ?)
2325 whd in match (\snd 〈preamble,instr_list〉) in H;
2326 whd in match (\fst 〈preamble,instr_list〉) in H;
2327 whd in match (\snd 〈final_pc,assembled〉) in H;
2328 whd in match (\snd 〈preamble,instr_list〉) in MAP;
2329 -s s'' labels costs final_ppc final_pc;
2330 letin ps' ≝ (execute_1_pseudo_instruction (ticks_of 〈preamble,instr_list〉 pol) ps)
2331 (* NICE STATEMENT HERE *)
2332 generalize in match (refl … ps') generalize in ⊢ (??%? → ?) normalize nodelta; -ps'; #ps'
2333 #K <K generalize in match K; -K;
2334 (* STATEMENT WITH EQUALITY HERE *)
2335 whd in ⊢ (???(?%?) → ?)
2336 whd in ⊢ (???% → ?) generalize in match (H (program_counter … ps)) -H; >EQ0 normalize nodelta;
2337 cases (fetch_pseudo_instruction instr_list (program_counter … ps)) in MAP ⊢ %
2338 #pi #newppc normalize nodelta; #MAP * #instructions *;
2339 cases pi in MAP; normalize nodelta;
2340  [2,3: (* Comment, Cost *) #ARG #MAP #H1 #H2 #EQ %[1,3:@0]
2341    generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP M';
2342    normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2343    #H2 >(eq_bv_eq … H2) >EQ %
2344(*  |6: (* Mov *) #arg1 #arg2
2345       #H1 #H2 #EQ %[@1]
2346       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2347       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2348       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2349       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2350       >H2b >(eq_instruction_to_eq … H2a)
2351       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2352       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?)
2353       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2354       normalize nodelta;
2355       [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2356       #result #flags
2357       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) % *)
2358(*  |5: (* Call *) #label #MAP
2359      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP;
2360      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2361       [ (* short *) #abs @⊥ destruct (abs)
2362       |3: (* long *) #H1 #H2 #EQ %[@1]
2363           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2364           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2365           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2366           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2367           >H2b >(eq_instruction_to_eq … H2a)
2368           generalize in match EQ; -EQ;
2369           whd in ⊢ (???% → ??%?);
2370           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;
2371           >(eq_bv_eq … H2c)
2372           change with
2373            ((?=let 〈ppc_bu,ppc_bl〉 ≝ split bool 8 8 newppc in ?) →
2374                (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2375           generalize in match (refl … (split … 8 8 newppc)) cases (split bool 8 8 newppc) in ⊢ (??%? → %) #ppc_bu #ppc_bl #EQppc
2376           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;
2377           >get_8051_sfr_write_at_stack_pointer >get_8051_sfr_write_at_stack_pointer
2378           >get_8051_sfr_set_8051_sfr >get_8051_sfr_set_8051_sfr
2379           generalize in match (refl … (half_add ? new_sp (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry' #new_sp' #EQ2 normalize nodelta;
2380           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2381           @split_eq_status;
2382            [ >code_memory_write_at_stack_pointer whd in ⊢ (??%?)
2383              >code_memory_write_at_stack_pointer %
2384            | >set_program_counter_set_low_internal_ram
2385              >set_clock_set_low_internal_ram
2386              @low_internal_ram_write_at_stack_pointer
2387               [ >EQ0 @pol | % | %
2388               | @(pair_destruct_2 … EQ1)
2389               | @(pair_destruct_2 … EQ2)
2390               | >(pair_destruct_1 ????? EQpc)
2391                 >(pair_destruct_2 ????? EQpc)
2392                 @split_elim #x #y #H <H -x y H;
2393                 >(pair_destruct_1 ????? EQppc)
2394                 >(pair_destruct_2 ????? EQppc)
2395                 @split_elim #x #y #H <H -x y H;
2396                 >EQ0 % ]
2397            | >set_low_internal_ram_set_high_internal_ram
2398              >set_program_counter_set_high_internal_ram
2399              >set_clock_set_high_internal_ram
2400              @high_internal_ram_write_at_stack_pointer
2401               [ >EQ0 @pol | % | %
2402               | @(pair_destruct_2 … EQ1)
2403               | @(pair_destruct_2 … EQ2)
2404               | >(pair_destruct_1 ????? EQpc)
2405                 >(pair_destruct_2 ????? EQpc)
2406                 @split_elim #x #y #H <H -x y H;
2407                 >(pair_destruct_1 ????? EQppc)
2408                 >(pair_destruct_2 ????? EQppc)
2409                 @split_elim #x #y #H <H -x y H;
2410                 >EQ0 % ]           
2411            | >external_ram_write_at_stack_pointer whd in ⊢ (??%?)
2412              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
2413              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
2414              >external_ram_write_at_stack_pointer %
2415            | change with (? = sigma ?? (address_of_word_labels_code_mem (\snd (code_memory ? ps)) ?))
2416              >EQ0 %
2417            | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (??%?)
2418              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
2419              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
2420              >special_function_registers_8051_write_at_stack_pointer %
2421            | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (??%?)
2422              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
2423              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
2424              >special_function_registers_8052_write_at_stack_pointer %
2425            | >p1_latch_write_at_stack_pointer whd in ⊢ (??%?)
2426              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
2427              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
2428              >p1_latch_write_at_stack_pointer %
2429            | >p3_latch_write_at_stack_pointer whd in ⊢ (??%?)
2430              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
2431              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
2432              >p3_latch_write_at_stack_pointer %
2433            | >clock_write_at_stack_pointer whd in ⊢ (??%?)
2434              >clock_write_at_stack_pointer whd in ⊢ (???%)
2435              >clock_write_at_stack_pointer whd in ⊢ (???%)
2436              >clock_write_at_stack_pointer %]
2437       (*| (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2438         @pair_elim' #fst_5_addr #rest_addr #EQ1
2439         @pair_elim' #fst_5_pc #rest_pc #EQ2
2440         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2441         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2442         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2443         change in ⊢ (? →??%?) with (execute_1_0 ??)
2444         @pair_elim' * #instr #newppc' #ticks #EQn
2445          * * #H2a #H2b whd in ⊢ (% → ?) #H2c >H2b >(eq_instruction_to_eq … H2a) whd in ⊢ (??%?)
2446          generalize in match EQ; -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2447          @pair_elim' #carry #new_sp change with (half_add ? (get_8051_sfr ? ps ?) ? = ? → ?) #EQ4
2448          @split_elim' #pc_bu #pc_bl >program_counter_set_8051_sfr XXX change with (newppc = ?) #EQ5
2449          @pair_elim' #carry' #new_sp' #EQ6 normalize nodelta; #EQx >EQx -EQx;
2450          change in ⊢ (??(match ????% with [_ ⇒ ?])?) with (sigma … newppc)
2451          @split_elim' #pc_bu' #pc_bl' #EQ7 change with (newppc' = ? → ?)
2452          >get_8051_sfr_set_8051_sfr
2453         
2454          whd in EQ:(???%) ⊢ ? >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2455           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?)
2456           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc)))
2457           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2458           generalize in match (refl … (split bool 4 4 pc_bu))
2459           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2460           generalize in match (refl … (split bool 3 8 rest_addr))
2461           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2462           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2463           generalize in match
2464            (refl …
2465             (half_add 16 (sigma 〈preamble,instr_list〉 newppc)
2466             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2467           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7
2468           @split_eq_status try %
2469            [ change with (? = sigma ? (address_of_word_labels ps label))
2470              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2471            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2472              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] *)]
2473  |4: (* Jmp *) #label #MAP
2474      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP >MAP -MAP;
2475      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2476       [3: (* long *) #H1 #H2 #EQ %[@1]
2477           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2478           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2479           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2480           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2481           >H2b >(eq_instruction_to_eq … H2a)
2482           generalize in match EQ; -EQ;
2483           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2484           cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
2485       |1: (* short *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2486           generalize in match
2487            (refl ?
2488             (sub_16_with_carry
2489              (sigma 〈preamble,instr_list〉 pol (program_counter … ps))
2490              (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))
2491              false))
2492           cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta;
2493           generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta;
2494           generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta;
2495           #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)]
2496           generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2497           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2498           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2499           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2500           >H2b >(eq_instruction_to_eq … H2a)
2501           generalize in match EQ; -EQ;
2502           whd in ⊢ (???% → ?);
2503           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2504           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ???) ? in ?) = ?)
2505           generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) (sign_extension lower)))
2506           cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4
2507           @split_eq_status try % change with (newpc = sigma ?? (address_of_word_labels ps label))
2508           (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2509       | (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2510         generalize in match
2511          (refl …
2512            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))))
2513         cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1
2514         generalize in match
2515          (refl …
2516            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (program_counter … ps))))
2517         cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2
2518         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2519         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2520         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2521         change in ⊢ (? →??%?) with (execute_1_0 ??)
2522           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2523           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2524           >H2b >(eq_instruction_to_eq … H2a)
2525           generalize in match EQ; -EQ;
2526           whd in ⊢ (???% → ?);
2527           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2528           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2529           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)))
2530           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2531           generalize in match (refl … (split bool 4 4 pc_bu))
2532           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2533           generalize in match (refl … (split bool 3 8 rest_addr))
2534           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2535           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2536           generalize in match
2537            (refl …
2538             (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc)
2539             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2540           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7   
2541           @split_eq_status try %
2542            [ change with (? = sigma ?? (address_of_word_labels ps label))
2543              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2544            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2545              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]]
2546*)  | (* Instruction *) -pi;  whd in ⊢ (? → ??%? → ?) *; normalize nodelta;
2547    [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1]
2548       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2549       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2550       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2551       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2552       >H2b >(eq_instruction_to_eq … H2a)
2553       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP;
2554       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1;
2555       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2556       normalize nodelta; #MAP; (*
2557       [1: change in ⊢ (? → %) with
2558        ((let 〈result,flags〉 ≝
2559          add_8_with_carry
2560           (get_arg_8 ? ps false ACC_A)
2561           (get_arg_8 ?
2562             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2563             false (DIRECT ARG2))
2564           ? in ?) = ?)
2565        [2,3: %]
2566        change in ⊢ (???% → ?) with
2567         (let 〈result,flags〉 ≝ add_8_with_carry ?(*(get_arg_8 ? ps false ACC_A)*) ?? in ?)
2568        >get_arg_8_set_clock *)
2569       [1,2: cases (addressing_mode_ok ???? ∧ addressing_mode_ok ????) in MAP ⊢ ?
2570         [2,4: #abs @⊥ normalize in abs; destruct (abs)
2571         |*:whd in ⊢ (??%? → ?) #H <(option_destruct_Some ??? H)]
2572       [ change in ⊢ (? → %) with
2573        ((let 〈result,flags〉 ≝
2574          add_8_with_carry
2575           (get_arg_8 ? ps false ACC_A)
2576           (get_arg_8 ?
2577             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2578             false (DIRECT ARG2))
2579           ? in ?) = ?)
2580          >get_arg_8_set_low_internal_ram
2581       
2582        cases (add_8_with_carry ???)
2583         
2584        [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2585       #result #flags
2586       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
2587    (*| (* INC *) #arg1 #H1 #H2 #EQ %[@1]
2588       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2589       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2590       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2591       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2592       >H2b >(eq_instruction_to_eq … H2a)
2593       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2594       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG]
2595       [1,2,3,4: cases (half_add ???) #carry #result
2596       | cases (half_add ???) #carry #bl normalize nodelta;
2597         cases (full_add ????) #carry' #bu normalize nodelta ]
2598        #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) -newppc';
2599        [5: %
2600        |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program
2601      (set_program_counter pseudo_assembly_program ps newppc)
2602      (\fst  (ticks_of0 〈preamble,instr_list〉
2603                   (program_counter pseudo_assembly_program ps)
2604                   (Instruction (INC Identifier (DIRECT ARG))))
2605       +clock pseudo_assembly_program
2606        (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG))
2607        [2,3: // ]
2608            <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://]
2609            whd in ⊢ (??%%)
2610            cases (split bool 4 4 ARG)
2611            #nu' #nl'
2612            normalize nodelta
2613            cases (split bool 1 3 nu')
2614            #bit_1' #ignore'
2615            normalize nodelta
2616            cases (get_index_v bool 4 nu' ? ?)
2617            [ normalize nodelta (* HERE *) whd in ⊢ (??%%) %
2618            |
2619            ] *)
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