source: src/ASM/AssemblyProof.ma @ 1043

Last change on this file since 1043 was 1043, checked in by sacerdot, 8 years ago

Axiom commented out.

File size: 86.4 KB
Line 
1include "ASM/Assembly.ma".
2include "ASM/Interpret.ma".
3include "ASM/StatusProofs.ma".
4
5definition bit_elim_prop: ∀P: bool → Prop. Prop ≝
6  λP.
7    P true ∧ P false.
8 
9let rec bitvector_elim_prop_internal
10  (n: nat) (P: BitVector n → Prop) (m: nat) on m: m ≤ n → BitVector (n - m) → Prop ≝
11  match m return λm. m ≤ n → BitVector (n - m) → Prop with
12  [ O    ⇒ λprf1. λprefix. P ?
13  | S n' ⇒ λprf2. λprefix. bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' ? ?)
14  ].
15  [ applyS prefix
16  | letin res ≝ (bit ::: prefix)
17    < (minus_S_S ? ?)
18    > (minus_Sn_m ? ?)
19  [ @ res
20  | @ prf2
21  ]
22  | /2/
23  ].
24qed.
25
26definition bitvector_elim_prop ≝
27  λn: nat.
28  λP: BitVector n → Prop.
29    bitvector_elim_prop_internal n P n ? ?.
30  [ @ (le_n ?)
31  | < (minus_n_n ?)
32    @ [[ ]]
33  ]
34qed.
35
36lemma bool_eq_internal_eq:
37  ∀b, c.
38    (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c.
39  #b #c
40  cases b
41  [ normalize //
42  | normalize
43    cases c
44    [ normalize //
45    | normalize //
46    ]
47  ]
48qed.
49
50definition bit_elim: ∀P: bool → bool. bool ≝
51  λP.
52    P true ∧ P false.
53
54let rec bitvector_elim_internal
55  (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝
56  match m return λm. m ≤ n → BitVector (n - m) → bool with
57  [ O    ⇒ λprf1. λprefix. P ?
58  | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?)
59  ].
60  [ applyS prefix
61  | letin res ≝ (bit ::: prefix)
62    < (minus_S_S ? ?)
63    > (minus_Sn_m ? ?)
64    [ @ res
65    | @ prf2
66    ]
67  | /2/
68  ].
69qed.
70
71definition bitvector_elim ≝
72  λn: nat.
73  λP: BitVector n → bool.
74    bitvector_elim_internal n P n ? ?.
75  [ @ (le_n ?)
76  | < (minus_n_n ?)
77    @ [[ ]]
78  ]
79qed.
80
81lemma super_rewrite2:
82 ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m.
83  ∀P: ∀m. Vector A m → Prop.
84   n=m → v1 ≃ v2 → P n v1 → P m v2.
85 #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ //
86qed.
87
88lemma vector_cons_append:
89  ∀A: Type[0].
90  ∀n: nat.
91  ∀e: A.
92  ∀v: Vector A n.
93    e ::: v = [[ e ]] @@ v.
94  # A # N # E # V
95  elim V
96  [ normalize %
97  | # NN # AA # VV # IH
98    normalize
99    %
100qed.
101
102lemma vector_cons_append2:
103  ∀A: Type[0].
104  ∀n, m: nat.
105  ∀v: Vector A n.
106  ∀q: Vector A m.
107  ∀hd: A.
108    hd:::(v@@q) = (hd:::v)@@q.
109  #A #n #m #v #q
110  elim v
111  [ #hd %
112  | #n' #hd' #tl' #ih #hd' <ih %
113  ]
114qed.
115
116lemma jmeq_cons_vector_monotone:
117  ∀A: Type[0].
118  ∀m, n: nat.
119  ∀v: Vector A m.
120  ∀q: Vector A n.
121  ∀prf: m = n.
122  ∀hd: A.
123    v ≃ q → hd:::v ≃ hd:::q.
124  #A #m #n #v #q #prf #hd #E
125  @(super_rewrite2 A … E)
126  [ assumption | @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
1138lemma 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 (whd in ⊢ (% → ?) * #H2)
1163 try (whd in ⊢ (% → ?) * #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:
1189  ∀i1,i2. eq_instruction i1 i2 = true → i1 = i2.
1190               
1191
1192notation < "❰ x ❱" with precedence 90 for @{'dp $x $y }.
1193interpretation "dp" 'dp x y = (dp x y).
1194
1195lemma fetch_assembly_pseudo':
1196 ∀program.∀pol:policy program.∀ppc,lookup_labels,lookup_datalabels.
1197  ∀pi,code_memory,len,assembled,instructions,pc.
1198   let expansion ≝ pol ppc in
1199   Some ? instructions = expand_pseudo_instruction_safe lookup_labels lookup_datalabels (bitvector_of_nat ? pc) expansion pi →
1200    Some … 〈len,assembled〉 = assembly_1_pseudoinstruction_safe program pol ppc (bitvector_of_nat ? pc) lookup_labels lookup_datalabels pi →
1201     encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
1202      fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions.
1203 #program #pol #ppc #lookup_labels #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc
1204 #EQ1 whd in ⊢ (???% → ?) <EQ1 whd in ⊢ (???% → ?) #EQ2
1205 cases (pair_destruct ?????? (option_destruct_Some … EQ2)) -EQ2; #EQ2a #EQ2b
1206 >EQ2a >EQ2b -EQ2a EQ2b;
1207  generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc
1208  generalize in match pc elim instructions
1209  [ #pc whd in ⊢ (% → %) #H >H @eq_bv_refl
1210  | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd
1211    generalize in match (fetch_assembly pc i code_memory … (refl …) H1)
1212    cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %)
1213    cases newi_pc #newi #newpc whd in ⊢ (% → %) #K cases (conjunction_true … K) -K; #K1
1214    cases (conjunction_true … K1) -K1; #K1 #K2 #K3 % try %
1215    [ @K1 | @eqb_true_to_eq <(eq_instruction_to_eq … K1) @K2 | >(eq_bv_eq … K3) @IH @H2 ]
1216qed.
1217
1218axiom bitvector_of_nat_nat_of_bitvector:
1219  ∀n,v.
1220    bitvector_of_nat n (nat_of_bitvector n v) = v.
1221
1222(* CSC: soo long next one; can we merge with previous one now? *)
1223lemma fetch_assembly_pseudo:
1224 ∀program.∀pol:policy program.∀ppc.
1225  ∀code_memory.
1226   let lookup_labels ≝ ? in
1227   let lookup_datalabels ≝ ? in
1228   let pc ≝ ? in
1229   let pi ≝  \fst  (fetch_pseudo_instruction (\snd  program) ppc) in
1230   let instructions ≝ expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels pc (refl …) (refl …) (refl …) in
1231   let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels pi (refl …) (refl …) (refl …) in
1232     encoding_check code_memory pc (bitvector_of_nat … (nat_of_bitvector ? pc + len)) assembled →
1233      fetch_many code_memory (bitvector_of_nat … (nat_of_bitvector ? pc + len)) pc instructions.
1234 #program #pol #ppc #code_memory
1235 letin lookup_labels ≝ (λx:Identifier
1236    .sigma program pol (address_of_word_labels_code_mem (\snd  program) x))
1237 letin lookup_datalabels ≝
1238  (λx:BitVector 16
1239    .lookup Identifier 16 x (construct_datalabels (\fst  program)) (zero 16))
1240 letin pc ≝ (sigma program pol ppc)
1241 letin pi ≝ (\fst  (fetch_pseudo_instruction (\snd  program) ppc))
1242 letin instructions ≝ (expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels pc (refl …) (refl …) (refl …))
1243 @pair_elim' #len #assembled #EQ1 #H
1244 generalize in match
1245  (fetch_assembly_pseudo' program pol ppc lookup_labels lookup_datalabels pi
1246  code_memory len assembled instructions (nat_of_bitvector ? pc) ???) in ⊢ ?;
1247 [ >bitvector_of_nat_nat_of_bitvector //
1248 | >bitvector_of_nat_nat_of_bitvector normalize nodelta >(sig2 ?? (expand_pseudo_instruction …)) %
1249 | >bitvector_of_nat_nat_of_bitvector <EQ1 @assembly_1_pseudoinstruction_ok1
1250 | //]
1251qed.
1252
1253(* This is a trivial consequence of fetch_assembly + the proof that the
1254   function that load the code in memory is correct. The latter is based
1255   on missing properties from the standard library on the BitVectorTrie
1256   data structrure. *)
1257axiom assembly_ok:
1258 ∀program,pol,assembled,costs'.
1259  let 〈labels,costs〉 ≝ build_maps program pol in
1260  〈assembled,costs'〉 = assembly program pol →
1261  costs = costs' ∧
1262  let code_memory ≝ load_code_memory assembled in
1263  let preamble ≝ \fst program in
1264  let datalabels ≝ construct_datalabels preamble in
1265  let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in
1266  let lookup_datalabels ≝ λx. lookup ?? x datalabels (zero ?) in
1267  ∀ppc.
1268  let pi_newppc ≝ fetch_pseudo_instruction (\snd program) ppc in
1269   ∀len,assembledi.
1270     〈len,assembledi〉 = assembly_1_pseudoinstruction program pol ppc lookup_labels lookup_datalabels (\fst pi_newppc) (refl …) (refl …) (refl …) →
1271      encoding_check code_memory (sigma program pol ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len)) assembledi ∧
1272       sigma program pol (\snd pi_newppc) = bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len).
1273
1274lemma fetch_assembly_pseudo2:
1275 ∀program,pol,ppc.
1276(*  let 〈labels,costs〉 ≝ build_maps program pol in *)
1277  let assembled ≝ \fst (assembly program pol) in
1278  let code_memory ≝ load_code_memory assembled in
1279  let data_labels ≝ construct_datalabels (\fst program) in
1280  let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in
1281  let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in
1282  let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1283  let instructions ≝ expand_pseudo_instruction program pol ppc lookup_labels lookup_datalabels (sigma program pol ppc) (refl …) (refl …) (refl …) in
1284   fetch_many code_memory (sigma program pol newppc) (sigma program pol ppc) instructions.
1285 * #preamble #instr_list #pol #ppc
1286 letin assembled ≝ (\fst (assembly 〈preamble,instr_list〉 pol))
1287 letin code_memory ≝ (load_code_memory assembled)
1288 letin data_labels ≝ (construct_datalabels preamble)
1289 letin lookup_labels ≝ (λx. sigma … pol (address_of_word_labels_code_mem instr_list x))
1290 letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?))
1291 @pair_elim' #pi #newppc #EQ1 change in EQ1 with (fetch_pseudo_instruction instr_list ? = ?)
1292 generalize in match (assembly_ok ? pol assembled (\snd (assembly 〈preamble,instr_list〉 pol)))
1293 @pair_elim' #labels #costs #EQ2 <eq_pair_fst_snd
1294 #H cases (H (refl \ldots)) -H; #EQ3
1295 generalize in match (refl … (assembly_1_pseudoinstruction … pol ppc lookup_labels lookup_datalabels ? (refl …) (refl …) (refl …)))
1296 cases (assembly_1_pseudoinstruction ?????????) in ⊢ (???% → ?) #len #assembledi #EQ4
1297 #H cases (H ppc len assembledi ?) [2: <EQ4 %] -H; #H1 #H2
1298 generalize in match (fetch_assembly_pseudo … pol ppc (load_code_memory assembled)) in ⊢ ?
1299 >EQ4 #H generalize in match (H H1) in ⊢ ? -H; >(pair_destruct_2 ????? (sym_eq … EQ1))
1300 >H2 #K @K
1301qed.
1302
1303(* OLD?
1304definition assembly_specification:
1305  ∀assembly_program: pseudo_assembly_program.
1306  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1307  λpseudo_assembly_program.
1308  λcode_mem.
1309    ∀pc: Word.
1310      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1311      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1312      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1313      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1314      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1315       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1316      match pre_assembled with
1317       [ None ⇒ True
1318       | Some pc_code ⇒
1319          let 〈new_pc,code〉 ≝ pc_code in
1320           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1321
1322axiom assembly_meets_specification:
1323  ∀pseudo_assembly_program.
1324    match assembly pseudo_assembly_program with
1325    [ None ⇒ True
1326    | Some code_mem_cost ⇒
1327      let 〈code_mem, cost〉 ≝ code_mem_cost in
1328        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1329    ].
1330(*
1331  # PROGRAM
1332  [ cases PROGRAM
1333    # PREAMBLE
1334    # INSTR_LIST
1335    elim INSTR_LIST
1336    [ whd
1337      whd in ⊢ (∀_. %)
1338      # PC
1339      whd
1340    | # INSTR
1341      # INSTR_LIST_TL
1342      # H
1343      whd
1344      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1345    ]
1346  | cases not_implemented
1347  ] *)
1348*)
1349
1350definition internal_pseudo_address_map ≝ list (BitVector 8).
1351
1352axiom low_internal_ram_of_pseudo_low_internal_ram:
1353 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1354
1355axiom high_internal_ram_of_pseudo_high_internal_ram:
1356 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1357
1358axiom low_internal_ram_of_pseudo_internal_ram_hit:
1359 ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀pol:policy (code_memory … s).∀addr:BitVector 7.
1360  member ? (eq_bv 8) (false:::addr) M = true →
1361   let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1362   let pbl ≝ lookup ? 7 addr (low_internal_ram ? s) (zero 8) in
1363   let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram ? s) (zero 8) in
1364   let bl ≝ lookup ? 7 addr ram (zero 8) in
1365   let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in
1366    bu@@bl = sigma (code_memory … s) pol (pbu@@pbl).
1367
1368(* changed from add to sub *)
1369axiom low_internal_ram_of_pseudo_internal_ram_miss:
1370 ∀T.∀M:internal_pseudo_address_map.∀s:PreStatus T.∀addr:BitVector 7.
1371  let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1372  let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in
1373  let carr ≝ get_index_v ? ? flags 1 ? in
1374  carr = false →
1375  member ? (eq_bv 8) (false:::Saddr) M = false →
1376   member ? (eq_bv 8) (false:::addr) M = false →
1377    lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?).
1378  //
1379qed.
1380
1381definition addressing_mode_ok ≝
1382 λT.λM:internal_pseudo_address_map.λs:PreStatus T.
1383  λaddr:addressing_mode.
1384   match addr with
1385    [ DIRECT d ⇒
1386       ¬(member ? (eq_bv 8) d M) ∧
1387       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1388    | INDIRECT i ⇒
1389       let d ≝ get_register ? s [[false;false;i]] in
1390       ¬(member ? (eq_bv 8) d M) ∧
1391       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1392    | EXT_INDIRECT _ ⇒ true
1393    | REGISTER _ ⇒ true
1394    | ACC_A ⇒ true
1395    | ACC_B ⇒ true
1396    | DPTR ⇒ true
1397    | DATA _ ⇒ true
1398    | DATA16 _ ⇒ true
1399    | ACC_DPTR ⇒ true
1400    | ACC_PC ⇒ true
1401    | EXT_INDIRECT_DPTR ⇒ true
1402    | INDIRECT_DPTR ⇒ true
1403    | CARRY ⇒ true
1404    | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1405    | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1406    | RELATIVE _ ⇒ true
1407    | ADDR11 _ ⇒ true
1408    | ADDR16 _ ⇒ true ].
1409   
1410definition next_internal_pseudo_address_map0 ≝
1411  λT.
1412  λfetched.
1413  λM: internal_pseudo_address_map.
1414  λs: PreStatus T.
1415   match fetched with
1416    [ Comment _ ⇒ Some ? M
1417    | Cost _ ⇒ Some … M
1418    | Jmp _ ⇒ Some … M
1419    | Call _ ⇒
1420       Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M)
1421    | Mov _ _ ⇒ Some … M
1422    | Instruction instr ⇒
1423       match instr with
1424        [ ADD addr1 addr2 ⇒
1425           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1426            Some ? M
1427           else
1428            None ?
1429        | ADDC addr1 addr2 ⇒
1430           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1431            Some ? M
1432           else
1433            None ?
1434        | SUBB addr1 addr2 ⇒
1435           if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then
1436            Some ? M
1437           else
1438            None ?       
1439        | _ ⇒ (* TO BE COMPLETED *) Some ? M ]].
1440 
1441
1442definition next_internal_pseudo_address_map ≝
1443 λM:internal_pseudo_address_map.
1444  λs:PseudoStatus.
1445    next_internal_pseudo_address_map0 ?
1446     (\fst (fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s))) M s.
1447   
1448definition status_of_pseudo_status:
1449 internal_pseudo_address_map → ∀ps:PseudoStatus. policy (code_memory … ps) → Status ≝
1450 λM,ps,pol.
1451  let pap ≝ code_memory … ps in
1452  let p ≝ assembly pap pol in
1453  let cm ≝ load_code_memory (\fst p) in
1454  let pc ≝ sigma pap pol (program_counter ? ps) in
1455  let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in
1456  let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in
1457     mk_PreStatus (BitVectorTrie Byte 16)
1458      cm
1459      lir
1460      hir
1461      (external_ram … ps)
1462      pc
1463      (special_function_registers_8051 … ps)
1464      (special_function_registers_8052 … ps)
1465      (p1_latch … ps)
1466      (p3_latch … ps)
1467      (clock … ps).
1468
1469(*
1470definition write_at_stack_pointer':
1471 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1472  λM: Type[0].
1473  λs: PreStatus M.
1474  λv: Byte.
1475    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1476    let bit_zero ≝ get_index_v… nu O ? in
1477    let bit_1 ≝ get_index_v… nu 1 ? in
1478    let bit_2 ≝ get_index_v… nu 2 ? in
1479    let bit_3 ≝ get_index_v… nu 3 ? in
1480      if bit_zero then
1481        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1482                              v (low_internal_ram ? s) in
1483          set_low_internal_ram ? s memory
1484      else
1485        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1486                              v (high_internal_ram ? s) in
1487          set_high_internal_ram ? s memory.
1488  [ cases l0 %
1489  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1490qed.
1491
1492definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1493 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1494
1495  λticks_of.
1496  λs.
1497  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1498  let ticks ≝ ticks_of (program_counter ? s) in
1499  let s ≝ set_clock ? s (clock ? s + ticks) in
1500  let s ≝ set_program_counter ? s pc in
1501    match instr with
1502    [ Instruction instr ⇒
1503       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1504    | Comment cmt ⇒ s
1505    | Cost cst ⇒ s
1506    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1507    | Call call ⇒
1508      let a ≝ address_of_word_labels s call in
1509      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1510      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1511      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1512      let s ≝ write_at_stack_pointer' ? s pc_bl in
1513      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1514      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1515      let s ≝ write_at_stack_pointer' ? s pc_bu in
1516        set_program_counter ? s a
1517    | Mov dptr ident ⇒
1518       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1519    ].
1520 [
1521 |2,3,4: %
1522 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1523 |
1524 | %
1525 ]
1526 cases not_implemented
1527qed.
1528*)
1529
1530axiom execute_1_pseudo_instruction_preserves_code_memory:
1531 ∀ticks_of,ps.
1532  code_memory … (execute_1_pseudo_instruction ticks_of ps) = code_memory … ps.
1533
1534(*
1535lemma execute_code_memory_unchanged:
1536 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1537 #ticks #ps whd in ⊢ (??? (??%))
1538 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1539  (program_counter pseudo_assembly_program ps)) #instr #pc
1540 whd in ⊢ (??? (??%)) cases instr
1541  [ #pre cases pre
1542     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1543       cases (split ????) #z1 #z2 %
1544     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1545       cases (split ????) #z1 #z2 %
1546     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1547       cases (split ????) #z1 #z2 %
1548     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1549       [ #x1 whd in ⊢ (??? (??%))
1550     | *: cases not_implemented
1551     ]
1552  | #comment %
1553  | #cost %
1554  | #label %
1555  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1556    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1557    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1558    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1559    (* CSC: ??? *)
1560  | #dptr #label (* CSC: ??? *)
1561  ]
1562  cases not_implemented
1563qed.
1564*)
1565
1566(* DEAD CODE?
1567lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1568 ∀M:internal_pseudo_address_map.
1569 ∀ps,ps': PseudoStatus.
1570 ∀pol.
1571  ∀prf:code_memory … ps = code_memory … ps'.
1572   let pol' ≝ ? in
1573   match status_of_pseudo_status M ps pol with
1574    [ None ⇒ status_of_pseudo_status M ps' pol' = None …
1575    | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w
1576    ].
1577 [2: <prf @pol]
1578 #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1579 generalize in match (refl … (assembly (code_memory … ps) pol))
1580 cases (assembly ??) in ⊢ (???% → %)
1581  [ #K whd whd in ⊢ (??%?) <H >K %
1582  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1583qed.
1584*)
1585
1586definition ticks_of0: ∀p:pseudo_assembly_program. policy p → Word → ? → nat × nat ≝
1587  λprogram: pseudo_assembly_program.λpol.
1588  λppc: Word.
1589  λfetched.
1590    match fetched with
1591    [ Instruction instr ⇒
1592      match instr with
1593      [ JC lbl ⇒
1594        match pol ppc with
1595        [ short_jump ⇒ 〈2, 2〉
1596        | medium_jump ⇒ ?
1597        | long_jump ⇒ 〈4, 4〉
1598        ]
1599      | JNC lbl ⇒
1600        match pol ppc with
1601        [ short_jump ⇒ 〈2, 2〉
1602        | medium_jump ⇒ ?
1603        | long_jump ⇒ 〈4, 4〉
1604        ]
1605      | JB bit lbl ⇒
1606        match pol ppc with
1607        [ short_jump ⇒ 〈2, 2〉
1608        | medium_jump ⇒ ?
1609        | long_jump ⇒ 〈4, 4〉
1610        ]
1611      | JNB bit lbl ⇒
1612        match pol ppc with
1613        [ short_jump ⇒ 〈2, 2〉
1614        | medium_jump ⇒ ?
1615        | long_jump ⇒ 〈4, 4〉
1616        ]
1617      | JBC bit lbl ⇒
1618        match pol ppc with
1619        [ short_jump ⇒ 〈2, 2〉
1620        | medium_jump ⇒ ?
1621        | long_jump ⇒ 〈4, 4〉
1622        ]
1623      | JZ lbl ⇒
1624        match pol ppc with
1625        [ short_jump ⇒ 〈2, 2〉
1626        | medium_jump ⇒ ?
1627        | long_jump ⇒ 〈4, 4〉
1628        ]
1629      | JNZ lbl ⇒
1630        match pol ppc with
1631        [ short_jump ⇒ 〈2, 2〉
1632        | medium_jump ⇒ ?
1633        | long_jump ⇒ 〈4, 4〉
1634        ]
1635      | CJNE arg lbl ⇒
1636        match pol ppc with
1637        [ short_jump ⇒ 〈2, 2〉
1638        | medium_jump ⇒ ?
1639        | long_jump ⇒ 〈4, 4〉
1640        ]
1641      | DJNZ arg lbl ⇒
1642        match pol ppc with
1643        [ short_jump ⇒ 〈2, 2〉
1644        | medium_jump ⇒ ?
1645        | long_jump ⇒ 〈4, 4〉
1646        ]
1647      | ADD arg1 arg2 ⇒
1648        let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in
1649         〈ticks, ticks〉
1650      | ADDC arg1 arg2 ⇒
1651        let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in
1652         〈ticks, ticks〉
1653      | SUBB arg1 arg2 ⇒
1654        let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in
1655         〈ticks, ticks〉
1656      | INC arg ⇒
1657        let ticks ≝ ticks_of_instruction (INC ? arg) in
1658         〈ticks, ticks〉
1659      | DEC arg ⇒
1660        let ticks ≝ ticks_of_instruction (DEC ? arg) in
1661         〈ticks, ticks〉
1662      | MUL arg1 arg2 ⇒
1663        let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in
1664         〈ticks, ticks〉
1665      | DIV arg1 arg2 ⇒
1666        let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in
1667         〈ticks, ticks〉
1668      | DA arg ⇒
1669        let ticks ≝ ticks_of_instruction (DA ? arg) in
1670         〈ticks, ticks〉
1671      | ANL arg ⇒
1672        let ticks ≝ ticks_of_instruction (ANL ? arg) in
1673         〈ticks, ticks〉
1674      | ORL arg ⇒
1675        let ticks ≝ ticks_of_instruction (ORL ? arg) in
1676         〈ticks, ticks〉
1677      | XRL arg ⇒
1678        let ticks ≝ ticks_of_instruction (XRL ? arg) in
1679         〈ticks, ticks〉
1680      | CLR arg ⇒
1681        let ticks ≝ ticks_of_instruction (CLR ? arg) in
1682         〈ticks, ticks〉
1683      | CPL arg ⇒
1684        let ticks ≝ ticks_of_instruction (CPL ? arg) in
1685         〈ticks, ticks〉
1686      | RL arg ⇒
1687        let ticks ≝ ticks_of_instruction (RL ? arg) in
1688         〈ticks, ticks〉
1689      | RLC arg ⇒
1690        let ticks ≝ ticks_of_instruction (RLC ? arg) in
1691         〈ticks, ticks〉
1692      | RR arg ⇒
1693        let ticks ≝ ticks_of_instruction (RR ? arg) in
1694         〈ticks, ticks〉
1695      | RRC arg ⇒
1696        let ticks ≝ ticks_of_instruction (RRC ? arg) in
1697         〈ticks, ticks〉
1698      | SWAP arg ⇒
1699        let ticks ≝ ticks_of_instruction (SWAP ? arg) in
1700         〈ticks, ticks〉
1701      | MOV arg ⇒
1702        let ticks ≝ ticks_of_instruction (MOV ? arg) in
1703         〈ticks, ticks〉
1704      | MOVX arg ⇒
1705        let ticks ≝ ticks_of_instruction (MOVX ? arg) in
1706         〈ticks, ticks〉
1707      | SETB arg ⇒
1708        let ticks ≝ ticks_of_instruction (SETB ? arg) in
1709         〈ticks, ticks〉
1710      | PUSH arg ⇒
1711        let ticks ≝ ticks_of_instruction (PUSH ? arg) in
1712         〈ticks, ticks〉
1713      | POP arg ⇒
1714        let ticks ≝ ticks_of_instruction (POP ? arg) in
1715         〈ticks, ticks〉
1716      | XCH arg1 arg2 ⇒
1717        let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in
1718         〈ticks, ticks〉
1719      | XCHD arg1 arg2 ⇒
1720        let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in
1721         〈ticks, ticks〉
1722      | RET ⇒
1723        let ticks ≝ ticks_of_instruction (RET ?) in
1724         〈ticks, ticks〉
1725      | RETI ⇒
1726        let ticks ≝ ticks_of_instruction (RETI ?) in
1727         〈ticks, ticks〉
1728      | NOP ⇒
1729        let ticks ≝ ticks_of_instruction (NOP ?) in
1730         〈ticks, ticks〉
1731      ]
1732    | Comment comment ⇒ 〈0, 0〉
1733    | Cost cost ⇒ 〈0, 0〉
1734    | Jmp jmp ⇒ 〈2, 2〉
1735    | Call call ⇒ 〈2, 2〉
1736    | Mov dptr tgt ⇒ 〈2, 2〉
1737    ].
1738  cases not_implemented (* policy returned medium_jump for conditional jumping = impossible *)
1739qed.
1740
1741definition ticks_of: ∀p:pseudo_assembly_program. policy p → Word → nat × nat ≝
1742  λprogram: pseudo_assembly_program.λpol.
1743  λppc: Word.
1744    let 〈preamble, pseudo〉 ≝ program in
1745    let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in
1746     ticks_of0 program pol ppc fetched.
1747
1748lemma eq_rect_Type1_r:
1749  ∀A: Type[1].
1750  ∀a:A.
1751  ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p.
1752  #A #a #P #H #x #p
1753  generalize in match H
1754  generalize in match P
1755  cases p
1756  //
1757qed.
1758
1759axiom split_append:
1760  ∀A: Type[0].
1761  ∀m, n: nat.
1762  ∀v, v': Vector A m.
1763  ∀q, q': Vector A n.
1764    let 〈v', q'〉 ≝ split A m n (v@@q) in
1765      v = v' ∧ q = q'.
1766
1767axiom split_vector_singleton:
1768  ∀A: Type[0].
1769  ∀n: nat.
1770  ∀v: Vector A (S n).
1771  ∀rest: Vector A n.
1772  ∀s: Vector A 1.
1773  ∀prf.
1774    v = s @@ rest →
1775    ((get_index_v A ? v 0 prf) ::: rest) = v.
1776
1777example sub_minus_one_seven_eight:
1778  ∀v: BitVector 7.
1779  false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) =
1780  \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false).
1781 cases daemon.
1782qed.
1783
1784(*
1785lemma blah:
1786  ∀m: internal_pseudo_address_map.
1787  ∀s: PseudoStatus.
1788  ∀arg: Byte.
1789  ∀b: bool.
1790    addressing_mode_ok m s (DIRECT arg) = true →
1791      get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) =
1792      get_arg_8 ? s b (DIRECT arg).
1793  [2, 3: normalize % ]
1794  #m #s #arg #b #hyp
1795  whd in ⊢ (??%%)
1796  @split_elim''
1797  #nu' #nl' #arg_nu_nl_eq
1798  normalize nodelta
1799  generalize in match (refl ? (get_index_v bool 4 nu' ? ?))
1800  cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %)
1801  #get_index_v_eq
1802  normalize nodelta
1803  [2:
1804    normalize nodelta
1805    @split_elim''
1806    #bit_one' #three_bits' #bit_one_three_bit_eq
1807    generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl'))
1808    normalize nodelta
1809    generalize in match (refl ? (sub_7_with_carry ? ? ?))
1810    cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %)
1811    #Saddr #carr' #Saddr_carr_eq
1812    normalize nodelta
1813    #carr_hyp'
1814    @carr_hyp'
1815    [1:
1816    |2: whd in hyp:(??%?); generalize in match hyp; -hyp;
1817        generalize in match (refl ? (¬(member (BitVector 8) ? arg m)))
1818        cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %)
1819        #member_eq
1820        normalize nodelta
1821        [2: #destr destruct(destr)
1822        |1: -carr_hyp';
1823            >arg_nu_nl_eq
1824            <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq)
1825            [1: >get_index_v_eq in ⊢ (??%? → ?)
1826            |2: @le_S @le_S @le_S @le_n
1827            ]
1828            cases (member (BitVector 8) ? (\fst ?) ?)
1829            [1: #destr normalize in destr; destruct(destr)
1830            |2:
1831            ]
1832        ]
1833    |3: >get_index_v_eq in ⊢ (??%?)
1834        change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl')
1835        >(split_vector_singleton … bit_one_three_bit_eq)
1836        <arg_nu_nl_eq
1837        whd in hyp:(??%?)
1838        cases (member (BitVector 8) (eq_bv 8) arg m) in hyp
1839        normalize nodelta [*: #ignore @sym_eq ]
1840    ]
1841  |
1842  ].
1843*)
1844(*
1845map_address0 ... (DIRECT arg) = Some .. →
1846  get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) =
1847  get_arg_8 (internal_ram ...) (DIRECT arg)
1848
1849((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) 
1850                     then Some internal_pseudo_address_map M 
1851                     else None internal_pseudo_address_map )
1852                    =Some internal_pseudo_address_map M')
1853*)
1854
1855axiom low_internal_ram_write_at_stack_pointer:
1856 ∀T1,T2,M,s1,s2,s3.∀pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
1857  get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP →
1858  low_internal_ram ? s2 = low_internal_ram T2 s3 →
1859  sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) →
1860  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
1861  bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) →
1862   low_internal_ram T1
1863     (write_at_stack_pointer ?
1864       (set_8051_sfr ?
1865         (write_at_stack_pointer ?
1866           (set_8051_sfr ?
1867             (set_low_internal_ram ? s1
1868               (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram ? s2)))
1869             SFR_SP sp1)
1870          bl)
1871        SFR_SP sp2)
1872      bu)
1873   = low_internal_ram_of_pseudo_low_internal_ram (sp1::M)
1874      (low_internal_ram ?
1875       (write_at_stack_pointer ?
1876         (set_8051_sfr ?
1877           (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl)
1878          SFR_SP sp2)
1879        pbu)).
1880
1881axiom high_internal_ram_write_at_stack_pointer:
1882 ∀T1,T2,M,s1,s2,s3,pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
1883  get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP →
1884  high_internal_ram ? s2 = high_internal_ram T2 s3 →
1885  sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) →
1886  sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) →
1887  bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) →
1888   high_internal_ram T1
1889     (write_at_stack_pointer ?
1890       (set_8051_sfr ?
1891         (write_at_stack_pointer ?
1892           (set_8051_sfr ?
1893             (set_high_internal_ram ? s1
1894               (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram ? s2)))
1895             SFR_SP sp1)
1896          bl)
1897        SFR_SP sp2)
1898      bu)
1899   = high_internal_ram_of_pseudo_high_internal_ram (sp1::M)
1900      (high_internal_ram ?
1901       (write_at_stack_pointer ?
1902         (set_8051_sfr ?
1903           (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl)
1904          SFR_SP sp2)
1905        pbu)).
1906
1907lemma Some_Some_elim:
1908 ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P.
1909 #T #x #y #P #H #K @H @option_destruct_Some //
1910qed.
1911
1912theorem main_thm:
1913 ∀M,M',ps,pol.
1914  next_internal_pseudo_address_map M ps = Some … M' →
1915   ∃n.
1916      execute n (status_of_pseudo_status M ps pol)
1917    = status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps) ?.
1918 [2: >execute_1_pseudo_instruction_preserves_code_memory @pol]
1919 #M #M' #ps #pol #SAFE
1920 cut
1921  (∀ps'.
1922    ∀prf:ps'=execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps.
1923    ∃n. execute n (status_of_pseudo_status M ps pol) = status_of_pseudo_status M' ps' ?)
1924 [ >prf >execute_1_pseudo_instruction_preserves_code_memory @pol |3: #K @(K ? (refl …))]
1925 #ps' #EQ
1926 whd in ⊢ (??(λ_.??(??%)?))
1927 change with
1928  (∃n.
1929    execute n
1930     (set_low_internal_ram ?
1931       (set_high_internal_ram ?
1932         (set_program_counter ?
1933           (set_code_memory ?? ps (load_code_memory ?))
1934          (sigma ? pol (program_counter ? ps)))
1935        (high_internal_ram_of_pseudo_high_internal_ram M ?))
1936      (low_internal_ram_of_pseudo_low_internal_ram M ?))
1937   = set_low_internal_ram ?
1938      (set_high_internal_ram ?
1939        (set_program_counter ?
1940          (set_code_memory ?? ? (load_code_memory ?))
1941         (sigma ???)) ?) ?)
1942 >EQ whd in match eq_rect_Type0_r normalize nodelta
1943 >execute_1_pseudo_instruction_preserves_code_memory normalize nodelta
1944 generalize in match EQ -EQ;
1945 generalize in match (refl … (code_memory pseudo_assembly_program ps))
1946 generalize in match pol -pol; generalize in ⊢ (∀_.??%? → ?)
1947 * #preamble #instr_list #pol #EQ1 generalize in match pol -pol <EQ1 #pol #EQps' <EQps'
1948 (* Dependent types madness ends here *)
1949 letin ppc ≝ (program_counter … ps)
1950 generalize in match (fetch_assembly_pseudo2 ? pol ppc) in ⊢ ?
1951 letin assembled ≝ (\fst (assembly ? pol))
1952 letin lookup_labels ≝ (λx. sigma ? pol (address_of_word_labels_code_mem instr_list x))
1953 letin lookup_datalabels ≝ (λx. lookup … x (construct_datalabels preamble) (zero 16))
1954 @pair_elim' #pi #newppc #EQ2
1955 letin instructions ≝
1956  (expand_pseudo_instruction ? pol ppc lookup_labels lookup_datalabels
1957    (sigma ? pol ppc) (refl …) (refl …) (refl …))
1958 change with (fetch_many ???? → ?) #H1
1959 change in EQ2 with (fetch_pseudo_instruction instr_list ppc = ?)
1960 change in SAFE with (next_internal_pseudo_address_map0 ???? = ?) <EQ1 in SAFE;
1961 >EQ2 whd in ⊢ (??(??%??)? → ?) #SAFE
1962 whd in EQps':(???%); <EQ1 in EQps'; >EQ2 normalize nodelta
1963 generalize in match H1; -H1; generalize in match instructions -instructions
1964 * #instructions >EQ2 change in match (\fst 〈pi,newppc〉) with pi
1965 whd in match ticks_of normalize nodelta <EQ1 >EQ2
1966 cases pi in SAFE
1967  [2,3: (* Comment, Cost *) #ARG whd in ⊢ (??%? → ???% → ?)
1968   @Some_Some_elim #MAP #EQ3 @(Some_Some_elim ????? EQ3) #EQ3'
1969   whd in match eject normalize nodelta >EQ3' in ⊢ (% → ?) whd in ⊢ (% → ?)
1970   #H2 #EQ %[1,3:@0]
1971   <MAP >(eq_bv_eq … H2) >EQ %
1972  |4: (* Jmp *) #label whd in ⊢ (??%? → ???% → ?)
1973   @Some_Some_elim #MAP cases (pol ?) normalize nodelta
1974       [3: (* long *) #EQ3 @(Some_Some_elim ????? EQ3) #EQ3'
1975         whd in match eject normalize nodelta >EQ3' in ⊢ (% → ?) whd in ⊢ (% → ?)
1976         @pair_elim' * #instr #newppc' #ticks #EQ4       
1977         * * #H2a #H2b whd in ⊢ (% → ?) #H2
1978         >H2b >(eq_instruction_to_eq … H2a)
1979         #EQ %[@1]
1980         <MAP >(eq_bv_eq … H2) >EQ
1981         whd in ⊢ (??%?) >EQ4 whd in ⊢ (??%?)
1982         cases ps in EQ4; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXX >XXX %
1983         whd in ⊢ (??%?)
1984         whd in ⊢ (??(match ?%? with [_ ⇒ ?])?)
1985         cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
1986  |6: (* Mov *) #arg1 #arg2
1987       #H1 #H2 #EQ %[@1]
1988       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
1989       change in ⊢ (? → ??%?) with (execute_1_0 ??)
1990       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
1991       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
1992       >H2b >(eq_instruction_to_eq … H2a)
1993       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
1994       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?)
1995       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
1996       normalize nodelta;
1997       [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
1998       #result #flags
1999       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
2000  |5: (* Call *) #label #MAP
2001      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP;
2002      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2003       [ (* short *) #abs @⊥ destruct (abs)
2004       |3: (* long *) #H1 #H2 #EQ %[@1]
2005           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2006           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2007           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2008           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2009           >H2b >(eq_instruction_to_eq … H2a)
2010           generalize in match EQ; -EQ;
2011           whd in ⊢ (???% → ??%?);
2012           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;
2013           >(eq_bv_eq … H2c)
2014           change with
2015            ((?=let 〈ppc_bu,ppc_bl〉 ≝ split bool 8 8 newppc in ?) →
2016                (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2017           generalize in match (refl … (split … 8 8 newppc)) cases (split bool 8 8 newppc) in ⊢ (??%? → %) #ppc_bu #ppc_bl #EQppc
2018           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;
2019           >get_8051_sfr_write_at_stack_pointer >get_8051_sfr_write_at_stack_pointer
2020           >get_8051_sfr_set_8051_sfr >get_8051_sfr_set_8051_sfr
2021           generalize in match (refl … (half_add ? new_sp (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry' #new_sp' #EQ2 normalize nodelta;
2022           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2023           @split_eq_status;
2024            [ >code_memory_write_at_stack_pointer whd in ⊢ (??%?)
2025              >code_memory_write_at_stack_pointer %
2026            | >set_program_counter_set_low_internal_ram
2027              >set_clock_set_low_internal_ram
2028              @low_internal_ram_write_at_stack_pointer
2029               [ >EQ0 @pol | % | %
2030               | @(pair_destruct_2 … EQ1)
2031               | @(pair_destruct_2 … EQ2)
2032               | >(pair_destruct_1 ????? EQpc)
2033                 >(pair_destruct_2 ????? EQpc)
2034                 @split_elim #x #y #H <H -x y H;
2035                 >(pair_destruct_1 ????? EQppc)
2036                 >(pair_destruct_2 ????? EQppc)
2037                 @split_elim #x #y #H <H -x y H;
2038                 >EQ0 % ]
2039            | >set_low_internal_ram_set_high_internal_ram
2040              >set_program_counter_set_high_internal_ram
2041              >set_clock_set_high_internal_ram
2042              @high_internal_ram_write_at_stack_pointer
2043               [ >EQ0 @pol | % | %
2044               | @(pair_destruct_2 … EQ1)
2045               | @(pair_destruct_2 … EQ2)
2046               | >(pair_destruct_1 ????? EQpc)
2047                 >(pair_destruct_2 ????? EQpc)
2048                 @split_elim #x #y #H <H -x y H;
2049                 >(pair_destruct_1 ????? EQppc)
2050                 >(pair_destruct_2 ????? EQppc)
2051                 @split_elim #x #y #H <H -x y H;
2052                 >EQ0 % ]           
2053            | >external_ram_write_at_stack_pointer whd in ⊢ (??%?)
2054              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
2055              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
2056              >external_ram_write_at_stack_pointer %
2057            | change with (? = sigma ?? (address_of_word_labels_code_mem (\snd (code_memory ? ps)) ?))
2058              >EQ0 %
2059            | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (??%?)
2060              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
2061              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
2062              >special_function_registers_8051_write_at_stack_pointer %
2063            | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (??%?)
2064              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
2065              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
2066              >special_function_registers_8052_write_at_stack_pointer %
2067            | >p1_latch_write_at_stack_pointer whd in ⊢ (??%?)
2068              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
2069              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
2070              >p1_latch_write_at_stack_pointer %
2071            | >p3_latch_write_at_stack_pointer whd in ⊢ (??%?)
2072              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
2073              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
2074              >p3_latch_write_at_stack_pointer %
2075            | >clock_write_at_stack_pointer whd in ⊢ (??%?)
2076              >clock_write_at_stack_pointer whd in ⊢ (???%)
2077              >clock_write_at_stack_pointer whd in ⊢ (???%)
2078              >clock_write_at_stack_pointer %]
2079       (*| (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2080         @pair_elim' #fst_5_addr #rest_addr #EQ1
2081         @pair_elim' #fst_5_pc #rest_pc #EQ2
2082         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2083         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2084         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2085         change in ⊢ (? →??%?) with (execute_1_0 ??)
2086         @pair_elim' * #instr #newppc' #ticks #EQn
2087          * * #H2a #H2b whd in ⊢ (% → ?) #H2c >H2b >(eq_instruction_to_eq … H2a) whd in ⊢ (??%?)
2088          generalize in match EQ; -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2089          @pair_elim' #carry #new_sp change with (half_add ? (get_8051_sfr ? ps ?) ? = ? → ?) #EQ4
2090          @split_elim' #pc_bu #pc_bl >program_counter_set_8051_sfr XXX change with (newppc = ?) #EQ5
2091          @pair_elim' #carry' #new_sp' #EQ6 normalize nodelta; #EQx >EQx -EQx;
2092          change in ⊢ (??(match ????% with [_ ⇒ ?])?) with (sigma … newppc)
2093          @split_elim' #pc_bu' #pc_bl' #EQ7 change with (newppc' = ? → ?)
2094          >get_8051_sfr_set_8051_sfr
2095         
2096          whd in EQ:(???%) ⊢ ? >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2097           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?)
2098           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc)))
2099           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2100           generalize in match (refl … (split bool 4 4 pc_bu))
2101           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2102           generalize in match (refl … (split bool 3 8 rest_addr))
2103           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2104           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2105           generalize in match
2106            (refl …
2107             (half_add 16 (sigma 〈preamble,instr_list〉 newppc)
2108             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2109           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7
2110           @split_eq_status try %
2111            [ change with (? = sigma ? (address_of_word_labels ps label))
2112              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2113            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2114              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] *)]
2115  |4: (* Jmp *) #label #MAP
2116      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP >MAP -MAP;
2117      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
2118       [3: (* long *) #H1 #H2 #EQ %[@1]
2119           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2120           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2121           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2122           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2123           >H2b >(eq_instruction_to_eq … H2a)
2124           generalize in match EQ; -EQ;
2125           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2126           cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
2127       |1: (* short *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2128           generalize in match
2129            (refl ?
2130             (sub_16_with_carry
2131              (sigma 〈preamble,instr_list〉 pol (program_counter … ps))
2132              (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))
2133              false))
2134           cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta;
2135           generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta;
2136           generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta;
2137           #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)]
2138           generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2139           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2140           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2141           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2142           >H2b >(eq_instruction_to_eq … H2a)
2143           generalize in match EQ; -EQ;
2144           whd in ⊢ (???% → ?);
2145           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
2146           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ???) ? in ?) = ?)
2147           generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) (sign_extension lower)))
2148           cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4
2149           @split_eq_status try % change with (newpc = sigma ?? (address_of_word_labels ps label))
2150           (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2151       | (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2152         generalize in match
2153          (refl …
2154            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))))
2155         cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1
2156         generalize in match
2157          (refl …
2158            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (program_counter … ps))))
2159         cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2
2160         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2161         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2162         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2163         change in ⊢ (? →??%?) with (execute_1_0 ??)
2164           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2165           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2166           >H2b >(eq_instruction_to_eq … H2a)
2167           generalize in match EQ; -EQ;
2168           whd in ⊢ (???% → ?);
2169           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
2170           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
2171           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)))
2172           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2173           generalize in match (refl … (split bool 4 4 pc_bu))
2174           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2175           generalize in match (refl … (split bool 3 8 rest_addr))
2176           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2177           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2178           generalize in match
2179            (refl …
2180             (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc)
2181             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2182           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7   
2183           @split_eq_status try %
2184            [ change with (? = sigma ?? (address_of_word_labels ps label))
2185              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2186            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2187              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]]
2188  | (* Instruction *) -pi;  whd in ⊢ (? → ??%? → ?) *; normalize nodelta;
2189    [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1]
2190       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2191       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2192       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2193       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2194       >H2b >(eq_instruction_to_eq … H2a)
2195       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP;
2196       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1;
2197       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2198       normalize nodelta; #MAP;
2199       [1: change in ⊢ (? → %) with
2200        ((let 〈result,flags〉 ≝
2201          add_8_with_carry
2202           (get_arg_8 ? ps false ACC_A)
2203           (get_arg_8 ?
2204             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2205             false (DIRECT ARG2))
2206           ? in ?) = ?)
2207        [2,3: %]
2208        change in ⊢ (???% → ?) with
2209         (let 〈result,flags〉 ≝ add_8_with_carry ?(*(get_arg_8 ? ps false ACC_A)*) ?? in ?)
2210        >get_arg_8_set_clock
2211       [1,2: cases (addressing_mode_ok ???? ∧ addressing_mode_ok ????) in MAP ⊢ ?
2212         [2,4: #abs @⊥ normalize in abs; destruct (abs)
2213         |*:whd in ⊢ (??%? → ?) #H <(option_destruct_Some ??? H)]
2214       [ change in ⊢ (? → %) with
2215        ((let 〈result,flags〉 ≝
2216          add_8_with_carry
2217           (get_arg_8 ? ps false ACC_A)
2218           (get_arg_8 ?
2219             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2220             false (DIRECT ARG2))
2221           ? in ?) = ?)
2222          >get_arg_8_set_low_internal_ram
2223       
2224        cases (add_8_with_carry ???)
2225         
2226        [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2227       #result #flags
2228       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
2229    | (* INC *) #arg1 #H1 #H2 #EQ %[@1]
2230       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2231       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2232       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2233       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2234       >H2b >(eq_instruction_to_eq … H2a)
2235       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2236       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG]
2237       [1,2,3,4: cases (half_add ???) #carry #result
2238       | cases (half_add ???) #carry #bl normalize nodelta;
2239         cases (full_add ????) #carry' #bu normalize nodelta ]
2240        #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) -newppc';
2241        [5: %
2242        |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program
2243      (set_program_counter pseudo_assembly_program ps newppc)
2244      (\fst  (ticks_of0 〈preamble,instr_list〉
2245                   (program_counter pseudo_assembly_program ps)
2246                   (Instruction (INC Identifier (DIRECT ARG))))
2247       +clock pseudo_assembly_program
2248        (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG))
2249        [2,3: // ]
2250            <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://]
2251            whd in ⊢ (??%%)
2252            cases (split bool 4 4 ARG)
2253            #nu' #nl'
2254            normalize nodelta
2255            cases (split bool 1 3 nu')
2256            #bit_1' #ignore'
2257            normalize nodelta
2258            cases (get_index_v bool 4 nu' ? ?)
2259            [ normalize nodelta (* HERE *) whd in ⊢ (??%%) %
2260            |
2261            ]
2262cases daemon (* EASY CASES TO BE COMPLETED *)
2263qed.
Note: See TracBrowser for help on using the repository browser.