source: src/ASM/AssemblyProof.ma @ 1983

Last change on this file since 1983 was 1983, checked in by mulligan, 7 years ago

Changes to simplify the simpler cases of the main_lemma.

File size: 75.8 KB
Line 
1include "ASM/Assembly.ma".
2include "ASM/Interpret.ma".
3include "ASM/StatusProofs.ma".
4include alias "arithmetics/nat.ma".
5
6definition bit_elim_prop: ∀P: bool → Prop. Prop ≝
7  λP.
8    P true ∧ P false.
9 
10let rec bitvector_elim_prop_internal
11  (n: nat) (P: BitVector n → Prop) (m: nat)
12    on m:
13      m ≤ n → BitVector (n - m) → Prop ≝
14  match m return λm. m ≤ n → BitVector (n - m) → Prop with
15  [ O    ⇒ λprf1. λprefix. P ?
16  | S n' ⇒ λprf2. λprefix.
17      bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' …)
18  ].
19  try applyS prefix
20  try (@le_S_to_le assumption)
21  letin res ≝ (bit ::: prefix)
22  <minus_S_S >minus_Sn_m
23  assumption
24qed.
25
26definition bitvector_elim_prop ≝
27  λn: nat.
28  λP: BitVector n → Prop.
29    bitvector_elim_prop_internal n P n ? ?.
30  try @le_n
31  <minus_n_n @[[ ]]
32qed.
33
34lemma bool_eq_internal_eq:
35  ∀b, c.
36    (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c.
37  #b #c
38  cases b cases c normalize nodelta
39  try (#_ % @I)
40  #assm destruct %
41qed.
42
43definition bit_elim: ∀P: bool → bool. bool ≝
44  λP.
45    P true ∧ P false.
46
47let rec bitvector_elim_internal
48  (n: nat) (P: BitVector n → bool) (m: nat)
49    on m:
50      m ≤ n → BitVector (n - m) → bool ≝
51  match m return λm. m ≤ n → BitVector (n - m) → bool with
52  [ O    ⇒ λprf1. λprefix. P ?
53  | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?)
54  ].
55  /2/
56qed.
57
58definition bitvector_elim ≝
59  λn: nat.
60  λP: BitVector n → bool.
61    bitvector_elim_internal n P n ? ?.
62  try @le_n
63  <minus_n_n @[[]]
64qed.
65
66lemma super_rewrite2:
67  ∀A:Type[0].
68  ∀n, m: nat.
69  ∀v1: Vector A n.
70  ∀v2: Vector A m.
71  ∀P: ∀m. Vector A m → Prop.
72    n = m → v1 ≃ v2 → P n v1 → P m v2.
73  #A #n #m #v1 #v2 #P #eq #jmeq
74  destruct #assm assumption
75qed.
76
77lemma vector_cons_append:
78  ∀A: Type[0].
79  ∀n: nat.
80  ∀e: A.
81  ∀v: Vector A n.
82    e ::: v = [[ e ]] @@ v.
83  #A #n #e #v
84  cases v try %
85  #n' #hd #tl %
86qed.
87
88lemma vector_cons_append2:
89  ∀A: Type[0].
90  ∀n, m: nat.
91  ∀v: Vector A n.
92  ∀q: Vector A m.
93  ∀hd: A.
94    hd:::(v@@q) = (hd:::v)@@q.
95  #A #n #m #v #q
96  elim v try (#hd %)
97  #n' #hd' #tl' #ih #hd'
98  <ih %
99qed.
100
101lemma jmeq_cons_vector_monotone:
102  ∀A: Type[0].
103  ∀m, n: nat.
104  ∀v: Vector A m.
105  ∀q: Vector A n.
106  ∀prf: m = n.
107  ∀hd: A.
108    v ≃ q → hd:::v ≃ hd:::q.
109  #A #m #n #v #q #prf #hd #E
110  @(super_rewrite2 A … E)
111  try assumption %
112qed.
113
114lemma vector_associative_append:
115  ∀A: Type[0].
116  ∀n, m, o:  nat.
117  ∀v: Vector A n.
118  ∀q: Vector A m.
119  ∀r: Vector A o.
120    (v @@ q) @@ r ≃ v @@ (q @@ r).
121  #A #n #m #o #v #q #r
122  elim v try %
123  #n' #hd #tl #ih
124  <(vector_cons_append2 A … hd)
125  @jmeq_cons_vector_monotone
126  try assumption
127  @associative_plus
128qed.
129
130lemma mem_middle_vector:
131  ∀A: Type[0].
132  ∀m, o: nat.
133  ∀eq: A → A → bool.
134  ∀reflex: ∀a. eq a a = true.
135  ∀p: Vector A m.
136  ∀a: A.
137  ∀r: Vector A o.
138    mem A eq ? (p@@(a:::r)) a = true.
139  #A #m #o #eq #reflex #p #a
140  elim p try (normalize >reflex #H %)
141  #m' #hd #tl #inductive_hypothesis
142  normalize
143  cases (eq ??) normalize nodelta
144  try (#irrelevant %)
145  @inductive_hypothesis
146qed.
147
148lemma mem_monotonic_wrt_append:
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 ? r a = true → mem A eq ? (p @@ r) a = true.
157  #A #m #o #eq #reflex #p #a
158  elim p try (#r #assm assumption)
159  #m' #hd #tl #inductive_hypothesis #r #assm
160  normalize
161  cases (eq ??) try %
162  @inductive_hypothesis assumption
163qed.
164
165lemma subvector_multiple_append:
166  ∀A: Type[0].
167  ∀o, n: nat.
168  ∀eq: A → A → bool.
169  ∀refl: ∀a. eq a a = true.
170  ∀h: Vector A o.
171  ∀v: Vector A n.
172  ∀m: nat.
173  ∀q: Vector A m.
174    bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)).
175  #A #o #n #eq #reflex #h #v
176  elim v try (normalize #m #irrelevant @I)
177  #m' #hd #tl #inductive_hypothesis #m #q
178  change with (bool_to_Prop (andb ??))
179  cut ((mem A eq (o + (m + S m')) (h@@q@@hd:::tl) hd) = true)
180  [1:
181    @mem_monotonic_wrt_append try assumption
182    @mem_monotonic_wrt_append try assumption
183    normalize >reflex %
184  |2:
185    #assm >assm
186    >vector_cons_append
187    change with (bool_to_Prop (subvector_with ??????))
188    @(super_rewrite2 … (vector_associative_append … q [[hd]] tl))
189    try @associative_plus
190    @inductive_hypothesis
191  ]
192qed.
193
194lemma vector_cons_empty:
195  ∀A: Type[0].
196  ∀n: nat.
197  ∀v: Vector A n.
198    [[ ]] @@ v = v.
199  #A #n #v
200  cases v try %
201  #n' #hd #tl %
202qed.
203
204corollary subvector_hd_tl:
205  ∀A: Type[0].
206  ∀o: nat.
207  ∀eq: A → A → bool.
208  ∀refl: ∀a. eq a a = true.
209  ∀h: A.
210  ∀v: Vector A o.
211    bool_to_Prop (subvector_with A ? ? eq v (h ::: v)).
212  #A #o #eq #reflex #h #v
213  >(vector_cons_append … h v)
214  <(vector_cons_empty … ([[h]] @@ v))
215  @(subvector_multiple_append … eq reflex [[ ]] v ? [[h]])
216qed.
217
218lemma eq_a_reflexive:
219  ∀a. eq_a a a = true.
220  #a cases a %
221qed.
222
223lemma is_in_monotonic_wrt_append:
224  ∀m, n: nat.
225  ∀p: Vector addressing_mode_tag m.
226  ∀q: Vector addressing_mode_tag n.
227  ∀to_search: addressing_mode.
228    bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search).
229  #m #n #p #q #to_search #assm
230  elim q try assumption
231  #n' #hd #tl #inductive_hypothesis
232  normalize
233  cases (is_a ??) try @I
234  >inductive_hypothesis @I
235qed.
236
237corollary is_in_hd_tl:
238  ∀to_search: addressing_mode.
239  ∀hd: addressing_mode_tag.
240  ∀n: nat.
241  ∀v: Vector addressing_mode_tag n.
242    bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search).
243  #to_search #hd #n #v
244  elim v
245  [1:
246    #absurd
247    normalize in absurd; cases absurd
248  |2:
249    #n' #hd' #tl #inductive_hypothesis #assm
250    >vector_cons_append >(vector_cons_append … hd' tl)
251    @(is_in_monotonic_wrt_append … ([[hd']]@@tl) [[hd]] to_search)
252    assumption
253  ]
254qed.
255 
256let rec list_addressing_mode_tags_elim
257  (n: nat) (l: Vector addressing_mode_tag (S n))
258    on l: (l → bool) → bool ≝
259  match l return λx.
260    match x with
261    [ O ⇒ λl: Vector … O. bool
262    | S x' ⇒ λl: Vector addressing_mode_tag (S x'). (l → bool) → bool
263    ] with
264  [ VEmpty      ⇒  true 
265  | VCons len hd tl ⇒ λP.
266    let process_hd ≝
267      match hd return λhd. ∀P: hd:::tl → bool. bool with
268      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
269      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
270      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
271      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
272      | acc_a ⇒ λP.P ACC_A
273      | acc_b ⇒ λP.P ACC_B
274      | dptr ⇒ λP.P DPTR
275      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
276      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
277      | acc_dptr ⇒ λP.P ACC_DPTR
278      | acc_pc ⇒ λP.P ACC_PC
279      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
280      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
281      | carry ⇒ λP.P CARRY
282      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
283      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
284      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
285      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
286      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
287      ]
288    in
289      andb (process_hd P)
290       (match len return λx. x = len → bool with
291         [ O ⇒ λprf. true
292         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
293  ].
294  try %
295  [2:
296    cases (sym_eq ??? prf); assumption
297  |1:
298    generalize in match H; generalize in match tl;
299    destruct #tl
300    normalize in ⊢ (∀_: %. ?);
301    #H
302    whd normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]);
303    cases (is_a hd (subaddressing_modeel y tl H))
304    whd try @I normalize nodelta //
305  ]
306qed.
307
308definition product_elim ≝
309  λm, n: nat.
310  λv: Vector addressing_mode_tag (S m).
311  λq: Vector addressing_mode_tag (S n).
312  λP: (v × q) → bool.
313    list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)).
314
315definition union_elim ≝
316  λA, B: Type[0].
317  λelimA: (A → bool) → bool.
318  λelimB: (B → bool) → bool.
319  λelimU: A ⊎ B → bool.
320    elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))).
321
322(*                           
323definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝
324  λP.
325    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧
326    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧
327    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧
328    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧
329    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧
330    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧
331    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧
332    list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
333    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧
334    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧
335    P (DA ? ACC_A) ∧
336    bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧
337    bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧
338    bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧
339    bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧
340    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧
341    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧
342    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧
343    list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
344    P (RL ? ACC_A) ∧
345    P (RLC ? ACC_A) ∧
346    P (RR ? ACC_A) ∧
347    P (RRC ? ACC_A) ∧
348    P (SWAP ? ACC_A) ∧
349    P (RET ?) ∧
350    P (RETI ?) ∧
351    P (NOP ?) ∧
352    bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧
353    list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧
354    bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧
355    bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧
356    union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]])
357                   (product_elim ? ? [[ registr; indirect ]] [[ data ]])
358                   (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧
359    list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧
360    union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]])
361                   (product_elim ? ? [[direct]] [[ acc_a ; data ]])
362                   (λd. P (XRL ? d)) ∧
363    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
364                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
365                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
366                   (λd. P (ANL ? d)) ∧
367    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]])
368                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
369                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
370                   (λd. P (ORL ? d)) ∧
371    union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]])
372                   (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]])
373                   (λd. P (MOVX ? d)) ∧
374    union_elim ? ? (
375      union_elim ? ? (
376        union_elim ? ? (
377          union_elim ? ? (
378            union_elim ? ?  (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
379                            (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]]))
380                            (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]]))
381                            (product_elim ? ? [[dptr]] [[data16]]))
382                            (product_elim ? ? [[carry]] [[bit_addr]]))
383                            (product_elim ? ? [[bit_addr]] [[carry]])
384                            (λd. P (MOV ? d)).
385  %
386qed.
387 
388definition instruction_elim: ∀P: instruction → bool. bool ≝
389  λP. (*
390    bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧
391    bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧
392    bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧
393    bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *)
394    bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (*  ∧
395    P (JMP INDIRECT_DPTR) ∧
396    list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧
397    preinstruction_elim (λp. P (RealInstruction p)). *)
398  %
399qed.
400
401
402axiom instruction_elim_complete:
403 ∀P. instruction_elim P = true → ∀i. P i = true.
404*)
405(*definition eq_instruction ≝
406  λi, j: instruction.
407    true.*)
408
409definition eq_addressing_mode: addressing_mode → addressing_mode → bool ≝
410  λa, b: addressing_mode.
411  match a with
412  [ DIRECT d ⇒
413    match b with
414    [ DIRECT e ⇒ eq_bv ? d e
415    | _ ⇒ false
416    ]
417  | INDIRECT b' ⇒
418    match b with
419    [ INDIRECT e ⇒ eq_b b' e
420    | _ ⇒ false
421    ]
422  | EXT_INDIRECT b' ⇒
423    match b with
424    [ EXT_INDIRECT e ⇒ eq_b b' e
425    | _ ⇒ false
426    ]
427  | REGISTER bv ⇒
428    match b with
429    [ REGISTER bv' ⇒ eq_bv ? bv bv'
430    | _ ⇒ false
431    ]
432  | ACC_A ⇒ match b with [ ACC_A ⇒ true | _ ⇒ false ]
433  | ACC_B ⇒ match b with [ ACC_B ⇒ true | _ ⇒ false ]
434  | DPTR ⇒ match b with [ DPTR ⇒ true | _ ⇒ false ]
435  | DATA b' ⇒
436    match b with
437    [ DATA e ⇒ eq_bv ? b' e
438    | _ ⇒ false
439    ]
440  | DATA16 w ⇒
441    match b with
442    [ DATA16 e ⇒ eq_bv ? w e
443    | _ ⇒ false
444    ]
445  | ACC_DPTR ⇒ match b with [ ACC_DPTR ⇒ true | _ ⇒ false ]
446  | ACC_PC ⇒ match b with [ ACC_PC ⇒ true | _ ⇒ false ]
447  | EXT_INDIRECT_DPTR ⇒ match b with [ EXT_INDIRECT_DPTR ⇒ true | _ ⇒ false ]
448  | INDIRECT_DPTR ⇒ match b with [ INDIRECT_DPTR ⇒ true | _ ⇒ false ]
449  | CARRY ⇒ match b with [ CARRY ⇒ true | _ ⇒ false ]
450  | BIT_ADDR b' ⇒
451    match b with
452    [ BIT_ADDR e ⇒ eq_bv ? b' e
453    | _ ⇒ false
454    ]
455  | N_BIT_ADDR b' ⇒
456    match b with
457    [ N_BIT_ADDR e ⇒ eq_bv ? b' e
458    | _ ⇒ false
459    ]
460  | RELATIVE n ⇒
461    match b with
462    [ RELATIVE e ⇒ eq_bv ? n e
463    | _ ⇒ false
464    ]
465  | ADDR11 w ⇒
466    match b with
467    [ ADDR11 e ⇒ eq_bv ? w e
468    | _ ⇒ false
469    ]
470  | ADDR16 w ⇒
471    match b with
472    [ ADDR16 e ⇒ eq_bv ? w e
473    | _ ⇒ false
474    ]
475  ].
476
477lemma eq_bv_refl:
478  ∀n, b.
479    eq_bv n b b = true.
480  #n #b cases b //
481qed.
482
483lemma eq_b_refl:
484  ∀b.
485    eq_b b b = true.
486  #b cases b //
487qed.
488
489lemma eq_addressing_mode_refl:
490  ∀a. eq_addressing_mode a a = true.
491  #a
492  cases a try #arg1 try #arg2
493  try @eq_bv_refl try @eq_b_refl
494  try normalize %
495qed.
496 
497definition eq_sum:
498    ∀A, B. (A → A → bool) → (B → B → bool) → (A ⊎ B) → (A ⊎ B) → bool ≝
499  λlt, rt, leq, req, left, right.
500    match left with
501    [ inl l ⇒
502      match right with
503      [ inl l' ⇒ leq l l'
504      | _ ⇒ false
505      ]
506    | inr r ⇒
507      match right with
508      [ inr r' ⇒ req r r'
509      | _ ⇒ false
510      ]
511    ].
512
513definition eq_prod: ∀A, B. (A → A → bool) → (B → B → bool) → (A × B) → (A × B) → bool ≝
514  λlt, rt, leq, req, left, right.
515    let 〈l, r〉 ≝ left in
516    let 〈l', r'〉 ≝ right in
517      leq l l' ∧ req r r'.
518
519definition eq_preinstruction: preinstruction [[relative]] → preinstruction [[relative]] → bool ≝
520  λi, j.
521  match i with
522  [ ADD arg1 arg2 ⇒
523    match j with
524    [ ADD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
525    | _ ⇒ false
526    ]
527  | ADDC arg1 arg2 ⇒
528    match j with
529    [ ADDC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
530    | _ ⇒ false
531    ]
532  | SUBB arg1 arg2 ⇒
533    match j with
534    [ SUBB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
535    | _ ⇒ false
536    ]
537  | INC arg ⇒
538    match j with
539    [ INC arg' ⇒ eq_addressing_mode arg arg'
540    | _ ⇒ false
541    ]
542  | DEC arg ⇒
543    match j with
544    [ DEC arg' ⇒ eq_addressing_mode arg arg'
545    | _ ⇒ false
546    ]
547  | MUL arg1 arg2 ⇒
548    match j with
549    [ MUL arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
550    | _ ⇒ false
551    ]
552  | DIV arg1 arg2 ⇒
553    match j with
554    [ DIV arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
555    | _ ⇒ false
556    ]
557  | DA arg ⇒
558    match j with
559    [ DA arg' ⇒ eq_addressing_mode arg arg'
560    | _ ⇒ false
561    ]
562  | JC arg ⇒
563    match j with
564    [ JC arg' ⇒ eq_addressing_mode arg arg'
565    | _ ⇒ false
566    ]
567  | JNC arg ⇒
568    match j with
569    [ JNC arg' ⇒ eq_addressing_mode arg arg'
570    | _ ⇒ false
571    ]
572  | JB arg1 arg2 ⇒
573    match j with
574    [ JB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
575    | _ ⇒ false
576    ]
577  | JNB arg1 arg2 ⇒
578    match j with
579    [ JNB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
580    | _ ⇒ false
581    ]
582  | JBC arg1 arg2 ⇒
583    match j with
584    [ JBC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
585    | _ ⇒ false
586    ]
587  | JZ arg ⇒
588    match j with
589    [ JZ arg' ⇒ eq_addressing_mode arg arg'
590    | _ ⇒ false
591    ]
592  | JNZ arg ⇒
593    match j with
594    [ JNZ arg' ⇒ eq_addressing_mode arg arg'
595    | _ ⇒ false
596    ]
597  | CJNE arg1 arg2 ⇒
598    match j with
599    [ CJNE arg1' arg2' ⇒
600      let prod_eq_left ≝ eq_prod [[acc_a]] [[direct; data]] eq_addressing_mode eq_addressing_mode in
601      let prod_eq_right ≝ eq_prod [[registr; indirect]] [[data]] eq_addressing_mode eq_addressing_mode in
602      let arg1_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
603        arg1_eq arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
604    | _ ⇒ false
605    ]
606  | DJNZ arg1 arg2 ⇒
607    match j with
608    [ DJNZ arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
609    | _ ⇒ false
610    ]
611  | CLR arg ⇒
612    match j with
613    [ CLR arg' ⇒ eq_addressing_mode arg arg'
614    | _ ⇒ false
615    ]
616  | CPL arg ⇒
617    match j with
618    [ CPL arg' ⇒ eq_addressing_mode arg arg'
619    | _ ⇒ false
620    ]
621  | RL arg ⇒
622    match j with
623    [ RL arg' ⇒ eq_addressing_mode arg arg'
624    | _ ⇒ false
625    ]
626  | RLC arg ⇒
627    match j with
628    [ RLC arg' ⇒ eq_addressing_mode arg arg'
629    | _ ⇒ false
630    ]
631  | RR arg ⇒
632    match j with
633    [ RR arg' ⇒ eq_addressing_mode arg arg'
634    | _ ⇒ false
635    ]
636  | RRC arg ⇒
637    match j with
638    [ RRC arg' ⇒ eq_addressing_mode arg arg'
639    | _ ⇒ false
640    ]
641  | SWAP arg ⇒
642    match j with
643    [ SWAP arg' ⇒ eq_addressing_mode arg arg'
644    | _ ⇒ false
645    ]
646  | SETB arg ⇒
647    match j with
648    [ SETB arg' ⇒ eq_addressing_mode arg arg'
649    | _ ⇒ false
650    ]
651  | PUSH arg ⇒
652    match j with
653    [ PUSH arg' ⇒ eq_addressing_mode arg arg'
654    | _ ⇒ false
655    ]
656  | POP arg ⇒
657    match j with
658    [ POP arg' ⇒ eq_addressing_mode arg arg'
659    | _ ⇒ false
660    ]
661  | XCH arg1 arg2 ⇒
662    match j with
663    [ XCH arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
664    | _ ⇒ false
665    ]
666  | XCHD arg1 arg2 ⇒
667    match j with
668    [ XCHD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
669    | _ ⇒ false
670    ]
671  | RET ⇒ match j with [ RET ⇒ true | _ ⇒ false ]
672  | RETI ⇒ match j with [ RETI ⇒ true | _ ⇒ false ]
673  | NOP ⇒ match j with [ NOP ⇒ true | _ ⇒ false ]
674  | MOVX arg ⇒
675    match j with
676    [ MOVX arg' ⇒
677      let prod_eq_left ≝ eq_prod [[acc_a]] [[ext_indirect; ext_indirect_dptr]] eq_addressing_mode eq_addressing_mode in
678      let prod_eq_right ≝ eq_prod [[ext_indirect; ext_indirect_dptr]] [[acc_a]] eq_addressing_mode eq_addressing_mode in
679      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
680        sum_eq arg arg'
681    | _ ⇒ false
682    ]
683  | XRL arg ⇒
684    match j with
685    [ XRL arg' ⇒
686      let prod_eq_left ≝ eq_prod [[acc_a]] [[ data ; registr ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in
687      let prod_eq_right ≝ eq_prod [[direct]] [[ acc_a ; data ]] eq_addressing_mode eq_addressing_mode in
688      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
689        sum_eq arg arg'
690    | _ ⇒ false
691    ]
692  | ORL arg ⇒
693    match j with
694    [ ORL arg' ⇒
695      let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; data ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in
696      let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in
697      let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in
698      let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in
699      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
700        sum_eq arg arg'
701    | _ ⇒ false
702    ]
703  | ANL arg ⇒
704    match j with
705    [ ANL arg' ⇒
706      let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; direct ; indirect ; data ]] eq_addressing_mode eq_addressing_mode in
707      let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in
708      let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in
709      let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in
710      let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in
711        sum_eq arg arg'
712    | _ ⇒ false
713    ]
714  | MOV arg ⇒
715    match j with
716    [ MOV arg' ⇒
717      let prod_eq_6 ≝ eq_prod [[acc_a]] [[registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in
718      let prod_eq_5 ≝ eq_prod [[registr; indirect]] [[acc_a; direct; data]] eq_addressing_mode eq_addressing_mode in
719      let prod_eq_4 ≝ eq_prod [[direct]] [[acc_a; registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in
720      let prod_eq_3 ≝ eq_prod [[dptr]] [[data16]] eq_addressing_mode eq_addressing_mode in
721      let prod_eq_2 ≝ eq_prod [[carry]] [[bit_addr]] eq_addressing_mode eq_addressing_mode in
722      let prod_eq_1 ≝ eq_prod [[bit_addr]] [[carry]] eq_addressing_mode eq_addressing_mode in
723      let sum_eq_1 ≝ eq_sum ? ? prod_eq_6 prod_eq_5 in
724      let sum_eq_2 ≝ eq_sum ? ? sum_eq_1 prod_eq_4 in
725      let sum_eq_3 ≝ eq_sum ? ? sum_eq_2 prod_eq_3 in
726      let sum_eq_4 ≝ eq_sum ? ? sum_eq_3 prod_eq_2 in
727      let sum_eq_5 ≝ eq_sum ? ? sum_eq_4 prod_eq_1 in
728        sum_eq_5 arg arg'
729    | _ ⇒ false
730    ]
731  ].
732
733lemma eq_sum_refl:
734  ∀A, B: Type[0].
735  ∀leq, req.
736  ∀s.
737  ∀leq_refl: (∀t. leq t t = true).
738  ∀req_refl: (∀u. req u u = true).
739    eq_sum A B leq req s s = true.
740  #A #B #leq #req #s #leq_refl #req_refl
741  cases s assumption
742qed.
743
744lemma eq_prod_refl:
745  ∀A, B: Type[0].
746  ∀leq, req.
747  ∀s.
748  ∀leq_refl: (∀t. leq t t = true).
749  ∀req_refl: (∀u. req u u = true).
750    eq_prod A B leq req s s = true.
751  #A #B #leq #req #s #leq_refl #req_refl
752  cases s
753  whd in ⊢ (? → ? → ??%?);
754  #l #r
755  >leq_refl @req_refl
756qed.
757
758lemma eq_preinstruction_refl:
759  ∀i.
760    eq_preinstruction i i = true.
761  #i cases i try #arg1 try #arg2
762  try @eq_addressing_mode_refl
763  [1,2,3,4,5,6,7,8,10,16,17,18,19,20:
764    whd in ⊢ (??%?); try %
765    >eq_addressing_mode_refl
766    >eq_addressing_mode_refl %
767  |13,15:
768    whd in ⊢ (??%?);
769    cases arg1
770    [*:
771      #arg1_left normalize nodelta
772      >eq_prod_refl [*: try % #argr @eq_addressing_mode_refl]
773    ]
774  |11,12:
775    whd in ⊢ (??%?);
776    cases arg1
777    [1:
778      #arg1_left normalize nodelta
779      >(eq_sum_refl …)
780      [1: % | 2,3: #arg @eq_prod_refl ]
781      @eq_addressing_mode_refl
782    |2:
783      #arg1_left normalize nodelta
784      @eq_prod_refl [*: @eq_addressing_mode_refl ]
785    |3:
786      #arg1_left normalize nodelta
787      >(eq_sum_refl …)
788      [1:
789        %
790      |2,3:
791        #arg @eq_prod_refl #arg @eq_addressing_mode_refl
792      ]
793    |4:
794      #arg1_left normalize nodelta
795      @eq_prod_refl [*: #arg @eq_addressing_mode_refl ]
796    ]
797  |14:
798    whd in ⊢ (??%?);
799    cases arg1
800    [1:
801      #arg1_left normalize nodelta
802      @eq_sum_refl
803      [1:
804        #arg @eq_sum_refl
805        [1:
806          #arg @eq_sum_refl
807          [1:
808            #arg @eq_sum_refl
809            [1:
810              #arg @eq_prod_refl
811              [*:
812                @eq_addressing_mode_refl
813              ]
814            |2:
815              #arg @eq_prod_refl
816              [*:
817                #arg @eq_addressing_mode_refl
818              ]
819            ]
820          |2:
821            #arg @eq_prod_refl
822            [*:
823              #arg @eq_addressing_mode_refl
824            ]
825          ]
826        |2:
827          #arg @eq_prod_refl
828          [*:
829            #arg @eq_addressing_mode_refl
830          ]
831        ]
832      |2:
833        #arg @eq_prod_refl
834        [*:
835          #arg @eq_addressing_mode_refl
836        ]
837      ]
838    |2:
839      #arg1_right normalize nodelta
840      @eq_prod_refl
841      [*:
842        #arg @eq_addressing_mode_refl
843      ]
844    ]
845  |*:
846    whd in ⊢ (??%?);
847    cases arg1
848    [*:
849      #arg1 >eq_sum_refl
850      [1,4:
851        @eq_addressing_mode_refl
852      |2,3,5,6:
853        #arg @eq_prod_refl
854        [*:
855          #arg @eq_addressing_mode_refl
856        ]
857      ]
858    ]
859  ]
860qed.
861
862definition eq_instruction: instruction → instruction → bool ≝
863  λi, j.
864  match i with
865  [ ACALL arg ⇒
866    match j with
867    [ ACALL arg' ⇒ eq_addressing_mode arg arg'
868    | _ ⇒ false
869    ]
870  | LCALL arg ⇒
871    match j with
872    [ LCALL arg' ⇒ eq_addressing_mode arg arg'
873    | _ ⇒ false
874    ]
875  | AJMP arg ⇒
876    match j with
877    [ AJMP arg' ⇒ eq_addressing_mode arg arg'
878    | _ ⇒ false
879    ]
880  | LJMP arg ⇒
881    match j with
882    [ LJMP arg' ⇒ eq_addressing_mode arg arg'
883    | _ ⇒ false
884    ]
885  | SJMP arg ⇒
886    match j with
887    [ SJMP arg' ⇒ eq_addressing_mode arg arg'
888    | _ ⇒ false
889    ]
890  | JMP arg ⇒
891    match j with
892    [ JMP arg' ⇒ eq_addressing_mode arg arg'
893    | _ ⇒ false
894    ]
895  | MOVC arg1 arg2 ⇒
896    match j with
897    [ MOVC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2'
898    | _ ⇒ false
899    ]
900  | RealInstruction instr ⇒
901    match j with
902    [ RealInstruction instr' ⇒ eq_preinstruction instr instr'
903    | _ ⇒ false
904    ]
905  ].
906 
907lemma eq_instruction_refl:
908  ∀i. eq_instruction i i = true.
909  #i cases i [*: #arg1 ]
910  try @eq_addressing_mode_refl
911  try @eq_preinstruction_refl
912  #arg2 whd in ⊢ (??%?);
913  >eq_addressing_mode_refl >eq_addressing_mode_refl %
914qed.
915
916let rec vect_member
917  (A: Type[0]) (n: nat) (eq: A → A → bool) (v: Vector A n) (a: A)
918    on v: bool ≝
919  match v with
920  [ VEmpty          ⇒ false
921  | VCons len hd tl ⇒
922      eq hd a ∨ (vect_member A ? eq tl a)
923  ].
924
925let rec list_addressing_mode_tags_elim_prop
926  (n: nat)
927  (l: Vector addressing_mode_tag (S n))
928  on l:
929  ∀P: l → Prop.
930  ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a.
931  ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a.
932  ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a.
933  ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a.
934  ∀x: l. P x ≝
935  match l return
936    λy.
937      match y with
938      [ O    ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True
939      | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop.
940               ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True.
941               ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True.
942               ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True.
943               ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True.
944               ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True.
945               ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True.
946               ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True.
947               ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True.
948               ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True.
949               ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True.
950               ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True.
951               ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True.
952               ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True.
953               ∀carry_a: if vect_member … eq_a l carry then P CARRY else True.
954               ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True.
955               ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True.
956               ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True.
957               ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True.
958               ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True.
959               ∀x:l. P x
960      ] with
961  [ VEmpty          ⇒ λAbsurd. ⊥
962  | VCons len hd tl ⇒ λProof. ?
963  ] (refl ? (S n)). cases daemon. qed. (*
964  [ destruct(Absurd)
965  | # A1 # A2 # A3 # A4 # A5 # A6 # A7
966    # A8 # A9 # A10 # A11 # A12 # A13 # A14
967    # A15 # A16 # A17 # A18 # A19 # X
968    cases X
969    # SUB cases daemon ] qed.
970    cases SUB
971    [ # BYTE
972    normalize
973  ].
974 
975 
976(*    let prepare_hd ≝
977      match hd with
978      [ direct ⇒ λdirect_prf. ?
979      | indirect ⇒ λindirect_prf. ?
980      | ext_indirect ⇒ λext_indirect_prf. ?
981      | registr ⇒ λregistr_prf. ?
982      | acc_a ⇒ λacc_a_prf. ?
983      | acc_b ⇒ λacc_b_prf. ?
984      | dptr ⇒ λdptr_prf. ?
985      | data ⇒ λdata_prf. ?
986      | data16 ⇒ λdata16_prf. ?
987      | acc_dptr ⇒ λacc_dptr_prf. ?
988      | acc_pc ⇒ λacc_pc_prf. ?
989      | ext_indirect_dptr ⇒ λext_indirect_prf. ?
990      | indirect_dptr ⇒ λindirect_prf. ?
991      | carry ⇒ λcarry_prf. ?
992      | bit_addr ⇒ λbit_addr_prf. ?
993      | n_bit_addr ⇒ λn_bit_addr_prf. ?
994      | relative ⇒ λrelative_prf. ?
995      | addr11 ⇒ λaddr11_prf. ?
996      | addr16 ⇒ λaddr16_prf. ?
997      ]
998    in ? *)
999  ].
1000  [ 1: destruct(absd)
1001  | 2: # A1 # A2 # A3 # A4 # A5 # A6
1002       # A7 # A8 # A9 # A10 # A11 # A12
1003       # A13 # A14 # A15 # A16 # A17 # A18
1004       # A19 *
1005  ].
1006
1007
1008  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
1009   (l → bool) → bool ] with
1010  [ VEmpty      ⇒  true 
1011  | VCons len hd tl ⇒ λP.
1012    let process_hd ≝
1013      match hd return λhd. ∀P: hd:::tl → bool. bool with
1014      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
1015      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
1016      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
1017      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
1018      | acc_a ⇒ λP.P ACC_A
1019      | acc_b ⇒ λP.P ACC_B
1020      | dptr ⇒ λP.P DPTR
1021      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
1022      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
1023      | acc_dptr ⇒ λP.P ACC_DPTR
1024      | acc_pc ⇒ λP.P ACC_PC
1025      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
1026      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
1027      | carry ⇒ λP.P CARRY
1028      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
1029      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
1030      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
1031      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
1032      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
1033      ]
1034    in
1035      andb (process_hd P)
1036       (match len return λx. x = len → bool with
1037         [ O ⇒ λprf. true
1038         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
1039  ].
1040  try %
1041  [ 2: cases (sym_eq ??? prf); @tl
1042  | generalize in match H; generalize in match tl; cases prf;
1043    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
1044    #tl
1045    normalize in ⊢ (∀_: %. ?)
1046    # H
1047    whd
1048    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1049    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
1050qed.
1051*)
1052
1053definition load_code_memory_aux ≝
1054 fold_left_i_aux … (
1055   λi, mem, v.
1056     insert … (bitvector_of_nat … i) v mem) (Stub Byte 16).
1057
1058lemma split_zero:
1059  ∀A,m.
1060  ∀v: Vector A m.
1061    〈[[]], v〉 = split A 0 m v.
1062  #A #m #v
1063  cases v try %
1064  #n #hd #tl %
1065qed.
1066
1067lemma Vector_O:
1068  ∀A: Type[0].
1069  ∀v: Vector A 0.
1070    v ≃ VEmpty A.
1071 #A #v
1072 generalize in match (refl … 0);
1073 cases v in ⊢ (??%? → ?%%??); //
1074 #n #hd #tl #absurd
1075 destruct(absurd)
1076qed.
1077
1078lemma Vector_Sn:
1079  ∀A: Type[0].
1080  ∀n: nat.
1081  ∀v: Vector A (S n).
1082    ∃hd: A. ∃tl: Vector A n.
1083      v ≃ VCons A n hd tl.
1084  #A #n #v
1085  generalize in match (refl … (S n));
1086  cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??)));
1087  [1:
1088    #absurd destruct(absurd)
1089  |2:
1090    #m #hd #tl #eq
1091    <(injective_S … eq)
1092    %{hd} %{tl} %
1093  ]
1094qed.
1095
1096lemma vector_append_zero:
1097  ∀A,m.
1098  ∀v: Vector A m.
1099  ∀q: Vector A 0.
1100    v = q@@v.
1101  #A #m #v #q
1102  >(Vector_O A q) %
1103qed.
1104
1105lemma prod_eq_left:
1106  ∀A: Type[0].
1107  ∀p, q, r: A.
1108    p = q → 〈p, r〉 = 〈q, r〉.
1109  #A #p #q #r #hyp
1110  destruct %
1111qed.
1112
1113lemma prod_eq_right:
1114  ∀A: Type[0].
1115  ∀p, q, r: A.
1116    p = q → 〈r, p〉 = 〈r, q〉.
1117  #A #p #q #r #hyp
1118  destruct %
1119qed.
1120
1121corollary prod_vector_zero_eq_left:
1122  ∀A, n.
1123  ∀q: Vector A O.
1124  ∀r: Vector A n.
1125    〈q, r〉 = 〈[[ ]], r〉.
1126  #A #n #q #r
1127  generalize in match (Vector_O A q …);
1128  #hyp destruct %
1129qed.
1130
1131lemma tail_head:
1132  ∀a: Type[0].
1133  ∀m, n: nat.
1134  ∀hd: a.
1135  ∀l: Vector a m.
1136  ∀r: Vector a n.
1137    tail a ? (hd:::(l@@r)) = l@@r.
1138  #a #m #n #hd #l #r
1139  cases l try %
1140  #m' #hd' #tl' %
1141qed.
1142
1143lemma head_head':
1144  ∀a: Type[0].
1145  ∀m: nat.
1146  ∀hd: a.
1147  ∀l: Vector a m.
1148    hd = head' … (hd:::l).
1149  #a #m #hd #l cases l try %
1150  #m' #hd' #tl %
1151qed.
1152
1153lemma split_succ:
1154  ∀A: Type[0].
1155  ∀m, n: nat.
1156  ∀l: Vector A m.
1157  ∀r: Vector A n.
1158  ∀v: Vector A (m + n).
1159  ∀hd: A.
1160    v = l@@r → (〈l, r〉 = split A m n v → 〈hd:::l, r〉 = split A (S m) n (hd:::v)).
1161  #A #m
1162  elim m
1163  [1:
1164    #n #l #r #v #hd #eq #hyp
1165    destruct >(Vector_O … l) %
1166  |2:
1167    #m' #inductive_hypothesis #n #l #r #v #hd #equal #hyp
1168    destruct
1169    cases (Vector_Sn … l) #hd' #tl'
1170    whd in ⊢ (???%);
1171    >tail_head
1172    <(? : split A (S m') n (l@@r) = split' A (S m') n (l@@r))
1173    try (<hyp <head_head' %)
1174    elim l normalize //
1175  ]
1176qed.
1177
1178lemma split_prod:
1179  ∀A: Type[0].
1180  ∀m, n: nat.
1181  ∀p: Vector A (m + n).
1182  ∀v: Vector A m.
1183  ∀q: Vector A n.
1184    p = v@@q → 〈v, q〉 = split A m n p.
1185  #A #m elim m
1186  [1:
1187    #n #p #v #q #hyp
1188    >hyp <(vector_append_zero A n q v)
1189    >(prod_vector_zero_eq_left A …)
1190    @split_zero
1191  |2:
1192    #r #ih #n #p #v #q #hyp
1193    >hyp
1194    cases (Vector_Sn A r v) #hd #exists
1195    cases exists #tl #jmeq
1196    >jmeq @split_succ try %
1197    @ih %
1198  ]
1199qed.
1200
1201(*
1202lemma split_prod_exists:
1203  ∀A, m, n.
1204  ∀p: Vector A (m + n).
1205  ∃v: Vector A m.
1206  ∃q: Vector A n.
1207    〈v, q〉 = split A m n p.
1208  #A #m #n #p
1209  elim m
1210  @ex_intro
1211  [1:
1212  |2: @ex_intro
1213      [1:
1214      |2:
1215      ]
1216  ]
1217*)
1218
1219definition split_elim:
1220  ∀A: Type[0].
1221  ∀l, m: nat.
1222  ∀v: Vector A (l + m).
1223  ∀P: (Vector A l) × (Vector A m) → Prop.
1224    (∀vl: Vector A l.
1225     ∀vm: Vector A m.
1226      v = vl@@vm → P 〈vl,vm〉) → P (split A l m v) ≝
1227  λa: Type[0].
1228  λl, m: nat.
1229  λv: Vector a (l + m).
1230  λP. ?.
1231  cases daemon
1232qed.
1233
1234(*
1235axiom not_eqvb_S:
1236 ∀pc.
1237 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))).
1238
1239axiom not_eqvb_SS:
1240 ∀pc.
1241 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))).
1242 
1243axiom bitvector_elim_complete:
1244 ∀n,P. bitvector_elim n P = true → ∀bv. P bv.
1245
1246lemma bitvector_elim_complete':
1247 ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true.
1248 #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv
1249 generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ //
1250qed.
1251*)
1252
1253(*
1254lemma andb_elim':
1255 ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true.
1256 #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize //
1257qed.
1258*)
1259
1260let rec encoding_check
1261  (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word)
1262    (encoding: list Byte)
1263      on encoding: Prop ≝
1264  match encoding with
1265  [ nil ⇒ final_pc = pc
1266  | cons hd tl ⇒
1267    let 〈new_pc, byte〉 ≝ next code_memory pc in
1268      hd = byte ∧ encoding_check code_memory new_pc final_pc tl
1269  ].
1270
1271axiom add_commutative:
1272  ∀n: nat.
1273  ∀l, r: BitVector n.
1274    add n l r = add n r l.
1275
1276axiom add_bitvector_of_nat_Sm:
1277  ∀n, m: nat.
1278    add … (bitvector_of_nat … 1) (bitvector_of_nat … m) =
1279      bitvector_of_nat n (S m).
1280
1281lemma encoding_check_append:
1282  ∀code_memory: BitVectorTrie Byte 16.
1283  ∀final_pc: Word.
1284  ∀l1: list Byte.
1285  ∀pc: Word.
1286  ∀l2: list Byte.
1287    encoding_check code_memory pc final_pc (l1@l2) →
1288      let pc_plus_len ≝ add … pc (bitvector_of_nat … (length … l1)) in
1289        encoding_check code_memory pc pc_plus_len l1 ∧
1290          encoding_check code_memory pc_plus_len final_pc l2.
1291  #code_memory #final_pc #l1 elim l1
1292  [1:
1293    #pc #l2
1294    whd in ⊢ (????% → ?); #H
1295    <add_zero
1296    whd whd in ⊢ (?%?); /2/
1297  |2:
1298    #hd #tl #IH #pc #l2 * #H1 #H2
1299(*    >add_SO in H2; #H2 *)
1300    cases (IH … H2) #E1 #E2 %
1301    [1:
1302      % try @H1
1303      <(add_bitvector_of_nat_Sm 16 (|tl|)) in E1;
1304      <add_associative #assm assumption
1305    |2:
1306      <add_associative in E2;
1307      <(add_bitvector_of_nat_Sm 16 (|tl|)) #assm
1308      assumption
1309    ]
1310  ]
1311qed.
1312
1313lemma destruct_bug_fix_1:
1314  ∀n: nat.
1315    S n = 0 → False.
1316  #n #absurd destruct(absurd)
1317qed.
1318
1319lemma destruct_bug_fix_2:
1320  ∀m, n: nat.
1321    S m = S n → m = n.
1322  #m #n #refl destruct %
1323qed.
1324
1325definition bitvector_3_cases:
1326  ∀b: BitVector 3.
1327    ∃l, c, r: bool.
1328      b ≃ [[l; c; r]].
1329  #b
1330  @(Vector_inv_ind bool 3 b (λn: nat. λv: Vector bool n. ∃l:bool.∃c:bool.∃r:bool. v ≃ [[l; c; r]]))
1331  [1:
1332    #absurd @⊥ -b @(destruct_bug_fix_1 2)
1333    >absurd %
1334  |2:
1335    #n #hd #tl #_ #size_refl #hd_tl_refl %{hd}
1336    cut (n = 2)
1337    [1:
1338      @destruct_bug_fix_2
1339      >size_refl %
1340    |2:
1341      #n_refl >n_refl in tl; #V
1342      @(Vector_inv_ind bool 2 V (λn: nat. λv: Vector bool n. ∃c:bool. ∃r:bool. hd:::v ≃ [[hd; c; r]]))
1343      [1:
1344        #absurd @⊥ -V @(destruct_bug_fix_1 1)
1345        >absurd %
1346      |2:
1347        #n' #hd' #tl' #_ #size_refl' #hd_tl_refl' %{hd'}
1348        cut (n' = 1)
1349        [1:
1350          @destruct_bug_fix_2 >size_refl' %
1351        |2:
1352          #n_refl' >n_refl' in tl'; #V'
1353          @(Vector_inv_ind bool 1 V' (λn: nat. λv: Vector bool n. ∃r: bool. hd:::hd':::v ≃ [[hd; hd'; r]]))
1354          [1:
1355            #absurd @⊥ -V' @(destruct_bug_fix_1 0)
1356            >absurd %
1357          |2:
1358            #n'' #hd'' #tl'' #_ #size_refl'' #hd_tl_refl'' %{hd''}
1359            cut (n'' = 0)
1360            [1:
1361              @destruct_bug_fix_2 >size_refl'' %
1362            |2:
1363              #n_refl'' >n_refl'' in tl''; #tl'''
1364              >(Vector_O … tl''') %
1365            ]
1366          ]
1367        ]
1368      ]
1369    ]
1370  ]
1371qed.
1372
1373lemma bitvector_3_elim_prop:
1374  ∀P: BitVector 3 → Prop.
1375    P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] →
1376    P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] →
1377    P [[true;true;false]] → P [[true;true;true]] → ∀v. P v.
1378  #P #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
1379  cases (bitvector_3_cases … H9) #l #assm cases assm
1380  -assm #c #assm cases assm
1381  -assm #r #assm cases assm destruct
1382  cases l cases c cases r assumption
1383qed.
1384
1385definition ticks_of_instruction ≝
1386  λi.
1387    let trivial_code_memory ≝ assembly1 i in
1388    let trivial_status ≝ load_code_memory trivial_code_memory in
1389      \snd (fetch trivial_status (zero ?)).
1390
1391lemma fetch_assembly:
1392  ∀pc: Word.
1393  ∀i: instruction.
1394  ∀code_memory: BitVectorTrie Byte 16.
1395  ∀assembled: list Byte.
1396    assembled = assembly1 i →
1397      let len ≝ length … assembled in
1398      let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1399        encoding_check code_memory pc pc_plus_len assembled →
1400          let 〈instr, pc', ticks〉 ≝ fetch code_memory pc in
1401           (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' pc_plus_len) = true.
1402  #pc #i #code_memory #assembled cases i [8: *]
1403 [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]]
1404 [47,48,49:
1405 |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg
1406  [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,
1407   59,60,63,64,65,66,67: #ARG]]
1408 [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,
1409  56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2
1410  [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,
1411   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,
1412   68,69,70,71: #ARG2]]
1413 [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3]
1414 normalize in ⊢ (???% → ?);
1415 [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)]
1416  normalize in ⊢ (???% → ?);]
1417 #H >H * #H1 try (whd in ⊢ (% → ?); * #H2)
1418 try (whd in ⊢ (% → ?); * #H3) whd in ⊢ (% → ?); #H4
1419 [ whd in match fetch; normalize nodelta <H1 ] cases daemon
1420(*
1421 whd in ⊢ (let ? ≝ ??% in ?); <H1 whd in ⊢ (let fetched ≝ % in ?)
1422 [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3]
1423 [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,
1424  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,
1425  69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2]
1426 whd >eq_instruction_refl >H4 @eq_bv_refl
1427*) (* XXX: not working! *)
1428qed.
1429
1430let rec fetch_many
1431  (code_memory: BitVectorTrie Byte 16) (final_pc: Word) (pc: Word)
1432    (expected: list instruction)
1433      on expected: Prop ≝
1434  match expected with
1435  [ nil ⇒ eq_bv … pc final_pc = true
1436  | cons i tl ⇒
1437      (∃pc': Word. 〈i, pc', ticks_of_instruction i〉 = fetch code_memory pc ∧
1438        fetch_many code_memory final_pc pc' tl)
1439  ].
1440
1441lemma option_destruct_Some:
1442  ∀A: Type[0].
1443  ∀a, b: A.
1444    Some A a = Some A b → a = b.
1445  #A #a #b #EQ
1446  destruct %
1447qed.
1448
1449lemma eq_instruction_to_eq:
1450  ∀i1, i2: instruction.
1451    eq_instruction i1 i2 = true → i1 ≃ i2.
1452  #i1 #i2
1453  cases i1 cases i2 cases daemon (*
1454  [1,10,19,28,37,46:
1455    #arg1 #arg2
1456    whd in match (eq_instruction ??);
1457    cases arg1 #subaddressing_mode
1458    cases subaddressing_mode
1459    try (#arg1' #arg2' normalize in ⊢ (% → ?); #absurd cases absurd @I)
1460    try (#arg1' normalize in ⊢ (% → ?); #absurd cases absurd @I)
1461    try (normalize in ⊢ (% → ?); #absurd cases absurd @I)
1462    #word11 #irrelevant
1463    cases arg2 #subaddressing_mode
1464    cases subaddressing_mode
1465    try (#arg1' #arg2' normalize in ⊢ (% → ?); #absurd cases absurd @I)
1466    try (#arg1' normalize in ⊢ (% → ?); #absurd cases absurd @I)
1467    try (normalize in ⊢ (% → ?); #absurd cases absurd @I)
1468    #word11' #irrelevant normalize nodelta
1469    #eq_bv_assm cases (eq_bv_eq … eq_bv_assm) % *)
1470qed.
1471         
1472lemma fetch_assembly_pseudo':
1473  ∀lookup_labels.
1474  ∀sigma: Word → Word.
1475  ∀policy: Word → bool.
1476  ∀ppc.
1477  ∀lookup_datalabels.
1478  ∀pi.
1479  ∀code_memory.
1480  ∀len.
1481  ∀assembled.
1482  ∀instructions.
1483    let pc ≝ sigma ppc in
1484      instructions = expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi →
1485        〈len,assembled〉 = assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi →
1486          let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1487            encoding_check code_memory pc pc_plus_len assembled →
1488              fetch_many code_memory pc_plus_len pc instructions.
1489  #lookup_labels #sigma #policy #ppc #lookup_datalabels #pi #code_memory #len #assembled #instructions
1490  normalize nodelta #instructions_refl whd in ⊢ (???% → ?); <instructions_refl whd in ⊢ (???% → ?); #assembled_refl
1491  cases (pair_destruct ?????? assembled_refl) -assembled_refl #len_refl #assembled_refl
1492  >len_refl >assembled_refl -len_refl
1493  generalize in match (add 16 (sigma ppc)
1494    (bitvector_of_nat 16
1495     (|flatten (Vector bool 8)
1496       (map instruction (list (Vector bool 8)) assembly1 instructions)|)));
1497  #final_pc
1498  generalize in match (sigma ppc); elim instructions
1499  [1:
1500    #pc whd in ⊢ (% → %); #H >H @eq_bv_refl
1501  |2:
1502    #i #tl #IH #pc #H whd
1503    cases (encoding_check_append ????? H) -H #H1 #H2
1504    lapply (fetch_assembly pc i code_memory (assembly1 i) (refl …)) whd in ⊢ (% → ?);   
1505    cases (fetch ??) * #instr #pc' #ticks
1506    #H3 lapply (H3 H1) -H3 normalize nodelta #H3
1507    lapply (conjunction_true ?? H3) * #H4 #H5
1508    lapply (conjunction_true … H4) * #B1 #B2
1509    %{pc'} <(eq_instruction_to_eq … B1) >(eq_bv_eq … H5)
1510    >(eqb_true_to_refl … B2) % try % @IH @H2
1511  ]
1512qed.
1513
1514lemma fetch_assembly_pseudo:
1515  ∀program: pseudo_assembly_program.
1516  ∀sigma: Word → Word.
1517  ∀policy: Word → bool.
1518  let lookup_labels ≝ λx:Identifier. sigma (address_of_word_labels_code_mem (\snd  program) x) in
1519  ∀ppc.
1520  ∀code_memory.
1521  let lookup_datalabels ≝ λx:Identifier.lookup_def … (construct_datalabels (\fst  program)) x (zero 16) in
1522  let pi ≝  \fst  (fetch_pseudo_instruction (\snd program) ppc) in
1523  let pc ≝ sigma ppc in
1524  let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in
1525  let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in
1526  let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1527    encoding_check code_memory pc pc_plus_len assembled →
1528      fetch_many code_memory pc_plus_len pc instructions.
1529  #program #sigma #policy letin lookup_labels ≝ (λx.?) #ppc #code_memory
1530  letin lookup_datalabels ≝ (λx.?)
1531  letin pi ≝ (fst ???)
1532  letin pc ≝ (sigma ?)
1533  letin instructions ≝ (expand_pseudo_instruction ??????)
1534  @pair_elim #len #assembled #assembled_refl normalize nodelta
1535  #H
1536  generalize in match
1537   (fetch_assembly_pseudo' lookup_labels sigma policy ppc lookup_datalabels pi code_memory len assembled instructions) in ⊢ ?;
1538  #X destruct normalize nodelta @X try % <assembled_refl try % assumption
1539qed.
1540
1541definition is_present_in_machine_code_image_p: ∀pseudo_instruction. Prop ≝
1542  λpseudo_instruction.
1543    match pseudo_instruction with
1544    [ Comment c ⇒ False
1545    | Cost c ⇒ False
1546    | _ ⇒ True
1547    ].
1548
1549definition sigma_policy_specification ≝
1550  λprogram: pseudo_assembly_program.
1551  λsigma: Word → Word.
1552  λpolicy: Word → bool.
1553  ∀ppc: Word.
1554    let 〈preamble, instr_list〉 ≝ program in
1555    let pc ≝ sigma ppc in
1556    let labels ≝ \fst (create_label_cost_map instr_list) in
1557    let lookup_labels ≝ λx. bitvector_of_nat 16 (lookup_def … labels x 0) in
1558    let instruction ≝ \fst (fetch_pseudo_instruction instr_list ppc) in
1559    let next_pc ≝ sigma (add 16 ppc (bitvector_of_nat 16 1)) in
1560      And (nat_of_bitvector … ppc ≤ |instr_list| →
1561        next_pc = add 16 pc (bitvector_of_nat …
1562          (instruction_size lookup_labels sigma policy ppc instruction)))
1563       (Or (nat_of_bitvector … ppc < |instr_list| →
1564         nat_of_bitvector … pc < nat_of_bitvector … next_pc)
1565        (nat_of_bitvector … ppc = |instr_list| → next_pc = (zero …))).
1566
1567(* This is a trivial consequence of fetch_assembly_pseudo + the proof that the
1568   function that load the code in memory is correct. The latter is based
1569   on missing properties from the standard library on the BitVectorTrie
1570   data structrure.
1571   
1572   Wrong at the moment, requires Jaap's precondition to ensure that the program
1573   does not overflow when put into code memory (i.e. shorter than 2^16 bytes).
1574*)
1575lemma assembly_ok:
1576  ∀program.
1577  ∀sigma: Word → Word.
1578  ∀policy: Word → bool.
1579  ∀sigma_policy_witness: sigma_policy_specification program sigma policy.
1580  ∀assembled.
1581  ∀costs'.
1582  let 〈preamble, instr_list〉 ≝ program in
1583  let 〈labels, costs〉 ≝ create_label_cost_map instr_list in
1584  let datalabels ≝ construct_datalabels preamble in
1585  let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero …) in
1586    〈assembled,costs'〉 = assembly program sigma policy →
1587      costs = costs' ∧
1588        let code_memory ≝ load_code_memory assembled in
1589        let lookup_labels ≝ λx. sigma (address_of_word_labels_code_mem instr_list x) in 
1590          ∀ppc.
1591            let 〈pi, newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in     
1592            let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in
1593            let pc ≝ sigma ppc in
1594            let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1595              encoding_check code_memory pc pc_plus_len assembled ∧
1596                  sigma newppc = add … pc (bitvector_of_nat … len).
1597  #program #sigma #policy #sigma_policy_witness #assembled #costs'
1598  @pair_elim #preamble #instr_list #program_refl
1599  @pair_elim #labels #costs #create_label_cost_refl
1600  #assembly_refl % cases daemon (*
1601  [1:
1602    (* XXX: lemma on BitVectorTries *)
1603    cases daemon
1604  |2:
1605    #ppc #sigma_policy_specification_witness
1606    @pair_elim #pi #newppc #fetch_pseudo_refl
1607    cases pi
1608    [2,3: (* Cost and Comment cases *)
1609      #comment_or_cost normalize in ⊢ (% → ?); #absurd cases absurd
1610    |1: (* PreInstruction cases *)
1611      #preinstruction #_
1612      @pair_elim #len #assembled' #assembly_1_pseudo_refl
1613      normalize nodelta %
1614      [1:
1615        cases assembled' normalize
1616      |2:
1617      ]
1618    ]
1619  ]
1620  cases daemon (* XXX: !!! *) *)
1621qed.
1622
1623(* XXX: should we add that costs = costs'? *)
1624lemma fetch_assembly_pseudo2:
1625  ∀program.
1626  ∀sigma.
1627  ∀policy.
1628  ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy.
1629  ∀ppc.
1630  let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in
1631  let 〈assembled, costs'〉 ≝ assembly program sigma policy in
1632  let code_memory ≝ load_code_memory assembled in
1633  let data_labels ≝ construct_datalabels (\fst program) in
1634  let lookup_labels ≝ λx. sigma (address_of_word_labels_code_mem (\snd program) x) in 
1635  let lookup_datalabels ≝ λx. lookup_def ? ? data_labels x (zero ?) in
1636  let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1637  let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in
1638    fetch_many code_memory (sigma newppc) (sigma ppc) instructions.
1639  * #preamble #instr_list #sigma #policy #sigma_policy_specification_witness #ppc
1640  @pair_elim #labels #costs #create_label_map_refl
1641  @pair_elim #assembled #costs' #assembled_refl
1642  letin code_memory ≝ (load_code_memory ?)
1643  letin data_labels ≝ (construct_datalabels ?)
1644  letin lookup_labels ≝ (λx. ?)
1645  letin lookup_datalabels ≝ (λx. ?)
1646  @pair_elim #pi #newppc #fetch_pseudo_refl
1647  lapply (assembly_ok 〈preamble, instr_list〉 sigma policy sigma_policy_specification_witness assembled costs')
1648  normalize nodelta
1649  @pair_elim #labels' #costs' #create_label_map_refl' #H
1650  cases (H (sym_eq … assembled_refl))
1651  #_
1652  lapply (refl … (assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi))
1653  cases (assembly_1_pseudoinstruction ??????) in ⊢ (???% → ?);
1654  #len #assembledi #EQ4 #H
1655  lapply (H ppc) >fetch_pseudo_refl #H
1656  lapply (fetch_assembly_pseudo 〈preamble,instr_list〉 sigma policy ppc (load_code_memory assembled))
1657  >EQ4 #H1 cases H
1658  #H2 #H3 >H3 normalize nodelta in H1; normalize nodelta
1659  >fetch_pseudo_refl in H1; #assm @assm assumption
1660qed.
1661
1662(* OLD?
1663definition assembly_specification:
1664  ∀assembly_program: pseudo_assembly_program.
1665  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1666  λpseudo_assembly_program.
1667  λcode_mem.
1668    ∀pc: Word.
1669      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1670      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1671      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1672      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1673      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1674       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1675      match pre_assembled with
1676       [ None ⇒ True
1677       | Some pc_code ⇒
1678          let 〈new_pc,code〉 ≝ pc_code in
1679           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1680
1681axiom assembly_meets_specification:
1682  ∀pseudo_assembly_program.
1683    match assembly pseudo_assembly_program with
1684    [ None ⇒ True
1685    | Some code_mem_cost ⇒
1686      let 〈code_mem, cost〉 ≝ code_mem_cost in
1687        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1688    ].
1689(*
1690  # PROGRAM
1691  [ cases PROGRAM
1692    # PREAMBLE
1693    # INSTR_LIST
1694    elim INSTR_LIST
1695    [ whd
1696      whd in ⊢ (∀_. %)
1697      # PC
1698      whd
1699    | # INSTR
1700      # INSTR_LIST_TL
1701      # H
1702      whd
1703      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1704    ]
1705  | cases not_implemented
1706  ] *)
1707*)
1708
1709definition internal_pseudo_address_map ≝ list (BitVector 8).
1710
1711axiom low_internal_ram_of_pseudo_low_internal_ram:
1712 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1713
1714axiom high_internal_ram_of_pseudo_high_internal_ram:
1715 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1716
1717axiom low_internal_ram_of_pseudo_internal_ram_hit:
1718 ∀M:internal_pseudo_address_map.∀cm.∀s:PseudoStatus cm.∀sigma:Word → Word × bool.∀addr:BitVector 7.
1719  member ? (eq_bv 8) (false:::addr) M = true →
1720   let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1721   let pbl ≝ lookup ? 7 addr (low_internal_ram … s) (zero 8) in
1722   let pbu ≝ lookup ? 7 (add ? addr (bitvector_of_nat 7 1)) (low_internal_ram … s) (zero 8) in
1723   let bl ≝ lookup ? 7 addr ram (zero 8) in
1724   let bu ≝ lookup ? 7 (add ? addr (bitvector_of_nat 7 1)) ram (zero 8) in
1725    bu@@bl = \fst (sigma (pbu@@pbl)).
1726
1727(* changed from add to sub *)
1728axiom low_internal_ram_of_pseudo_internal_ram_miss:
1729 ∀T.∀M:internal_pseudo_address_map.∀cm.∀s:PreStatus T cm.∀addr:BitVector 7.
1730  let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1731  let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in
1732  let carr ≝ get_index_v ? ? flags 1 ? in
1733  carr = false →
1734  member ? (eq_bv 8) (false:::Saddr) M = false →
1735   member ? (eq_bv 8) (false:::addr) M = false →
1736    lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?).
1737  //
1738qed.
1739
1740definition addressing_mode_ok ≝
1741 λT.λM:internal_pseudo_address_map.λcm.λs:PreStatus T cm.
1742  λaddr:addressing_mode.
1743   match addr with
1744    [ DIRECT d ⇒
1745       ¬(member ? (eq_bv 8) d M) ∧
1746       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1747    | INDIRECT i ⇒
1748       let d ≝ get_register … s [[false;false;i]] in
1749       ¬(member ? (eq_bv 8) d M) ∧
1750       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1751    | EXT_INDIRECT _ ⇒ true
1752    | REGISTER _ ⇒ true
1753    | ACC_A ⇒ true
1754    | ACC_B ⇒ true
1755    | DPTR ⇒ true
1756    | DATA _ ⇒ true
1757    | DATA16 _ ⇒ true
1758    | ACC_DPTR ⇒ true
1759    | ACC_PC ⇒ true
1760    | EXT_INDIRECT_DPTR ⇒ true
1761    | INDIRECT_DPTR ⇒ true
1762    | CARRY ⇒ true
1763    | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1764    | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1765    | RELATIVE _ ⇒ true
1766    | ADDR11 _ ⇒ true
1767    | ADDR16 _ ⇒ true ].
1768   
1769definition next_internal_pseudo_address_map0 ≝
1770  λT.
1771  λfetched.
1772  λM: internal_pseudo_address_map.
1773  λcm:T.
1774  λs: PreStatus T cm.
1775   match fetched with
1776    [ Comment _ ⇒ Some ? M
1777    | Cost _ ⇒ Some … M
1778    | Jmp _ ⇒ Some … M
1779    | Call _ ⇒
1780       Some … (add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1)::M)
1781    | Mov _ _ ⇒ Some … M
1782    | Instruction instr ⇒
1783       match instr with
1784        [ ADD addr1 addr2 ⇒
1785           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1786            Some ? M
1787           else
1788            None ?
1789        | ADDC addr1 addr2 ⇒
1790           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1791            Some ? M
1792           else
1793            None ?
1794        | SUBB addr1 addr2 ⇒
1795           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1796            Some ? M
1797           else
1798            None ?       
1799        | _ ⇒ (* TO BE COMPLETED *) Some ? M ]].
1800 
1801
1802definition next_internal_pseudo_address_map ≝
1803 λM:internal_pseudo_address_map.
1804 λcm.
1805  λs:PseudoStatus cm.
1806    next_internal_pseudo_address_map0 ?
1807     (\fst (fetch_pseudo_instruction (\snd cm) (program_counter … s))) M cm s.
1808
1809definition code_memory_of_pseudo_assembly_program:
1810    ∀pap:pseudo_assembly_program.
1811      (Word → Word) → (Word → bool) → BitVectorTrie Byte 16 ≝
1812  λpap.
1813  λsigma.
1814  λpolicy.
1815    let p ≝ assembly pap sigma policy in
1816      load_code_memory (\fst p).
1817
1818definition status_of_pseudo_status:
1819    internal_pseudo_address_map → ∀pap. ∀ps: PseudoStatus pap.
1820      ∀sigma: Word → Word. ∀policy: Word → bool.
1821        Status (code_memory_of_pseudo_assembly_program pap sigma policy) ≝
1822  λM.
1823  λpap.
1824  λps.
1825  λsigma.
1826  λpolicy.
1827  let cm ≝ code_memory_of_pseudo_assembly_program … sigma policy in
1828  let pc ≝ sigma (program_counter … ps) in
1829  let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in
1830  let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in
1831     mk_PreStatus (BitVectorTrie Byte 16)
1832      cm
1833      lir
1834      hir
1835      (external_ram … ps)
1836      pc
1837      (special_function_registers_8051 … ps)
1838      (special_function_registers_8052 … ps)
1839      (p1_latch … ps)
1840      (p3_latch … ps)
1841      (clock … ps).
1842
1843(*
1844definition write_at_stack_pointer':
1845 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1846  λM: Type[0].
1847  λs: PreStatus M.
1848  λv: Byte.
1849    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1850    let bit_zero ≝ get_index_v… nu O ? in
1851    let bit_1 ≝ get_index_v… nu 1 ? in
1852    let bit_2 ≝ get_index_v… nu 2 ? in
1853    let bit_3 ≝ get_index_v… nu 3 ? in
1854      if bit_zero then
1855        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1856                              v (low_internal_ram ? s) in
1857          set_low_internal_ram ? s memory
1858      else
1859        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1860                              v (high_internal_ram ? s) in
1861          set_high_internal_ram ? s memory.
1862  [ cases l0 %
1863  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1864qed.
1865
1866definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1867 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1868
1869  λticks_of.
1870  λs.
1871  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1872  let ticks ≝ ticks_of (program_counter ? s) in
1873  let s ≝ set_clock ? s (clock ? s + ticks) in
1874  let s ≝ set_program_counter ? s pc in
1875    match instr with
1876    [ Instruction instr ⇒
1877       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1878    | Comment cmt ⇒ s
1879    | Cost cst ⇒ s
1880    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1881    | Call call ⇒
1882      let a ≝ address_of_word_labels s call in
1883      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1884      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1885      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1886      let s ≝ write_at_stack_pointer' ? s pc_bl in
1887      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1888      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1889      let s ≝ write_at_stack_pointer' ? s pc_bu in
1890        set_program_counter ? s a
1891    | Mov dptr ident ⇒
1892       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1893    ].
1894 [
1895 |2,3,4: %
1896 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1897 |
1898 | %
1899 ]
1900 cases not_implemented
1901qed.
1902*)
1903
1904(*
1905lemma execute_code_memory_unchanged:
1906 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1907 #ticks #ps whd in ⊢ (??? (??%))
1908 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1909  (program_counter pseudo_assembly_program ps)) #instr #pc
1910 whd in ⊢ (??? (??%)) cases instr
1911  [ #pre cases pre
1912     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1913       cases (split ????) #z1 #z2 %
1914     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1915       cases (split ????) #z1 #z2 %
1916     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1917       cases (split ????) #z1 #z2 %
1918     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1919       [ #x1 whd in ⊢ (??? (??%))
1920     | *: cases not_implemented
1921     ]
1922  | #comment %
1923  | #cost %
1924  | #label %
1925  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1926    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1927    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1928    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1929    (* CSC: ??? *)
1930  | #dptr #label (* CSC: ??? *)
1931  ]
1932  cases not_implemented
1933qed.
1934*)
1935
1936(* DEAD CODE?
1937lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1938 ∀M:internal_pseudo_address_map.
1939 ∀ps,ps': PseudoStatus.
1940 ∀pol.
1941  ∀prf:code_memory … ps = code_memory … ps'.
1942   let pol' ≝ ? in
1943   match status_of_pseudo_status M ps pol with
1944    [ None ⇒ status_of_pseudo_status M ps' pol' = None …
1945    | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w
1946    ].
1947 [2: <prf @pol]
1948 #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1949 generalize in match (refl … (assembly (code_memory … ps) pol))
1950 cases (assembly ??) in ⊢ (???% → %)
1951  [ #K whd whd in ⊢ (??%?) <H >K %
1952  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1953qed.
1954*)
1955
1956definition ticks_of0:
1957    ∀p:pseudo_assembly_program.
1958      (Word → Word) → (Word → bool) → Word → pseudo_instruction → nat × nat ≝
1959  λprogram: pseudo_assembly_program.
1960  λsigma.
1961  λpolicy.
1962  λppc: Word.
1963  λfetched.
1964    match fetched with
1965    [ Instruction instr ⇒
1966      match instr with
1967      [ JC lbl ⇒ ? (*
1968        match pol lookup_labels ppc with
1969        [ short_jump ⇒ 〈2, 2〉
1970        | medium_jump ⇒ ?
1971        | long_jump ⇒ 〈4, 4〉
1972        ] *)
1973      | JNC lbl ⇒ ? (*
1974        match pol lookup_labels ppc with
1975        [ short_jump ⇒ 〈2, 2〉
1976        | medium_jump ⇒ ?
1977        | long_jump ⇒ 〈4, 4〉
1978        ] *)
1979      | JB bit lbl ⇒ ? (*
1980        match pol lookup_labels ppc with
1981        [ short_jump ⇒ 〈2, 2〉
1982        | medium_jump ⇒ ?
1983        | long_jump ⇒ 〈4, 4〉
1984        ] *)
1985      | JNB bit lbl ⇒ ? (*
1986        match pol lookup_labels ppc with
1987        [ short_jump ⇒ 〈2, 2〉
1988        | medium_jump ⇒ ?
1989        | long_jump ⇒ 〈4, 4〉
1990        ] *)
1991      | JBC bit lbl ⇒ ? (*
1992        match pol lookup_labels ppc with
1993        [ short_jump ⇒ 〈2, 2〉
1994        | medium_jump ⇒ ?
1995        | long_jump ⇒ 〈4, 4〉
1996        ] *)
1997      | JZ lbl ⇒ ? (*
1998        match pol lookup_labels ppc with
1999        [ short_jump ⇒ 〈2, 2〉
2000        | medium_jump ⇒ ?
2001        | long_jump ⇒ 〈4, 4〉
2002        ] *)
2003      | JNZ lbl ⇒ ? (*
2004        match pol lookup_labels  ppc with
2005        [ short_jump ⇒ 〈2, 2〉
2006        | medium_jump ⇒ ?
2007        | long_jump ⇒ 〈4, 4〉
2008        ] *)
2009      | CJNE arg lbl ⇒ ? (*
2010        match pol lookup_labels ppc with
2011        [ short_jump ⇒ 〈2, 2〉
2012        | medium_jump ⇒ ?
2013        | long_jump ⇒ 〈4, 4〉
2014        ] *)
2015      | DJNZ arg lbl ⇒ ? (*
2016        match pol lookup_labels ppc with
2017        [ short_jump ⇒ 〈2, 2〉
2018        | medium_jump ⇒ ?
2019        | long_jump ⇒ 〈4, 4〉
2020        ] *)
2021      | ADD arg1 arg2 ⇒
2022        let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in
2023         〈ticks, ticks〉
2024      | ADDC arg1 arg2 ⇒
2025        let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in
2026         〈ticks, ticks〉
2027      | SUBB arg1 arg2 ⇒
2028        let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in
2029         〈ticks, ticks〉
2030      | INC arg ⇒
2031        let ticks ≝ ticks_of_instruction (INC ? arg) in
2032         〈ticks, ticks〉
2033      | DEC arg ⇒
2034        let ticks ≝ ticks_of_instruction (DEC ? arg) in
2035         〈ticks, ticks〉
2036      | MUL arg1 arg2 ⇒
2037        let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in
2038         〈ticks, ticks〉
2039      | DIV arg1 arg2 ⇒
2040        let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in
2041         〈ticks, ticks〉
2042      | DA arg ⇒
2043        let ticks ≝ ticks_of_instruction (DA ? arg) in
2044         〈ticks, ticks〉
2045      | ANL arg ⇒
2046        let ticks ≝ ticks_of_instruction (ANL ? arg) in
2047         〈ticks, ticks〉
2048      | ORL arg ⇒
2049        let ticks ≝ ticks_of_instruction (ORL ? arg) in
2050         〈ticks, ticks〉
2051      | XRL arg ⇒
2052        let ticks ≝ ticks_of_instruction (XRL ? arg) in
2053         〈ticks, ticks〉
2054      | CLR arg ⇒
2055        let ticks ≝ ticks_of_instruction (CLR ? arg) in
2056         〈ticks, ticks〉
2057      | CPL arg ⇒
2058        let ticks ≝ ticks_of_instruction (CPL ? arg) in
2059         〈ticks, ticks〉
2060      | RL arg ⇒
2061        let ticks ≝ ticks_of_instruction (RL ? arg) in
2062         〈ticks, ticks〉
2063      | RLC arg ⇒
2064        let ticks ≝ ticks_of_instruction (RLC ? arg) in
2065         〈ticks, ticks〉
2066      | RR arg ⇒
2067        let ticks ≝ ticks_of_instruction (RR ? arg) in
2068         〈ticks, ticks〉
2069      | RRC arg ⇒
2070        let ticks ≝ ticks_of_instruction (RRC ? arg) in
2071         〈ticks, ticks〉
2072      | SWAP arg ⇒
2073        let ticks ≝ ticks_of_instruction (SWAP ? arg) in
2074         〈ticks, ticks〉
2075      | MOV arg ⇒
2076        let ticks ≝ ticks_of_instruction (MOV ? arg) in
2077         〈ticks, ticks〉
2078      | MOVX arg ⇒
2079        let ticks ≝ ticks_of_instruction (MOVX ? arg) in
2080         〈ticks, ticks〉
2081      | SETB arg ⇒
2082        let ticks ≝ ticks_of_instruction (SETB ? arg) in
2083         〈ticks, ticks〉
2084      | PUSH arg ⇒
2085        let ticks ≝ ticks_of_instruction (PUSH ? arg) in
2086         〈ticks, ticks〉
2087      | POP arg ⇒
2088        let ticks ≝ ticks_of_instruction (POP ? arg) in
2089         〈ticks, ticks〉
2090      | XCH arg1 arg2 ⇒
2091        let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in
2092         〈ticks, ticks〉
2093      | XCHD arg1 arg2 ⇒
2094        let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in
2095         〈ticks, ticks〉
2096      | RET ⇒
2097        let ticks ≝ ticks_of_instruction (RET ?) in
2098         〈ticks, ticks〉
2099      | RETI ⇒
2100        let ticks ≝ ticks_of_instruction (RETI ?) in
2101         〈ticks, ticks〉
2102      | NOP ⇒
2103        let ticks ≝ ticks_of_instruction (NOP ?) in
2104         〈ticks, ticks〉
2105      ]
2106    | Comment comment ⇒ 〈0, 0〉
2107    | Cost cost ⇒ 〈0, 0〉
2108    | Jmp jmp ⇒ 〈2, 2〉
2109    | Call call ⇒ 〈2, 2〉
2110    | Mov dptr tgt ⇒ 〈2, 2〉
2111    ].
2112    cases daemon
2113qed.
2114
2115definition ticks_of:
2116    ∀p:pseudo_assembly_program.
2117      (Word → Word) → (Word → bool) → Word → nat × nat ≝
2118  λprogram: pseudo_assembly_program.
2119  λsigma.
2120  λpolicy.
2121  λppc: Word.
2122    let 〈preamble, pseudo〉 ≝ program in
2123    let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in
2124     ticks_of0 program sigma policy ppc fetched.
2125
2126lemma eq_rect_Type1_r:
2127  ∀A: Type[1].
2128  ∀a: A.
2129  ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p.
2130  #A #a #P #H #x #p
2131  generalize in match H;
2132  generalize in match P;
2133  cases p //
2134qed.
2135
2136axiom split_append:
2137  ∀A: Type[0].
2138  ∀m, n: nat.
2139  ∀v, v': Vector A m.
2140  ∀q, q': Vector A n.
2141    let 〈v', q'〉 ≝ split A m n (v@@q) in
2142      v = v' ∧ q = q'.
2143
2144lemma split_vector_singleton:
2145  ∀A: Type[0].
2146  ∀n: nat.
2147  ∀v: Vector A (S n).
2148  ∀rest: Vector A n.
2149  ∀s: Vector A 1.
2150    v = s @@ rest →
2151    ((get_index_v A ? v 0 ?) ::: rest) = v.
2152  [1:
2153    #A #n #v cases daemon (* XXX: !!! *)
2154  |2:
2155    @le_S_S @le_O_n
2156  ]
2157qed.
2158
2159example sub_minus_one_seven_eight:
2160  ∀v: BitVector 7.
2161  false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) =
2162  \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false).
2163 cases daemon.
2164qed.
2165
2166(*
2167lemma blah:
2168  ∀m: internal_pseudo_address_map.
2169  ∀s: PseudoStatus.
2170  ∀arg: Byte.
2171  ∀b: bool.
2172    addressing_mode_ok m s (DIRECT arg) = true →
2173      get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) =
2174      get_arg_8 ? s b (DIRECT arg).
2175  [2, 3: normalize % ]
2176  #m #s #arg #b #hyp
2177  whd in ⊢ (??%%)
2178  @split_elim''
2179  #nu' #nl' #arg_nu_nl_eq
2180  normalize nodelta
2181  generalize in match (refl ? (get_index_v bool 4 nu' ? ?))
2182  cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %)
2183  #get_index_v_eq
2184  normalize nodelta
2185  [2:
2186    normalize nodelta
2187    @split_elim''
2188    #bit_one' #three_bits' #bit_one_three_bit_eq
2189    generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl'))
2190    normalize nodelta
2191    generalize in match (refl ? (sub_7_with_carry ? ? ?))
2192    cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %)
2193    #Saddr #carr' #Saddr_carr_eq
2194    normalize nodelta
2195    #carr_hyp'
2196    @carr_hyp'
2197    [1:
2198    |2: whd in hyp:(??%?); generalize in match hyp; -hyp;
2199        generalize in match (refl ? (¬(member (BitVector 8) ? arg m)))
2200        cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %)
2201        #member_eq
2202        normalize nodelta
2203        [2: #destr destruct(destr)
2204        |1: -carr_hyp';
2205            >arg_nu_nl_eq
2206            <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq)
2207            [1: >get_index_v_eq in ⊢ (??%? → ?)
2208            |2: @le_S @le_S @le_S @le_n
2209            ]
2210            cases (member (BitVector 8) ? (\fst ?) ?)
2211            [1: #destr normalize in destr; destruct(destr)
2212            |2:
2213            ]
2214        ]
2215    |3: >get_index_v_eq in ⊢ (??%?)
2216        change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl')
2217        >(split_vector_singleton … bit_one_three_bit_eq)
2218        <arg_nu_nl_eq
2219        whd in hyp:(??%?)
2220        cases (member (BitVector 8) (eq_bv 8) arg m) in hyp
2221        normalize nodelta [*: #ignore @sym_eq ]
2222    ]
2223  |
2224  ].
2225*)
2226(*
2227map_address0 ... (DIRECT arg) = Some .. →
2228  get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) =
2229  get_arg_8 (internal_ram ...) (DIRECT arg)
2230
2231((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) 
2232                     then Some internal_pseudo_address_map M 
2233                     else None internal_pseudo_address_map )
2234                    =Some internal_pseudo_address_map M')
2235*)
2236
2237axiom low_internal_ram_write_at_stack_pointer:
2238 ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma: Word → Word.∀policy: Word → bool.
2239 ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2240  get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP →
2241  low_internal_ram ? cm2 s2 = low_internal_ram T2 cm3 s3 →
2242  sp1 = add ? (get_8051_sfr … cm1 s1 SFR_SP) (bitvector_of_nat 8 1) →
2243  sp2 = add ? sp1 (bitvector_of_nat 8 1) →
2244  bu@@bl = sigma (pbu@@pbl) →
2245   low_internal_ram T1 cm1
2246     (write_at_stack_pointer …
2247       (set_8051_sfr …
2248         (write_at_stack_pointer …
2249           (set_8051_sfr …
2250             (set_low_internal_ram … s1
2251               (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s2)))
2252             SFR_SP sp1)
2253          bl)
2254        SFR_SP sp2)
2255      bu)
2256   = low_internal_ram_of_pseudo_low_internal_ram (sp1::M)
2257      (low_internal_ram …
2258       (write_at_stack_pointer …
2259         (set_8051_sfr …
2260           (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl)
2261          SFR_SP sp2)
2262        pbu)).
2263
2264lemma high_internal_ram_write_at_stack_pointer:
2265 ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma:Word → Word.∀policy: Word → bool.
2266 ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2267  get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP →
2268  high_internal_ram ?? s2 = high_internal_ram T2 cm3 s3 →
2269  sp1 = add ? (get_8051_sfr ? cm1 s1 SFR_SP) (bitvector_of_nat 8 1) →
2270  sp2 = add ? sp1 (bitvector_of_nat 8 1) →
2271  bu@@bl = sigma (pbu@@pbl) →
2272   high_internal_ram T1 cm1
2273     (write_at_stack_pointer …
2274       (set_8051_sfr …
2275         (write_at_stack_pointer …
2276           (set_8051_sfr …
2277             (set_high_internal_ram … s1
2278               (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … s2)))
2279             SFR_SP sp1)
2280          bl)
2281        SFR_SP sp2)
2282      bu)
2283   = high_internal_ram_of_pseudo_high_internal_ram (sp1::M)
2284      (high_internal_ram …
2285       (write_at_stack_pointer …
2286         (set_8051_sfr …
2287           (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl)
2288          SFR_SP sp2)
2289        pbu)).
2290  #T1 #T2 #M #cm1 #s1 #cm2 #s2 #cm3 #s3 #sigma #policy #pbu #pbl #bu #bl #sp1 #sp2
2291  #get_8051_sfr_refl #high_internal_ram_refl #sp1_refl #sp2_refl #sigma_refl
2292  cases daemon (* XXX: !!! *)
2293qed.
2294
2295lemma Some_Some_elim:
2296 ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P.
2297 #T #x #y #P #H #K @H @option_destruct_Some //
2298qed.
2299
2300lemma pair_destruct_right:
2301  ∀A: Type[0].
2302  ∀B: Type[0].
2303  ∀a, c: A.
2304  ∀b, d: B.
2305    〈a, b〉 = 〈c, d〉 → b = d.
2306  #A #B #a #b #c #d //
2307qed.
2308   
2309(*CSC: ???*)
2310lemma snd_assembly_1_pseudoinstruction_ok:
2311  ∀program: pseudo_assembly_program.
2312  ∀sigma: Word → Word.
2313  ∀policy: Word → bool.
2314  ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy.
2315  ∀ppc: Word.
2316  ∀pi.
2317  ∀lookup_labels.
2318  ∀lookup_datalabels.
2319    lookup_labels = (λx. sigma (address_of_word_labels_code_mem (\snd program) x)) →
2320    lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
2321    \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
2322    let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels sigma policy (*(sigma ppc)*) ppc lookup_datalabels  pi) in
2323      sigma (add … ppc (bitvector_of_nat ? 1)) = add … (sigma ppc) (bitvector_of_nat ? len).
2324  #program #sigma #policy #sigma_policy_specification_witness #ppc #pi
2325  #lookup_labels #lookup_datalabels
2326  #lookup_labels_refl #lookup_datalabels_refl #fetch_pseudo_refl
2327  normalize nodelta
2328  generalize in match fetch_pseudo_refl; -fetch_pseudo_refl
2329  #fetch_pseudo_refl
2330  letin assembled ≝ (\fst (assembly program sigma policy))
2331  letin costs ≝ (\snd (assembly program sigma policy))
2332  lapply (assembly_ok program sigma policy sigma_policy_specification_witness assembled costs)
2333  @pair_elim #preamble #instr_list #program_refl
2334  @pair_elim #labels #costs' #create_label_cost_map_refl
2335  <eq_pair_fst_snd #H cases (H (refl …)) -H #costs_refl #H
2336  lapply (H ppc) -H
2337  @pair_elim #pi' #newppc #fetch_pseudo_refl'
2338  @pair_elim #len #assembled #assembly1_refl #H
2339  cases H
2340  #encoding_check_assm #sigma_newppc_refl
2341  >fetch_pseudo_refl' in fetch_pseudo_refl; #pi_refl'
2342  >pi_refl' in assembly1_refl; #assembly1_refl
2343  >lookup_labels_refl >lookup_datalabels_refl
2344  >program_refl normalize nodelta
2345  >assembly1_refl
2346  <sigma_newppc_refl
2347  generalize in match fetch_pseudo_refl';
2348  whd in match (fetch_pseudo_instruction ??);
2349  @pair_elim #lbl #instr #nth_refl normalize nodelta
2350  #G cases (pair_destruct_right ?????? G) %
2351qed.
2352
2353lemma pose: ∀A:Type[0].∀B:A → Type[0].∀a:A. (∀a':A. a'=a → B a') → B a.
2354  /2/
2355qed.
2356
2357(* To be moved in ProofStatus *)
2358lemma program_counter_set_program_counter:
2359  ∀T.
2360  ∀cm.
2361  ∀s.
2362  ∀x.
2363    program_counter T cm (set_program_counter T cm s x) = x.
2364  //
2365qed.
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