source: src/ASM/AssemblyProof.ma @ 1966

Last change on this file since 1966 was 1966, checked in by mulligan, 8 years ago

Progress made on main_thm proof: trying to find a pattern to use across all ~150 goals

File size: 74.2 KB
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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:
1314  ∀n: nat.
1315    S n = 0 → False.
1316  #n #absurd destruct(absurd)
1317qed.
1318
1319definition bitvector_3_cases:
1320  ∀b: BitVector 3.
1321    ∃l, c, r: bool.
1322      b ≃ [[l; c; r]].
1323  #b
1324  @(Vector_inv_ind bool 3 b (λn: nat. λv: Vector bool n. ∃l:bool.∃c:bool.∃r:bool. v ≃ [[l; c; r]]))
1325  [1:
1326    #absurd @⊥ -b @(destruct_bug_fix 2)
1327    >absurd %
1328  |2:
1329    #n #hd #tl #_ #size_refl #hd_tl_refl %{hd}
1330    cut (n = 2)
1331    [1:
1332    |2:
1333      #n_refl >n_refl in tl;
1334    cases daemon
1335  ]
1336  cases daemon
1337qed.
1338
1339lemma bitvector_3_elim_prop:
1340  ∀P: BitVector 3 → Prop.
1341    P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] →
1342    P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] →
1343    P [[true;true;false]] → P [[true;true;true]] → ∀v. P v.
1344  #P #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
1345  cases (bitvector_3_cases … H9) #l #assm cases assm
1346  -assm #c #assm cases assm
1347  -assm #r #assm cases assm destruct
1348  cases l cases c cases r //
1349qed.
1350
1351definition ticks_of_instruction ≝
1352  λi.
1353    let trivial_code_memory ≝ assembly1 i in
1354    let trivial_status ≝ load_code_memory trivial_code_memory in
1355      \snd (fetch trivial_status (zero ?)).
1356
1357lemma fetch_assembly:
1358  ∀pc: Word.
1359  ∀i: instruction.
1360  ∀code_memory: BitVectorTrie Byte 16.
1361  ∀assembled: list Byte.
1362    assembled = assembly1 i →
1363      let len ≝ length … assembled in
1364      let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1365        encoding_check code_memory pc pc_plus_len assembled →
1366          let 〈instr, pc', ticks〉 ≝ fetch code_memory pc in
1367           (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' pc_plus_len) = true.
1368  #pc #i #code_memory #assembled cases i [8: *]
1369 [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]]
1370 [47,48,49:
1371 |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg
1372  [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,
1373   59,60,63,64,65,66,67: #ARG]]
1374 [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,
1375  56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2
1376  [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,
1377   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,
1378   68,69,70,71: #ARG2]]
1379 [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3]
1380 normalize in ⊢ (???% → ?);
1381 [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)]
1382  normalize in ⊢ (???% → ?);]
1383 #H >H * #H1 try (whd in ⊢ (% → ?); * #H2)
1384 try (whd in ⊢ (% → ?); * #H3) whd in ⊢ (% → ?); #H4
1385 [ whd in match fetch; normalize nodelta <H1 ] cases daemon
1386(*
1387 whd in ⊢ (let ? ≝ ??% in ?); <H1 whd in ⊢ (let fetched ≝ % in ?)
1388 [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3]
1389 [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,
1390  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,
1391  69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2]
1392 whd >eq_instruction_refl >H4 @eq_bv_refl
1393*) (* XXX: not working! *)
1394qed.
1395
1396let rec fetch_many
1397  (code_memory: BitVectorTrie Byte 16) (final_pc: Word) (pc: Word)
1398    (expected: list instruction)
1399      on expected: Prop ≝
1400  match expected with
1401  [ nil ⇒ eq_bv … pc final_pc = true
1402  | cons i tl ⇒
1403    let fetched ≝ fetch code_memory pc in
1404    let 〈instr_pc, ticks〉 ≝ fetched in
1405    let 〈instr,pc'〉 ≝ instr_pc in
1406      eq_instruction instr i = true ∧
1407        ticks = (ticks_of_instruction i) ∧
1408        fetch_many code_memory final_pc pc' tl
1409  ].
1410
1411lemma option_destruct_Some:
1412  ∀A: Type[0].
1413  ∀a, b: A.
1414    Some A a = Some A b → a = b.
1415  #A #a #b #EQ
1416  destruct %
1417qed.
1418
1419axiom eq_instruction_to_eq:
1420  ∀i1, i2: instruction.
1421    eq_instruction i1 i2 = true → i1 ≃ i2.
1422         
1423lemma fetch_assembly_pseudo':
1424  ∀lookup_labels.
1425  ∀sigma: Word → Word.
1426  ∀policy: Word → bool.
1427  ∀ppc.
1428  ∀lookup_datalabels.
1429  ∀pi.
1430  ∀code_memory.
1431  ∀len.
1432  ∀assembled.
1433  ∀instructions.
1434    let pc ≝ sigma ppc in
1435      instructions = expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi →
1436        〈len,assembled〉 = assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi →
1437          let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1438            encoding_check code_memory pc pc_plus_len assembled →
1439              fetch_many code_memory pc_plus_len pc instructions.
1440  #lookup_labels #sigma #policy #ppc #lookup_datalabels #pi #code_memory #len #assembled #instructions
1441  normalize nodelta #instructions_refl whd in ⊢ (???% → ?); <instructions_refl whd in ⊢ (???% → ?); #assembled_refl
1442  cases (pair_destruct ?????? assembled_refl) -assembled_refl #len_refl #assembled_refl
1443  >len_refl >assembled_refl -len_refl
1444  generalize in match (add 16 (sigma ppc)
1445    (bitvector_of_nat 16
1446     (|flatten (Vector bool 8)
1447       (map instruction (list (Vector bool 8)) assembly1 instructions)|)));
1448  #final_pc
1449  generalize in match (sigma ppc); elim instructions
1450  [1:
1451    #pc whd in ⊢ (% → %); #H >H @eq_bv_refl
1452  |2:
1453    #i #tl #IH #pc #H whd
1454    cases (encoding_check_append ????? H) -H #H1 #H2
1455    @pair_elim #instr_pc #ticks #fetch_refl normalize nodelta
1456    @pair_elim #instr #pc' #instr_pc_refl normalize nodelta
1457    lapply (fetch_assembly pc i code_memory (assembly1 i) (refl …)) whd in ⊢ (% → ?);
1458    #H3 lapply (H3 H1) -H3 >fetch_refl >instr_pc_refl normalize nodelta
1459    #H3 lapply (conjunction_true ?? H3) * #H4 #H5 %
1460    [1:
1461      lapply (conjunction_true … H4) * #B1 #B2
1462      % try assumption @eqb_true_to_eq
1463      <(eq_instruction_to_eq … B1) assumption
1464    |2:
1465      >(eq_bv_eq … H5) @IH @H2
1466    ]
1467  ]
1468qed.
1469
1470lemma fetch_assembly_pseudo:
1471  ∀program: pseudo_assembly_program.
1472  ∀sigma: Word → Word.
1473  ∀policy: Word → bool.
1474  let lookup_labels ≝ λx:Identifier. sigma (address_of_word_labels_code_mem (\snd  program) x) in
1475  ∀ppc.
1476  ∀code_memory.
1477  let lookup_datalabels ≝ λx:Identifier.lookup_def … (construct_datalabels (\fst  program)) x (zero 16) in
1478  let pi ≝  \fst  (fetch_pseudo_instruction (\snd program) ppc) in
1479  let pc ≝ sigma ppc in
1480  let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in
1481  let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in
1482  let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1483    encoding_check code_memory pc pc_plus_len assembled →
1484      fetch_many code_memory pc_plus_len pc instructions.
1485  #program #sigma #policy letin lookup_labels ≝ (λx.?) #ppc #code_memory
1486  letin lookup_datalabels ≝ (λx.?)
1487  letin pi ≝ (fst ???)
1488  letin pc ≝ (sigma ?)
1489  letin instructions ≝ (expand_pseudo_instruction ??????)
1490  @pair_elim #len #assembled #assembled_refl normalize nodelta
1491  #H
1492  generalize in match
1493   (fetch_assembly_pseudo' lookup_labels sigma policy ppc lookup_datalabels pi code_memory len assembled instructions) in ⊢ ?;
1494  #X destruct normalize nodelta @X try % <assembled_refl try % assumption
1495qed.
1496
1497definition is_present_in_machine_code_image_p: ∀pseudo_instruction. Prop ≝
1498  λpseudo_instruction.
1499    match pseudo_instruction with
1500    [ Comment c ⇒ False
1501    | Cost c ⇒ False
1502    | _ ⇒ True
1503    ].
1504
1505definition sigma_policy_specification ≝
1506  λprogram: pseudo_assembly_program.
1507  λsigma: Word → Word.
1508  λpolicy: Word → bool.
1509  ∀ppc: Word.
1510    let 〈preamble, instr_list〉 ≝ program in
1511    let pc ≝ sigma ppc in
1512    let labels ≝ \fst (create_label_cost_map instr_list) in
1513    let lookup_labels ≝ λx. bitvector_of_nat 16 (lookup_def … labels x 0) in
1514    let instruction ≝ \fst (fetch_pseudo_instruction instr_list ppc) in
1515    let next_pc ≝ sigma (add 16 ppc (bitvector_of_nat 16 1)) in
1516      And (nat_of_bitvector … ppc ≤ |instr_list| →
1517        next_pc = add 16 pc (bitvector_of_nat …
1518          (instruction_size lookup_labels sigma policy ppc instruction)))
1519       (Or (nat_of_bitvector … ppc < |instr_list| →
1520         nat_of_bitvector … pc < nat_of_bitvector … next_pc)
1521        (nat_of_bitvector … ppc = |instr_list| → next_pc = (zero …))).
1522
1523(* This is a trivial consequence of fetch_assembly_pseudo + the proof that the
1524   function that load the code in memory is correct. The latter is based
1525   on missing properties from the standard library on the BitVectorTrie
1526   data structrure.
1527   
1528   Wrong at the moment, requires Jaap's precondition to ensure that the program
1529   does not overflow when put into code memory (i.e. shorter than 2^16 bytes).
1530*)
1531lemma assembly_ok:
1532  ∀program.
1533  ∀sigma: Word → Word.
1534  ∀policy: Word → bool.
1535  ∀sigma_policy_witness: sigma_policy_specification program sigma policy.
1536  ∀assembled.
1537  ∀costs'.
1538  let 〈preamble, instr_list〉 ≝ program in
1539  let 〈labels, costs〉 ≝ create_label_cost_map instr_list in
1540  let datalabels ≝ construct_datalabels preamble in
1541  let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero …) in
1542    〈assembled,costs'〉 = assembly program sigma policy →
1543      costs = costs' ∧
1544        let code_memory ≝ load_code_memory assembled in
1545        let lookup_labels ≝ λx. sigma (address_of_word_labels_code_mem instr_list x) in 
1546          ∀ppc.
1547            let 〈pi, newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in     
1548            let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in
1549            let pc ≝ sigma ppc in
1550            let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1551              encoding_check code_memory pc pc_plus_len assembled ∧
1552                  sigma newppc = add … pc (bitvector_of_nat … len).
1553  #program #sigma #policy #sigma_policy_witness #assembled #costs'
1554  @pair_elim #preamble #instr_list #program_refl
1555  @pair_elim #labels #costs #create_label_cost_refl
1556  #assembly_refl % cases daemon (*
1557  [1:
1558    (* XXX: lemma on BitVectorTries *)
1559    cases daemon
1560  |2:
1561    #ppc #sigma_policy_specification_witness
1562    @pair_elim #pi #newppc #fetch_pseudo_refl
1563    cases pi
1564    [2,3: (* Cost and Comment cases *)
1565      #comment_or_cost normalize in ⊢ (% → ?); #absurd cases absurd
1566    |1: (* PreInstruction cases *)
1567      #preinstruction #_
1568      @pair_elim #len #assembled' #assembly_1_pseudo_refl
1569      normalize nodelta %
1570      [1:
1571        cases assembled' normalize
1572      |2:
1573      ]
1574    ]
1575  ]
1576  cases daemon (* XXX: !!! *) *)
1577qed.
1578
1579(* XXX: should we add that costs = costs'? *)
1580lemma fetch_assembly_pseudo2:
1581  ∀program.
1582  ∀sigma.
1583  ∀policy.
1584  ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy.
1585  ∀ppc.
1586  let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in
1587  let 〈assembled, costs'〉 ≝ assembly program sigma policy in
1588  let code_memory ≝ load_code_memory assembled in
1589  let data_labels ≝ construct_datalabels (\fst program) in
1590  let lookup_labels ≝ λx. sigma (address_of_word_labels_code_mem (\snd program) x) in 
1591  let lookup_datalabels ≝ λx. lookup_def ? ? data_labels x (zero ?) in
1592  let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1593  let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in
1594    fetch_many code_memory (sigma newppc) (sigma ppc) instructions.
1595  * #preamble #instr_list #sigma #policy #sigma_policy_specification_witness #ppc
1596  @pair_elim #labels #costs #create_label_map_refl
1597  @pair_elim #assembled #costs' #assembled_refl
1598  letin code_memory ≝ (load_code_memory ?)
1599  letin data_labels ≝ (construct_datalabels ?)
1600  letin lookup_labels ≝ (λx. ?)
1601  letin lookup_datalabels ≝ (λx. ?)
1602  @pair_elim #pi #newppc #fetch_pseudo_refl
1603  lapply (assembly_ok 〈preamble, instr_list〉 sigma policy sigma_policy_specification_witness assembled costs')
1604  normalize nodelta
1605  @pair_elim #labels' #costs' #create_label_map_refl' #H
1606  cases (H (sym_eq … assembled_refl))
1607  #_
1608  lapply (refl … (assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi))
1609  cases (assembly_1_pseudoinstruction ??????) in ⊢ (???% → ?);
1610  #len #assembledi #EQ4 #H
1611  lapply (H ppc) >fetch_pseudo_refl #H
1612  lapply (fetch_assembly_pseudo 〈preamble,instr_list〉 sigma policy ppc (load_code_memory assembled))
1613  >EQ4 #H1 cases H
1614  #H2 #H3 >H3 normalize nodelta in H1; normalize nodelta
1615  >fetch_pseudo_refl in H1; #assm @assm assumption
1616qed.
1617
1618(* OLD?
1619definition assembly_specification:
1620  ∀assembly_program: pseudo_assembly_program.
1621  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1622  λpseudo_assembly_program.
1623  λcode_mem.
1624    ∀pc: Word.
1625      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1626      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1627      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1628      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1629      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1630       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1631      match pre_assembled with
1632       [ None ⇒ True
1633       | Some pc_code ⇒
1634          let 〈new_pc,code〉 ≝ pc_code in
1635           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1636
1637axiom assembly_meets_specification:
1638  ∀pseudo_assembly_program.
1639    match assembly pseudo_assembly_program with
1640    [ None ⇒ True
1641    | Some code_mem_cost ⇒
1642      let 〈code_mem, cost〉 ≝ code_mem_cost in
1643        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1644    ].
1645(*
1646  # PROGRAM
1647  [ cases PROGRAM
1648    # PREAMBLE
1649    # INSTR_LIST
1650    elim INSTR_LIST
1651    [ whd
1652      whd in ⊢ (∀_. %)
1653      # PC
1654      whd
1655    | # INSTR
1656      # INSTR_LIST_TL
1657      # H
1658      whd
1659      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1660    ]
1661  | cases not_implemented
1662  ] *)
1663*)
1664
1665definition internal_pseudo_address_map ≝ list (BitVector 8).
1666
1667axiom low_internal_ram_of_pseudo_low_internal_ram:
1668 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1669
1670axiom high_internal_ram_of_pseudo_high_internal_ram:
1671 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1672
1673axiom low_internal_ram_of_pseudo_internal_ram_hit:
1674 ∀M:internal_pseudo_address_map.∀cm.∀s:PseudoStatus cm.∀sigma:Word → Word × bool.∀addr:BitVector 7.
1675  member ? (eq_bv 8) (false:::addr) M = true →
1676   let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1677   let pbl ≝ lookup ? 7 addr (low_internal_ram … s) (zero 8) in
1678   let pbu ≝ lookup ? 7 (add ? addr (bitvector_of_nat 7 1)) (low_internal_ram … s) (zero 8) in
1679   let bl ≝ lookup ? 7 addr ram (zero 8) in
1680   let bu ≝ lookup ? 7 (add ? addr (bitvector_of_nat 7 1)) ram (zero 8) in
1681    bu@@bl = \fst (sigma (pbu@@pbl)).
1682
1683(* changed from add to sub *)
1684axiom low_internal_ram_of_pseudo_internal_ram_miss:
1685 ∀T.∀M:internal_pseudo_address_map.∀cm.∀s:PreStatus T cm.∀addr:BitVector 7.
1686  let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1687  let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in
1688  let carr ≝ get_index_v ? ? flags 1 ? in
1689  carr = false →
1690  member ? (eq_bv 8) (false:::Saddr) M = false →
1691   member ? (eq_bv 8) (false:::addr) M = false →
1692    lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?).
1693  //
1694qed.
1695
1696definition addressing_mode_ok ≝
1697 λT.λM:internal_pseudo_address_map.λcm.λs:PreStatus T cm.
1698  λaddr:addressing_mode.
1699   match addr with
1700    [ DIRECT d ⇒
1701       ¬(member ? (eq_bv 8) d M) ∧
1702       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1703    | INDIRECT i ⇒
1704       let d ≝ get_register … s [[false;false;i]] in
1705       ¬(member ? (eq_bv 8) d M) ∧
1706       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1707    | EXT_INDIRECT _ ⇒ true
1708    | REGISTER _ ⇒ true
1709    | ACC_A ⇒ true
1710    | ACC_B ⇒ true
1711    | DPTR ⇒ true
1712    | DATA _ ⇒ true
1713    | DATA16 _ ⇒ true
1714    | ACC_DPTR ⇒ true
1715    | ACC_PC ⇒ true
1716    | EXT_INDIRECT_DPTR ⇒ true
1717    | INDIRECT_DPTR ⇒ true
1718    | CARRY ⇒ true
1719    | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1720    | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1721    | RELATIVE _ ⇒ true
1722    | ADDR11 _ ⇒ true
1723    | ADDR16 _ ⇒ true ].
1724   
1725definition next_internal_pseudo_address_map0 ≝
1726  λT.
1727  λfetched.
1728  λM: internal_pseudo_address_map.
1729  λcm:T.
1730  λs: PreStatus T cm.
1731   match fetched with
1732    [ Comment _ ⇒ Some ? M
1733    | Cost _ ⇒ Some … M
1734    | Jmp _ ⇒ Some … M
1735    | Call _ ⇒
1736       Some … (add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1)::M)
1737    | Mov _ _ ⇒ Some … M
1738    | Instruction instr ⇒
1739       match instr with
1740        [ ADD addr1 addr2 ⇒
1741           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1742            Some ? M
1743           else
1744            None ?
1745        | ADDC addr1 addr2 ⇒
1746           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1747            Some ? M
1748           else
1749            None ?
1750        | SUBB addr1 addr2 ⇒
1751           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1752            Some ? M
1753           else
1754            None ?       
1755        | _ ⇒ (* TO BE COMPLETED *) Some ? M ]].
1756 
1757
1758definition next_internal_pseudo_address_map ≝
1759 λM:internal_pseudo_address_map.
1760 λcm.
1761  λs:PseudoStatus cm.
1762    next_internal_pseudo_address_map0 ?
1763     (\fst (fetch_pseudo_instruction (\snd cm) (program_counter … s))) M cm s.
1764
1765definition code_memory_of_pseudo_assembly_program:
1766    ∀pap:pseudo_assembly_program.
1767      (Word → Word) → (Word → bool) → BitVectorTrie Byte 16 ≝
1768  λpap.
1769  λsigma.
1770  λpolicy.
1771    let p ≝ assembly pap sigma policy in
1772      load_code_memory (\fst p).
1773
1774definition status_of_pseudo_status:
1775    internal_pseudo_address_map → ∀pap. ∀ps: PseudoStatus pap.
1776      ∀sigma: Word → Word. ∀policy: Word → bool.
1777        Status (code_memory_of_pseudo_assembly_program pap sigma policy) ≝
1778  λM.
1779  λpap.
1780  λps.
1781  λsigma.
1782  λpolicy.
1783  let cm ≝ code_memory_of_pseudo_assembly_program … sigma policy in
1784  let pc ≝ sigma (program_counter … ps) in
1785  let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in
1786  let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in
1787     mk_PreStatus (BitVectorTrie Byte 16)
1788      cm
1789      lir
1790      hir
1791      (external_ram … ps)
1792      pc
1793      (special_function_registers_8051 … ps)
1794      (special_function_registers_8052 … ps)
1795      (p1_latch … ps)
1796      (p3_latch … ps)
1797      (clock … ps).
1798
1799(*
1800definition write_at_stack_pointer':
1801 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1802  λM: Type[0].
1803  λs: PreStatus M.
1804  λv: Byte.
1805    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1806    let bit_zero ≝ get_index_v… nu O ? in
1807    let bit_1 ≝ get_index_v… nu 1 ? in
1808    let bit_2 ≝ get_index_v… nu 2 ? in
1809    let bit_3 ≝ get_index_v… nu 3 ? in
1810      if bit_zero then
1811        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1812                              v (low_internal_ram ? s) in
1813          set_low_internal_ram ? s memory
1814      else
1815        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1816                              v (high_internal_ram ? s) in
1817          set_high_internal_ram ? s memory.
1818  [ cases l0 %
1819  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1820qed.
1821
1822definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1823 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1824
1825  λticks_of.
1826  λs.
1827  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1828  let ticks ≝ ticks_of (program_counter ? s) in
1829  let s ≝ set_clock ? s (clock ? s + ticks) in
1830  let s ≝ set_program_counter ? s pc in
1831    match instr with
1832    [ Instruction instr ⇒
1833       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1834    | Comment cmt ⇒ s
1835    | Cost cst ⇒ s
1836    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1837    | Call call ⇒
1838      let a ≝ address_of_word_labels s call in
1839      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1840      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1841      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1842      let s ≝ write_at_stack_pointer' ? s pc_bl in
1843      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1844      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1845      let s ≝ write_at_stack_pointer' ? s pc_bu in
1846        set_program_counter ? s a
1847    | Mov dptr ident ⇒
1848       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1849    ].
1850 [
1851 |2,3,4: %
1852 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1853 |
1854 | %
1855 ]
1856 cases not_implemented
1857qed.
1858*)
1859
1860(*
1861lemma execute_code_memory_unchanged:
1862 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1863 #ticks #ps whd in ⊢ (??? (??%))
1864 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1865  (program_counter pseudo_assembly_program ps)) #instr #pc
1866 whd in ⊢ (??? (??%)) cases instr
1867  [ #pre cases pre
1868     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1869       cases (split ????) #z1 #z2 %
1870     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1871       cases (split ????) #z1 #z2 %
1872     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1873       cases (split ????) #z1 #z2 %
1874     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1875       [ #x1 whd in ⊢ (??? (??%))
1876     | *: cases not_implemented
1877     ]
1878  | #comment %
1879  | #cost %
1880  | #label %
1881  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1882    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1883    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1884    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1885    (* CSC: ??? *)
1886  | #dptr #label (* CSC: ??? *)
1887  ]
1888  cases not_implemented
1889qed.
1890*)
1891
1892(* DEAD CODE?
1893lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1894 ∀M:internal_pseudo_address_map.
1895 ∀ps,ps': PseudoStatus.
1896 ∀pol.
1897  ∀prf:code_memory … ps = code_memory … ps'.
1898   let pol' ≝ ? in
1899   match status_of_pseudo_status M ps pol with
1900    [ None ⇒ status_of_pseudo_status M ps' pol' = None …
1901    | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w
1902    ].
1903 [2: <prf @pol]
1904 #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1905 generalize in match (refl … (assembly (code_memory … ps) pol))
1906 cases (assembly ??) in ⊢ (???% → %)
1907  [ #K whd whd in ⊢ (??%?) <H >K %
1908  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1909qed.
1910*)
1911
1912definition ticks_of0:
1913    ∀p:pseudo_assembly_program.
1914      (Word → Word) → (Word → bool) → Word → pseudo_instruction → nat × nat ≝
1915  λprogram: pseudo_assembly_program.
1916  λsigma.
1917  λpolicy.
1918  λppc: Word.
1919  λfetched.
1920    match fetched with
1921    [ Instruction instr ⇒
1922      match instr with
1923      [ JC lbl ⇒ ? (*
1924        match pol lookup_labels ppc with
1925        [ short_jump ⇒ 〈2, 2〉
1926        | medium_jump ⇒ ?
1927        | long_jump ⇒ 〈4, 4〉
1928        ] *)
1929      | JNC lbl ⇒ ? (*
1930        match pol lookup_labels ppc with
1931        [ short_jump ⇒ 〈2, 2〉
1932        | medium_jump ⇒ ?
1933        | long_jump ⇒ 〈4, 4〉
1934        ] *)
1935      | JB bit lbl ⇒ ? (*
1936        match pol lookup_labels ppc with
1937        [ short_jump ⇒ 〈2, 2〉
1938        | medium_jump ⇒ ?
1939        | long_jump ⇒ 〈4, 4〉
1940        ] *)
1941      | JNB bit lbl ⇒ ? (*
1942        match pol lookup_labels ppc with
1943        [ short_jump ⇒ 〈2, 2〉
1944        | medium_jump ⇒ ?
1945        | long_jump ⇒ 〈4, 4〉
1946        ] *)
1947      | JBC bit lbl ⇒ ? (*
1948        match pol lookup_labels ppc with
1949        [ short_jump ⇒ 〈2, 2〉
1950        | medium_jump ⇒ ?
1951        | long_jump ⇒ 〈4, 4〉
1952        ] *)
1953      | JZ lbl ⇒ ? (*
1954        match pol lookup_labels ppc with
1955        [ short_jump ⇒ 〈2, 2〉
1956        | medium_jump ⇒ ?
1957        | long_jump ⇒ 〈4, 4〉
1958        ] *)
1959      | JNZ lbl ⇒ ? (*
1960        match pol lookup_labels  ppc with
1961        [ short_jump ⇒ 〈2, 2〉
1962        | medium_jump ⇒ ?
1963        | long_jump ⇒ 〈4, 4〉
1964        ] *)
1965      | CJNE arg lbl ⇒ ? (*
1966        match pol lookup_labels ppc with
1967        [ short_jump ⇒ 〈2, 2〉
1968        | medium_jump ⇒ ?
1969        | long_jump ⇒ 〈4, 4〉
1970        ] *)
1971      | DJNZ arg lbl ⇒ ? (*
1972        match pol lookup_labels ppc with
1973        [ short_jump ⇒ 〈2, 2〉
1974        | medium_jump ⇒ ?
1975        | long_jump ⇒ 〈4, 4〉
1976        ] *)
1977      | ADD arg1 arg2 ⇒
1978        let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in
1979         〈ticks, ticks〉
1980      | ADDC arg1 arg2 ⇒
1981        let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in
1982         〈ticks, ticks〉
1983      | SUBB arg1 arg2 ⇒
1984        let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in
1985         〈ticks, ticks〉
1986      | INC arg ⇒
1987        let ticks ≝ ticks_of_instruction (INC ? arg) in
1988         〈ticks, ticks〉
1989      | DEC arg ⇒
1990        let ticks ≝ ticks_of_instruction (DEC ? arg) in
1991         〈ticks, ticks〉
1992      | MUL arg1 arg2 ⇒
1993        let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in
1994         〈ticks, ticks〉
1995      | DIV arg1 arg2 ⇒
1996        let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in
1997         〈ticks, ticks〉
1998      | DA arg ⇒
1999        let ticks ≝ ticks_of_instruction (DA ? arg) in
2000         〈ticks, ticks〉
2001      | ANL arg ⇒
2002        let ticks ≝ ticks_of_instruction (ANL ? arg) in
2003         〈ticks, ticks〉
2004      | ORL arg ⇒
2005        let ticks ≝ ticks_of_instruction (ORL ? arg) in
2006         〈ticks, ticks〉
2007      | XRL arg ⇒
2008        let ticks ≝ ticks_of_instruction (XRL ? arg) in
2009         〈ticks, ticks〉
2010      | CLR arg ⇒
2011        let ticks ≝ ticks_of_instruction (CLR ? arg) in
2012         〈ticks, ticks〉
2013      | CPL arg ⇒
2014        let ticks ≝ ticks_of_instruction (CPL ? arg) in
2015         〈ticks, ticks〉
2016      | RL arg ⇒
2017        let ticks ≝ ticks_of_instruction (RL ? arg) in
2018         〈ticks, ticks〉
2019      | RLC arg ⇒
2020        let ticks ≝ ticks_of_instruction (RLC ? arg) in
2021         〈ticks, ticks〉
2022      | RR arg ⇒
2023        let ticks ≝ ticks_of_instruction (RR ? arg) in
2024         〈ticks, ticks〉
2025      | RRC arg ⇒
2026        let ticks ≝ ticks_of_instruction (RRC ? arg) in
2027         〈ticks, ticks〉
2028      | SWAP arg ⇒
2029        let ticks ≝ ticks_of_instruction (SWAP ? arg) in
2030         〈ticks, ticks〉
2031      | MOV arg ⇒
2032        let ticks ≝ ticks_of_instruction (MOV ? arg) in
2033         〈ticks, ticks〉
2034      | MOVX arg ⇒
2035        let ticks ≝ ticks_of_instruction (MOVX ? arg) in
2036         〈ticks, ticks〉
2037      | SETB arg ⇒
2038        let ticks ≝ ticks_of_instruction (SETB ? arg) in
2039         〈ticks, ticks〉
2040      | PUSH arg ⇒
2041        let ticks ≝ ticks_of_instruction (PUSH ? arg) in
2042         〈ticks, ticks〉
2043      | POP arg ⇒
2044        let ticks ≝ ticks_of_instruction (POP ? arg) in
2045         〈ticks, ticks〉
2046      | XCH arg1 arg2 ⇒
2047        let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in
2048         〈ticks, ticks〉
2049      | XCHD arg1 arg2 ⇒
2050        let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in
2051         〈ticks, ticks〉
2052      | RET ⇒
2053        let ticks ≝ ticks_of_instruction (RET ?) in
2054         〈ticks, ticks〉
2055      | RETI ⇒
2056        let ticks ≝ ticks_of_instruction (RETI ?) in
2057         〈ticks, ticks〉
2058      | NOP ⇒
2059        let ticks ≝ ticks_of_instruction (NOP ?) in
2060         〈ticks, ticks〉
2061      ]
2062    | Comment comment ⇒ 〈0, 0〉
2063    | Cost cost ⇒ 〈0, 0〉
2064    | Jmp jmp ⇒ 〈2, 2〉
2065    | Call call ⇒ 〈2, 2〉
2066    | Mov dptr tgt ⇒ 〈2, 2〉
2067    ].
2068    cases daemon
2069qed.
2070
2071definition ticks_of:
2072    ∀p:pseudo_assembly_program.
2073      (Word → Word) → (Word → bool) → Word → nat × nat ≝
2074  λprogram: pseudo_assembly_program.
2075  λsigma.
2076  λpolicy.
2077  λppc: Word.
2078    let 〈preamble, pseudo〉 ≝ program in
2079    let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in
2080     ticks_of0 program sigma policy ppc fetched.
2081
2082lemma eq_rect_Type1_r:
2083  ∀A: Type[1].
2084  ∀a: A.
2085  ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p.
2086  #A #a #P #H #x #p
2087  generalize in match H;
2088  generalize in match P;
2089  cases p //
2090qed.
2091
2092axiom split_append:
2093  ∀A: Type[0].
2094  ∀m, n: nat.
2095  ∀v, v': Vector A m.
2096  ∀q, q': Vector A n.
2097    let 〈v', q'〉 ≝ split A m n (v@@q) in
2098      v = v' ∧ q = q'.
2099
2100lemma split_vector_singleton:
2101  ∀A: Type[0].
2102  ∀n: nat.
2103  ∀v: Vector A (S n).
2104  ∀rest: Vector A n.
2105  ∀s: Vector A 1.
2106    v = s @@ rest →
2107    ((get_index_v A ? v 0 ?) ::: rest) = v.
2108  [1:
2109    #A #n #v cases daemon (* XXX: !!! *)
2110  |2:
2111    @le_S_S @le_O_n
2112  ]
2113qed.
2114
2115example sub_minus_one_seven_eight:
2116  ∀v: BitVector 7.
2117  false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) =
2118  \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false).
2119 cases daemon.
2120qed.
2121
2122(*
2123lemma blah:
2124  ∀m: internal_pseudo_address_map.
2125  ∀s: PseudoStatus.
2126  ∀arg: Byte.
2127  ∀b: bool.
2128    addressing_mode_ok m s (DIRECT arg) = true →
2129      get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) =
2130      get_arg_8 ? s b (DIRECT arg).
2131  [2, 3: normalize % ]
2132  #m #s #arg #b #hyp
2133  whd in ⊢ (??%%)
2134  @split_elim''
2135  #nu' #nl' #arg_nu_nl_eq
2136  normalize nodelta
2137  generalize in match (refl ? (get_index_v bool 4 nu' ? ?))
2138  cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %)
2139  #get_index_v_eq
2140  normalize nodelta
2141  [2:
2142    normalize nodelta
2143    @split_elim''
2144    #bit_one' #three_bits' #bit_one_three_bit_eq
2145    generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl'))
2146    normalize nodelta
2147    generalize in match (refl ? (sub_7_with_carry ? ? ?))
2148    cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %)
2149    #Saddr #carr' #Saddr_carr_eq
2150    normalize nodelta
2151    #carr_hyp'
2152    @carr_hyp'
2153    [1:
2154    |2: whd in hyp:(??%?); generalize in match hyp; -hyp;
2155        generalize in match (refl ? (¬(member (BitVector 8) ? arg m)))
2156        cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %)
2157        #member_eq
2158        normalize nodelta
2159        [2: #destr destruct(destr)
2160        |1: -carr_hyp';
2161            >arg_nu_nl_eq
2162            <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq)
2163            [1: >get_index_v_eq in ⊢ (??%? → ?)
2164            |2: @le_S @le_S @le_S @le_n
2165            ]
2166            cases (member (BitVector 8) ? (\fst ?) ?)
2167            [1: #destr normalize in destr; destruct(destr)
2168            |2:
2169            ]
2170        ]
2171    |3: >get_index_v_eq in ⊢ (??%?)
2172        change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl')
2173        >(split_vector_singleton … bit_one_three_bit_eq)
2174        <arg_nu_nl_eq
2175        whd in hyp:(??%?)
2176        cases (member (BitVector 8) (eq_bv 8) arg m) in hyp
2177        normalize nodelta [*: #ignore @sym_eq ]
2178    ]
2179  |
2180  ].
2181*)
2182(*
2183map_address0 ... (DIRECT arg) = Some .. →
2184  get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) =
2185  get_arg_8 (internal_ram ...) (DIRECT arg)
2186
2187((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) 
2188                     then Some internal_pseudo_address_map M 
2189                     else None internal_pseudo_address_map )
2190                    =Some internal_pseudo_address_map M')
2191*)
2192
2193axiom low_internal_ram_write_at_stack_pointer:
2194 ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma: Word → Word.∀policy: Word → bool.
2195 ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2196  get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP →
2197  low_internal_ram ? cm2 s2 = low_internal_ram T2 cm3 s3 →
2198  sp1 = add ? (get_8051_sfr … cm1 s1 SFR_SP) (bitvector_of_nat 8 1) →
2199  sp2 = add ? sp1 (bitvector_of_nat 8 1) →
2200  bu@@bl = sigma (pbu@@pbl) →
2201   low_internal_ram T1 cm1
2202     (write_at_stack_pointer …
2203       (set_8051_sfr …
2204         (write_at_stack_pointer …
2205           (set_8051_sfr …
2206             (set_low_internal_ram … s1
2207               (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s2)))
2208             SFR_SP sp1)
2209          bl)
2210        SFR_SP sp2)
2211      bu)
2212   = low_internal_ram_of_pseudo_low_internal_ram (sp1::M)
2213      (low_internal_ram …
2214       (write_at_stack_pointer …
2215         (set_8051_sfr …
2216           (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl)
2217          SFR_SP sp2)
2218        pbu)).
2219
2220lemma high_internal_ram_write_at_stack_pointer:
2221 ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma:Word → Word.∀policy: Word → bool.
2222 ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2223  get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP →
2224  high_internal_ram ?? s2 = high_internal_ram T2 cm3 s3 →
2225  sp1 = add ? (get_8051_sfr ? cm1 s1 SFR_SP) (bitvector_of_nat 8 1) →
2226  sp2 = add ? sp1 (bitvector_of_nat 8 1) →
2227  bu@@bl = sigma (pbu@@pbl) →
2228   high_internal_ram T1 cm1
2229     (write_at_stack_pointer …
2230       (set_8051_sfr …
2231         (write_at_stack_pointer …
2232           (set_8051_sfr …
2233             (set_high_internal_ram … s1
2234               (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … s2)))
2235             SFR_SP sp1)
2236          bl)
2237        SFR_SP sp2)
2238      bu)
2239   = high_internal_ram_of_pseudo_high_internal_ram (sp1::M)
2240      (high_internal_ram …
2241       (write_at_stack_pointer …
2242         (set_8051_sfr …
2243           (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl)
2244          SFR_SP sp2)
2245        pbu)).
2246  #T1 #T2 #M #cm1 #s1 #cm2 #s2 #cm3 #s3 #sigma #policy #pbu #pbl #bu #bl #sp1 #sp2
2247  #get_8051_sfr_refl #high_internal_ram_refl #sp1_refl #sp2_refl #sigma_refl
2248  cases daemon (* XXX: !!! *)
2249qed.
2250
2251lemma Some_Some_elim:
2252 ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P.
2253 #T #x #y #P #H #K @H @option_destruct_Some //
2254qed.
2255
2256lemma pair_destruct_right:
2257  ∀A: Type[0].
2258  ∀B: Type[0].
2259  ∀a, c: A.
2260  ∀b, d: B.
2261    〈a, b〉 = 〈c, d〉 → b = d.
2262  #A #B #a #b #c #d //
2263qed.
2264   
2265(*CSC: ???*)
2266lemma snd_assembly_1_pseudoinstruction_ok:
2267  ∀program: pseudo_assembly_program.
2268  ∀sigma: Word → Word.
2269  ∀policy: Word → bool.
2270  ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy.
2271  ∀ppc: Word.
2272  ∀pi.
2273  ∀lookup_labels.
2274  ∀lookup_datalabels.
2275    lookup_labels = (λx. sigma (address_of_word_labels_code_mem (\snd program) x)) →
2276    lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
2277    \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
2278    let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels sigma policy (*(sigma ppc)*) ppc lookup_datalabels  pi) in
2279      sigma (add … ppc (bitvector_of_nat ? 1)) = add … (sigma ppc) (bitvector_of_nat ? len).
2280  #program #sigma #policy #sigma_policy_specification_witness #ppc #pi
2281  #lookup_labels #lookup_datalabels
2282  #lookup_labels_refl #lookup_datalabels_refl #fetch_pseudo_refl
2283  normalize nodelta
2284  generalize in match fetch_pseudo_refl; -fetch_pseudo_refl
2285  #fetch_pseudo_refl
2286  letin assembled ≝ (\fst (assembly program sigma policy))
2287  letin costs ≝ (\snd (assembly program sigma policy))
2288  lapply (assembly_ok program sigma policy sigma_policy_specification_witness assembled costs)
2289  @pair_elim #preamble #instr_list #program_refl
2290  @pair_elim #labels #costs' #create_label_cost_map_refl
2291  <eq_pair_fst_snd #H cases (H (refl …)) -H #costs_refl #H
2292  lapply (H ppc) -H
2293  @pair_elim #pi' #newppc #fetch_pseudo_refl'
2294  @pair_elim #len #assembled #assembly1_refl #H
2295  cases H
2296  #encoding_check_assm #sigma_newppc_refl
2297  >fetch_pseudo_refl' in fetch_pseudo_refl; #pi_refl'
2298  >pi_refl' in assembly1_refl; #assembly1_refl
2299  >lookup_labels_refl >lookup_datalabels_refl
2300  >program_refl normalize nodelta
2301  >assembly1_refl
2302  <sigma_newppc_refl
2303  generalize in match fetch_pseudo_refl';
2304  whd in match (fetch_pseudo_instruction ??);
2305  @pair_elim #lbl #instr #nth_refl normalize nodelta
2306  #G cases (pair_destruct_right ?????? G) %
2307qed.
2308
2309lemma pose: ∀A:Type[0].∀B:A → Type[0].∀a:A. (∀a':A. a'=a → B a') → B a.
2310  /2/
2311qed.
2312
2313(* To be moved in ProofStatus *)
2314lemma program_counter_set_program_counter:
2315  ∀T.
2316  ∀cm.
2317  ∀s.
2318  ∀x.
2319    program_counter T cm (set_program_counter T cm s x) = x.
2320  //
2321qed.
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