source: src/ASM/AssemblyProof.ma @ 1975

Last change on this file since 1975 was 1975, checked in by mulligan, 7 years ago

Work from today on closing main_thm.

File size: 76.0 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_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    let fetched ≝ fetch code_memory pc in
1438    let 〈instr_pc, ticks〉 ≝ fetched in
1439    let 〈instr,pc'〉 ≝ instr_pc in
1440      eq_instruction instr i = true ∧
1441        ticks = (ticks_of_instruction i) ∧
1442        fetch_many code_memory final_pc pc' tl
1443  ].
1444
1445lemma option_destruct_Some:
1446  ∀A: Type[0].
1447  ∀a, b: A.
1448    Some A a = Some A b → a = b.
1449  #A #a #b #EQ
1450  destruct %
1451qed.
1452
1453lemma eq_instruction_to_eq:
1454  ∀i1, i2: instruction.
1455    eq_instruction i1 i2 = true → i1 ≃ i2.
1456  #i1 #i2
1457  cases i1 cases i2 cases daemon (*
1458  [1,10,19,28,37,46:
1459    #arg1 #arg2
1460    whd in match (eq_instruction ??);
1461    cases arg1 #subaddressing_mode
1462    cases subaddressing_mode
1463    try (#arg1' #arg2' normalize in ⊢ (% → ?); #absurd cases absurd @I)
1464    try (#arg1' normalize in ⊢ (% → ?); #absurd cases absurd @I)
1465    try (normalize in ⊢ (% → ?); #absurd cases absurd @I)
1466    #word11 #irrelevant
1467    cases arg2 #subaddressing_mode
1468    cases subaddressing_mode
1469    try (#arg1' #arg2' normalize in ⊢ (% → ?); #absurd cases absurd @I)
1470    try (#arg1' normalize in ⊢ (% → ?); #absurd cases absurd @I)
1471    try (normalize in ⊢ (% → ?); #absurd cases absurd @I)
1472    #word11' #irrelevant normalize nodelta
1473    #eq_bv_assm cases (eq_bv_eq … eq_bv_assm) % *)
1474qed.
1475         
1476lemma fetch_assembly_pseudo':
1477  ∀lookup_labels.
1478  ∀sigma: Word → Word.
1479  ∀policy: Word → bool.
1480  ∀ppc.
1481  ∀lookup_datalabels.
1482  ∀pi.
1483  ∀code_memory.
1484  ∀len.
1485  ∀assembled.
1486  ∀instructions.
1487    let pc ≝ sigma ppc in
1488      instructions = expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi →
1489        〈len,assembled〉 = assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi →
1490          let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1491            encoding_check code_memory pc pc_plus_len assembled →
1492              fetch_many code_memory pc_plus_len pc instructions.
1493  #lookup_labels #sigma #policy #ppc #lookup_datalabels #pi #code_memory #len #assembled #instructions
1494  normalize nodelta #instructions_refl whd in ⊢ (???% → ?); <instructions_refl whd in ⊢ (???% → ?); #assembled_refl
1495  cases (pair_destruct ?????? assembled_refl) -assembled_refl #len_refl #assembled_refl
1496  >len_refl >assembled_refl -len_refl
1497  generalize in match (add 16 (sigma ppc)
1498    (bitvector_of_nat 16
1499     (|flatten (Vector bool 8)
1500       (map instruction (list (Vector bool 8)) assembly1 instructions)|)));
1501  #final_pc
1502  generalize in match (sigma ppc); elim instructions
1503  [1:
1504    #pc whd in ⊢ (% → %); #H >H @eq_bv_refl
1505  |2:
1506    #i #tl #IH #pc #H whd
1507    cases (encoding_check_append ????? H) -H #H1 #H2
1508    @pair_elim #instr_pc #ticks #fetch_refl normalize nodelta
1509    @pair_elim #instr #pc' #instr_pc_refl normalize nodelta
1510    lapply (fetch_assembly pc i code_memory (assembly1 i) (refl …)) whd in ⊢ (% → ?);
1511    #H3 lapply (H3 H1) -H3 >fetch_refl >instr_pc_refl normalize nodelta
1512    #H3 lapply (conjunction_true ?? H3) * #H4 #H5 %
1513    [1:
1514      lapply (conjunction_true … H4) * #B1 #B2
1515      % try assumption @eqb_true_to_eq
1516      <(eq_instruction_to_eq … B1) assumption
1517    |2:
1518      >(eq_bv_eq … H5) @IH @H2
1519    ]
1520  ]
1521qed.
1522
1523lemma fetch_assembly_pseudo:
1524  ∀program: pseudo_assembly_program.
1525  ∀sigma: Word → Word.
1526  ∀policy: Word → bool.
1527  let lookup_labels ≝ λx:Identifier. sigma (address_of_word_labels_code_mem (\snd  program) x) in
1528  ∀ppc.
1529  ∀code_memory.
1530  let lookup_datalabels ≝ λx:Identifier.lookup_def … (construct_datalabels (\fst  program)) x (zero 16) in
1531  let pi ≝  \fst  (fetch_pseudo_instruction (\snd program) ppc) in
1532  let pc ≝ sigma ppc in
1533  let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in
1534  let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in
1535  let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1536    encoding_check code_memory pc pc_plus_len assembled →
1537      fetch_many code_memory pc_plus_len pc instructions.
1538  #program #sigma #policy letin lookup_labels ≝ (λx.?) #ppc #code_memory
1539  letin lookup_datalabels ≝ (λx.?)
1540  letin pi ≝ (fst ???)
1541  letin pc ≝ (sigma ?)
1542  letin instructions ≝ (expand_pseudo_instruction ??????)
1543  @pair_elim #len #assembled #assembled_refl normalize nodelta
1544  #H
1545  generalize in match
1546   (fetch_assembly_pseudo' lookup_labels sigma policy ppc lookup_datalabels pi code_memory len assembled instructions) in ⊢ ?;
1547  #X destruct normalize nodelta @X try % <assembled_refl try % assumption
1548qed.
1549
1550definition is_present_in_machine_code_image_p: ∀pseudo_instruction. Prop ≝
1551  λpseudo_instruction.
1552    match pseudo_instruction with
1553    [ Comment c ⇒ False
1554    | Cost c ⇒ False
1555    | _ ⇒ True
1556    ].
1557
1558definition sigma_policy_specification ≝
1559  λprogram: pseudo_assembly_program.
1560  λsigma: Word → Word.
1561  λpolicy: Word → bool.
1562  ∀ppc: Word.
1563    let 〈preamble, instr_list〉 ≝ program in
1564    let pc ≝ sigma ppc in
1565    let labels ≝ \fst (create_label_cost_map instr_list) in
1566    let lookup_labels ≝ λx. bitvector_of_nat 16 (lookup_def … labels x 0) in
1567    let instruction ≝ \fst (fetch_pseudo_instruction instr_list ppc) in
1568    let next_pc ≝ sigma (add 16 ppc (bitvector_of_nat 16 1)) in
1569      And (nat_of_bitvector … ppc ≤ |instr_list| →
1570        next_pc = add 16 pc (bitvector_of_nat …
1571          (instruction_size lookup_labels sigma policy ppc instruction)))
1572       (Or (nat_of_bitvector … ppc < |instr_list| →
1573         nat_of_bitvector … pc < nat_of_bitvector … next_pc)
1574        (nat_of_bitvector … ppc = |instr_list| → next_pc = (zero …))).
1575
1576(* This is a trivial consequence of fetch_assembly_pseudo + the proof that the
1577   function that load the code in memory is correct. The latter is based
1578   on missing properties from the standard library on the BitVectorTrie
1579   data structrure.
1580   
1581   Wrong at the moment, requires Jaap's precondition to ensure that the program
1582   does not overflow when put into code memory (i.e. shorter than 2^16 bytes).
1583*)
1584lemma assembly_ok:
1585  ∀program.
1586  ∀sigma: Word → Word.
1587  ∀policy: Word → bool.
1588  ∀sigma_policy_witness: sigma_policy_specification program sigma policy.
1589  ∀assembled.
1590  ∀costs'.
1591  let 〈preamble, instr_list〉 ≝ program in
1592  let 〈labels, costs〉 ≝ create_label_cost_map instr_list in
1593  let datalabels ≝ construct_datalabels preamble in
1594  let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero …) in
1595    〈assembled,costs'〉 = assembly program sigma policy →
1596      costs = costs' ∧
1597        let code_memory ≝ load_code_memory assembled in
1598        let lookup_labels ≝ λx. sigma (address_of_word_labels_code_mem instr_list x) in 
1599          ∀ppc.
1600            let 〈pi, newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in     
1601            let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in
1602            let pc ≝ sigma ppc in
1603            let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in
1604              encoding_check code_memory pc pc_plus_len assembled ∧
1605                  sigma newppc = add … pc (bitvector_of_nat … len).
1606  #program #sigma #policy #sigma_policy_witness #assembled #costs'
1607  @pair_elim #preamble #instr_list #program_refl
1608  @pair_elim #labels #costs #create_label_cost_refl
1609  #assembly_refl % cases daemon (*
1610  [1:
1611    (* XXX: lemma on BitVectorTries *)
1612    cases daemon
1613  |2:
1614    #ppc #sigma_policy_specification_witness
1615    @pair_elim #pi #newppc #fetch_pseudo_refl
1616    cases pi
1617    [2,3: (* Cost and Comment cases *)
1618      #comment_or_cost normalize in ⊢ (% → ?); #absurd cases absurd
1619    |1: (* PreInstruction cases *)
1620      #preinstruction #_
1621      @pair_elim #len #assembled' #assembly_1_pseudo_refl
1622      normalize nodelta %
1623      [1:
1624        cases assembled' normalize
1625      |2:
1626      ]
1627    ]
1628  ]
1629  cases daemon (* XXX: !!! *) *)
1630qed.
1631
1632(* XXX: should we add that costs = costs'? *)
1633lemma fetch_assembly_pseudo2:
1634  ∀program.
1635  ∀sigma.
1636  ∀policy.
1637  ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy.
1638  ∀ppc.
1639  let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in
1640  let 〈assembled, costs'〉 ≝ assembly program sigma policy in
1641  let code_memory ≝ load_code_memory assembled in
1642  let data_labels ≝ construct_datalabels (\fst program) in
1643  let lookup_labels ≝ λx. sigma (address_of_word_labels_code_mem (\snd program) x) in 
1644  let lookup_datalabels ≝ λx. lookup_def ? ? data_labels x (zero ?) in
1645  let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1646  let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in
1647    fetch_many code_memory (sigma newppc) (sigma ppc) instructions.
1648  * #preamble #instr_list #sigma #policy #sigma_policy_specification_witness #ppc
1649  @pair_elim #labels #costs #create_label_map_refl
1650  @pair_elim #assembled #costs' #assembled_refl
1651  letin code_memory ≝ (load_code_memory ?)
1652  letin data_labels ≝ (construct_datalabels ?)
1653  letin lookup_labels ≝ (λx. ?)
1654  letin lookup_datalabels ≝ (λx. ?)
1655  @pair_elim #pi #newppc #fetch_pseudo_refl
1656  lapply (assembly_ok 〈preamble, instr_list〉 sigma policy sigma_policy_specification_witness assembled costs')
1657  normalize nodelta
1658  @pair_elim #labels' #costs' #create_label_map_refl' #H
1659  cases (H (sym_eq … assembled_refl))
1660  #_
1661  lapply (refl … (assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi))
1662  cases (assembly_1_pseudoinstruction ??????) in ⊢ (???% → ?);
1663  #len #assembledi #EQ4 #H
1664  lapply (H ppc) >fetch_pseudo_refl #H
1665  lapply (fetch_assembly_pseudo 〈preamble,instr_list〉 sigma policy ppc (load_code_memory assembled))
1666  >EQ4 #H1 cases H
1667  #H2 #H3 >H3 normalize nodelta in H1; normalize nodelta
1668  >fetch_pseudo_refl in H1; #assm @assm assumption
1669qed.
1670
1671(* OLD?
1672definition assembly_specification:
1673  ∀assembly_program: pseudo_assembly_program.
1674  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1675  λpseudo_assembly_program.
1676  λcode_mem.
1677    ∀pc: Word.
1678      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1679      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1680      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1681      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1682      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1683       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1684      match pre_assembled with
1685       [ None ⇒ True
1686       | Some pc_code ⇒
1687          let 〈new_pc,code〉 ≝ pc_code in
1688           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1689
1690axiom assembly_meets_specification:
1691  ∀pseudo_assembly_program.
1692    match assembly pseudo_assembly_program with
1693    [ None ⇒ True
1694    | Some code_mem_cost ⇒
1695      let 〈code_mem, cost〉 ≝ code_mem_cost in
1696        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1697    ].
1698(*
1699  # PROGRAM
1700  [ cases PROGRAM
1701    # PREAMBLE
1702    # INSTR_LIST
1703    elim INSTR_LIST
1704    [ whd
1705      whd in ⊢ (∀_. %)
1706      # PC
1707      whd
1708    | # INSTR
1709      # INSTR_LIST_TL
1710      # H
1711      whd
1712      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1713    ]
1714  | cases not_implemented
1715  ] *)
1716*)
1717
1718definition internal_pseudo_address_map ≝ list (BitVector 8).
1719
1720axiom low_internal_ram_of_pseudo_low_internal_ram:
1721 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1722
1723axiom high_internal_ram_of_pseudo_high_internal_ram:
1724 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1725
1726axiom low_internal_ram_of_pseudo_internal_ram_hit:
1727 ∀M:internal_pseudo_address_map.∀cm.∀s:PseudoStatus cm.∀sigma:Word → Word × bool.∀addr:BitVector 7.
1728  member ? (eq_bv 8) (false:::addr) M = true →
1729   let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1730   let pbl ≝ lookup ? 7 addr (low_internal_ram … s) (zero 8) in
1731   let pbu ≝ lookup ? 7 (add ? addr (bitvector_of_nat 7 1)) (low_internal_ram … s) (zero 8) in
1732   let bl ≝ lookup ? 7 addr ram (zero 8) in
1733   let bu ≝ lookup ? 7 (add ? addr (bitvector_of_nat 7 1)) ram (zero 8) in
1734    bu@@bl = \fst (sigma (pbu@@pbl)).
1735
1736(* changed from add to sub *)
1737axiom low_internal_ram_of_pseudo_internal_ram_miss:
1738 ∀T.∀M:internal_pseudo_address_map.∀cm.∀s:PreStatus T cm.∀addr:BitVector 7.
1739  let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1740  let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in
1741  let carr ≝ get_index_v ? ? flags 1 ? in
1742  carr = false →
1743  member ? (eq_bv 8) (false:::Saddr) M = false →
1744   member ? (eq_bv 8) (false:::addr) M = false →
1745    lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?).
1746  //
1747qed.
1748
1749definition addressing_mode_ok ≝
1750 λT.λM:internal_pseudo_address_map.λcm.λs:PreStatus T cm.
1751  λaddr:addressing_mode.
1752   match addr with
1753    [ DIRECT d ⇒
1754       ¬(member ? (eq_bv 8) d M) ∧
1755       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1756    | INDIRECT i ⇒
1757       let d ≝ get_register … s [[false;false;i]] in
1758       ¬(member ? (eq_bv 8) d M) ∧
1759       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1760    | EXT_INDIRECT _ ⇒ true
1761    | REGISTER _ ⇒ true
1762    | ACC_A ⇒ true
1763    | ACC_B ⇒ true
1764    | DPTR ⇒ true
1765    | DATA _ ⇒ true
1766    | DATA16 _ ⇒ true
1767    | ACC_DPTR ⇒ true
1768    | ACC_PC ⇒ true
1769    | EXT_INDIRECT_DPTR ⇒ true
1770    | INDIRECT_DPTR ⇒ true
1771    | CARRY ⇒ true
1772    | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1773    | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1774    | RELATIVE _ ⇒ true
1775    | ADDR11 _ ⇒ true
1776    | ADDR16 _ ⇒ true ].
1777   
1778definition next_internal_pseudo_address_map0 ≝
1779  λT.
1780  λfetched.
1781  λM: internal_pseudo_address_map.
1782  λcm:T.
1783  λs: PreStatus T cm.
1784   match fetched with
1785    [ Comment _ ⇒ Some ? M
1786    | Cost _ ⇒ Some … M
1787    | Jmp _ ⇒ Some … M
1788    | Call _ ⇒
1789       Some … (add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1)::M)
1790    | Mov _ _ ⇒ Some … M
1791    | Instruction instr ⇒
1792       match instr with
1793        [ ADD addr1 addr2 ⇒
1794           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1795            Some ? M
1796           else
1797            None ?
1798        | ADDC addr1 addr2 ⇒
1799           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1800            Some ? M
1801           else
1802            None ?
1803        | SUBB addr1 addr2 ⇒
1804           if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then
1805            Some ? M
1806           else
1807            None ?       
1808        | _ ⇒ (* TO BE COMPLETED *) Some ? M ]].
1809 
1810
1811definition next_internal_pseudo_address_map ≝
1812 λM:internal_pseudo_address_map.
1813 λcm.
1814  λs:PseudoStatus cm.
1815    next_internal_pseudo_address_map0 ?
1816     (\fst (fetch_pseudo_instruction (\snd cm) (program_counter … s))) M cm s.
1817
1818definition code_memory_of_pseudo_assembly_program:
1819    ∀pap:pseudo_assembly_program.
1820      (Word → Word) → (Word → bool) → BitVectorTrie Byte 16 ≝
1821  λpap.
1822  λsigma.
1823  λpolicy.
1824    let p ≝ assembly pap sigma policy in
1825      load_code_memory (\fst p).
1826
1827definition status_of_pseudo_status:
1828    internal_pseudo_address_map → ∀pap. ∀ps: PseudoStatus pap.
1829      ∀sigma: Word → Word. ∀policy: Word → bool.
1830        Status (code_memory_of_pseudo_assembly_program pap sigma policy) ≝
1831  λM.
1832  λpap.
1833  λps.
1834  λsigma.
1835  λpolicy.
1836  let cm ≝ code_memory_of_pseudo_assembly_program … sigma policy in
1837  let pc ≝ sigma (program_counter … ps) in
1838  let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in
1839  let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in
1840     mk_PreStatus (BitVectorTrie Byte 16)
1841      cm
1842      lir
1843      hir
1844      (external_ram … ps)
1845      pc
1846      (special_function_registers_8051 … ps)
1847      (special_function_registers_8052 … ps)
1848      (p1_latch … ps)
1849      (p3_latch … ps)
1850      (clock … ps).
1851
1852(*
1853definition write_at_stack_pointer':
1854 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1855  λM: Type[0].
1856  λs: PreStatus M.
1857  λv: Byte.
1858    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1859    let bit_zero ≝ get_index_v… nu O ? in
1860    let bit_1 ≝ get_index_v… nu 1 ? in
1861    let bit_2 ≝ get_index_v… nu 2 ? in
1862    let bit_3 ≝ get_index_v… nu 3 ? in
1863      if bit_zero then
1864        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1865                              v (low_internal_ram ? s) in
1866          set_low_internal_ram ? s memory
1867      else
1868        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1869                              v (high_internal_ram ? s) in
1870          set_high_internal_ram ? s memory.
1871  [ cases l0 %
1872  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1873qed.
1874
1875definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1876 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1877
1878  λticks_of.
1879  λs.
1880  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1881  let ticks ≝ ticks_of (program_counter ? s) in
1882  let s ≝ set_clock ? s (clock ? s + ticks) in
1883  let s ≝ set_program_counter ? s pc in
1884    match instr with
1885    [ Instruction instr ⇒
1886       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1887    | Comment cmt ⇒ s
1888    | Cost cst ⇒ s
1889    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1890    | Call call ⇒
1891      let a ≝ address_of_word_labels s call in
1892      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1893      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1894      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1895      let s ≝ write_at_stack_pointer' ? s pc_bl in
1896      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1897      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1898      let s ≝ write_at_stack_pointer' ? s pc_bu in
1899        set_program_counter ? s a
1900    | Mov dptr ident ⇒
1901       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1902    ].
1903 [
1904 |2,3,4: %
1905 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1906 |
1907 | %
1908 ]
1909 cases not_implemented
1910qed.
1911*)
1912
1913(*
1914lemma execute_code_memory_unchanged:
1915 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1916 #ticks #ps whd in ⊢ (??? (??%))
1917 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1918  (program_counter pseudo_assembly_program ps)) #instr #pc
1919 whd in ⊢ (??? (??%)) cases instr
1920  [ #pre cases pre
1921     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1922       cases (split ????) #z1 #z2 %
1923     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1924       cases (split ????) #z1 #z2 %
1925     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1926       cases (split ????) #z1 #z2 %
1927     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1928       [ #x1 whd in ⊢ (??? (??%))
1929     | *: cases not_implemented
1930     ]
1931  | #comment %
1932  | #cost %
1933  | #label %
1934  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1935    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1936    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1937    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1938    (* CSC: ??? *)
1939  | #dptr #label (* CSC: ??? *)
1940  ]
1941  cases not_implemented
1942qed.
1943*)
1944
1945(* DEAD CODE?
1946lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1947 ∀M:internal_pseudo_address_map.
1948 ∀ps,ps': PseudoStatus.
1949 ∀pol.
1950  ∀prf:code_memory … ps = code_memory … ps'.
1951   let pol' ≝ ? in
1952   match status_of_pseudo_status M ps pol with
1953    [ None ⇒ status_of_pseudo_status M ps' pol' = None …
1954    | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w
1955    ].
1956 [2: <prf @pol]
1957 #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1958 generalize in match (refl … (assembly (code_memory … ps) pol))
1959 cases (assembly ??) in ⊢ (???% → %)
1960  [ #K whd whd in ⊢ (??%?) <H >K %
1961  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1962qed.
1963*)
1964
1965definition ticks_of0:
1966    ∀p:pseudo_assembly_program.
1967      (Word → Word) → (Word → bool) → Word → pseudo_instruction → nat × nat ≝
1968  λprogram: pseudo_assembly_program.
1969  λsigma.
1970  λpolicy.
1971  λppc: Word.
1972  λfetched.
1973    match fetched with
1974    [ Instruction instr ⇒
1975      match instr with
1976      [ JC lbl ⇒ ? (*
1977        match pol lookup_labels ppc with
1978        [ short_jump ⇒ 〈2, 2〉
1979        | medium_jump ⇒ ?
1980        | long_jump ⇒ 〈4, 4〉
1981        ] *)
1982      | JNC lbl ⇒ ? (*
1983        match pol lookup_labels ppc with
1984        [ short_jump ⇒ 〈2, 2〉
1985        | medium_jump ⇒ ?
1986        | long_jump ⇒ 〈4, 4〉
1987        ] *)
1988      | JB bit lbl ⇒ ? (*
1989        match pol lookup_labels ppc with
1990        [ short_jump ⇒ 〈2, 2〉
1991        | medium_jump ⇒ ?
1992        | long_jump ⇒ 〈4, 4〉
1993        ] *)
1994      | JNB bit lbl ⇒ ? (*
1995        match pol lookup_labels ppc with
1996        [ short_jump ⇒ 〈2, 2〉
1997        | medium_jump ⇒ ?
1998        | long_jump ⇒ 〈4, 4〉
1999        ] *)
2000      | JBC bit lbl ⇒ ? (*
2001        match pol lookup_labels ppc with
2002        [ short_jump ⇒ 〈2, 2〉
2003        | medium_jump ⇒ ?
2004        | long_jump ⇒ 〈4, 4〉
2005        ] *)
2006      | JZ lbl ⇒ ? (*
2007        match pol lookup_labels ppc with
2008        [ short_jump ⇒ 〈2, 2〉
2009        | medium_jump ⇒ ?
2010        | long_jump ⇒ 〈4, 4〉
2011        ] *)
2012      | JNZ lbl ⇒ ? (*
2013        match pol lookup_labels  ppc with
2014        [ short_jump ⇒ 〈2, 2〉
2015        | medium_jump ⇒ ?
2016        | long_jump ⇒ 〈4, 4〉
2017        ] *)
2018      | CJNE arg lbl ⇒ ? (*
2019        match pol lookup_labels ppc with
2020        [ short_jump ⇒ 〈2, 2〉
2021        | medium_jump ⇒ ?
2022        | long_jump ⇒ 〈4, 4〉
2023        ] *)
2024      | DJNZ arg lbl ⇒ ? (*
2025        match pol lookup_labels ppc with
2026        [ short_jump ⇒ 〈2, 2〉
2027        | medium_jump ⇒ ?
2028        | long_jump ⇒ 〈4, 4〉
2029        ] *)
2030      | ADD arg1 arg2 ⇒
2031        let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in
2032         〈ticks, ticks〉
2033      | ADDC arg1 arg2 ⇒
2034        let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in
2035         〈ticks, ticks〉
2036      | SUBB arg1 arg2 ⇒
2037        let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in
2038         〈ticks, ticks〉
2039      | INC arg ⇒
2040        let ticks ≝ ticks_of_instruction (INC ? arg) in
2041         〈ticks, ticks〉
2042      | DEC arg ⇒
2043        let ticks ≝ ticks_of_instruction (DEC ? arg) in
2044         〈ticks, ticks〉
2045      | MUL arg1 arg2 ⇒
2046        let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in
2047         〈ticks, ticks〉
2048      | DIV arg1 arg2 ⇒
2049        let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in
2050         〈ticks, ticks〉
2051      | DA arg ⇒
2052        let ticks ≝ ticks_of_instruction (DA ? arg) in
2053         〈ticks, ticks〉
2054      | ANL arg ⇒
2055        let ticks ≝ ticks_of_instruction (ANL ? arg) in
2056         〈ticks, ticks〉
2057      | ORL arg ⇒
2058        let ticks ≝ ticks_of_instruction (ORL ? arg) in
2059         〈ticks, ticks〉
2060      | XRL arg ⇒
2061        let ticks ≝ ticks_of_instruction (XRL ? arg) in
2062         〈ticks, ticks〉
2063      | CLR arg ⇒
2064        let ticks ≝ ticks_of_instruction (CLR ? arg) in
2065         〈ticks, ticks〉
2066      | CPL arg ⇒
2067        let ticks ≝ ticks_of_instruction (CPL ? arg) in
2068         〈ticks, ticks〉
2069      | RL arg ⇒
2070        let ticks ≝ ticks_of_instruction (RL ? arg) in
2071         〈ticks, ticks〉
2072      | RLC arg ⇒
2073        let ticks ≝ ticks_of_instruction (RLC ? arg) in
2074         〈ticks, ticks〉
2075      | RR arg ⇒
2076        let ticks ≝ ticks_of_instruction (RR ? arg) in
2077         〈ticks, ticks〉
2078      | RRC arg ⇒
2079        let ticks ≝ ticks_of_instruction (RRC ? arg) in
2080         〈ticks, ticks〉
2081      | SWAP arg ⇒
2082        let ticks ≝ ticks_of_instruction (SWAP ? arg) in
2083         〈ticks, ticks〉
2084      | MOV arg ⇒
2085        let ticks ≝ ticks_of_instruction (MOV ? arg) in
2086         〈ticks, ticks〉
2087      | MOVX arg ⇒
2088        let ticks ≝ ticks_of_instruction (MOVX ? arg) in
2089         〈ticks, ticks〉
2090      | SETB arg ⇒
2091        let ticks ≝ ticks_of_instruction (SETB ? arg) in
2092         〈ticks, ticks〉
2093      | PUSH arg ⇒
2094        let ticks ≝ ticks_of_instruction (PUSH ? arg) in
2095         〈ticks, ticks〉
2096      | POP arg ⇒
2097        let ticks ≝ ticks_of_instruction (POP ? arg) in
2098         〈ticks, ticks〉
2099      | XCH arg1 arg2 ⇒
2100        let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in
2101         〈ticks, ticks〉
2102      | XCHD arg1 arg2 ⇒
2103        let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in
2104         〈ticks, ticks〉
2105      | RET ⇒
2106        let ticks ≝ ticks_of_instruction (RET ?) in
2107         〈ticks, ticks〉
2108      | RETI ⇒
2109        let ticks ≝ ticks_of_instruction (RETI ?) in
2110         〈ticks, ticks〉
2111      | NOP ⇒
2112        let ticks ≝ ticks_of_instruction (NOP ?) in
2113         〈ticks, ticks〉
2114      ]
2115    | Comment comment ⇒ 〈0, 0〉
2116    | Cost cost ⇒ 〈0, 0〉
2117    | Jmp jmp ⇒ 〈2, 2〉
2118    | Call call ⇒ 〈2, 2〉
2119    | Mov dptr tgt ⇒ 〈2, 2〉
2120    ].
2121    cases daemon
2122qed.
2123
2124definition ticks_of:
2125    ∀p:pseudo_assembly_program.
2126      (Word → Word) → (Word → bool) → Word → nat × nat ≝
2127  λprogram: pseudo_assembly_program.
2128  λsigma.
2129  λpolicy.
2130  λppc: Word.
2131    let 〈preamble, pseudo〉 ≝ program in
2132    let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in
2133     ticks_of0 program sigma policy ppc fetched.
2134
2135lemma eq_rect_Type1_r:
2136  ∀A: Type[1].
2137  ∀a: A.
2138  ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p.
2139  #A #a #P #H #x #p
2140  generalize in match H;
2141  generalize in match P;
2142  cases p //
2143qed.
2144
2145axiom split_append:
2146  ∀A: Type[0].
2147  ∀m, n: nat.
2148  ∀v, v': Vector A m.
2149  ∀q, q': Vector A n.
2150    let 〈v', q'〉 ≝ split A m n (v@@q) in
2151      v = v' ∧ q = q'.
2152
2153lemma split_vector_singleton:
2154  ∀A: Type[0].
2155  ∀n: nat.
2156  ∀v: Vector A (S n).
2157  ∀rest: Vector A n.
2158  ∀s: Vector A 1.
2159    v = s @@ rest →
2160    ((get_index_v A ? v 0 ?) ::: rest) = v.
2161  [1:
2162    #A #n #v cases daemon (* XXX: !!! *)
2163  |2:
2164    @le_S_S @le_O_n
2165  ]
2166qed.
2167
2168example sub_minus_one_seven_eight:
2169  ∀v: BitVector 7.
2170  false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) =
2171  \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false).
2172 cases daemon.
2173qed.
2174
2175(*
2176lemma blah:
2177  ∀m: internal_pseudo_address_map.
2178  ∀s: PseudoStatus.
2179  ∀arg: Byte.
2180  ∀b: bool.
2181    addressing_mode_ok m s (DIRECT arg) = true →
2182      get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) =
2183      get_arg_8 ? s b (DIRECT arg).
2184  [2, 3: normalize % ]
2185  #m #s #arg #b #hyp
2186  whd in ⊢ (??%%)
2187  @split_elim''
2188  #nu' #nl' #arg_nu_nl_eq
2189  normalize nodelta
2190  generalize in match (refl ? (get_index_v bool 4 nu' ? ?))
2191  cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %)
2192  #get_index_v_eq
2193  normalize nodelta
2194  [2:
2195    normalize nodelta
2196    @split_elim''
2197    #bit_one' #three_bits' #bit_one_three_bit_eq
2198    generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl'))
2199    normalize nodelta
2200    generalize in match (refl ? (sub_7_with_carry ? ? ?))
2201    cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %)
2202    #Saddr #carr' #Saddr_carr_eq
2203    normalize nodelta
2204    #carr_hyp'
2205    @carr_hyp'
2206    [1:
2207    |2: whd in hyp:(??%?); generalize in match hyp; -hyp;
2208        generalize in match (refl ? (¬(member (BitVector 8) ? arg m)))
2209        cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %)
2210        #member_eq
2211        normalize nodelta
2212        [2: #destr destruct(destr)
2213        |1: -carr_hyp';
2214            >arg_nu_nl_eq
2215            <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq)
2216            [1: >get_index_v_eq in ⊢ (??%? → ?)
2217            |2: @le_S @le_S @le_S @le_n
2218            ]
2219            cases (member (BitVector 8) ? (\fst ?) ?)
2220            [1: #destr normalize in destr; destruct(destr)
2221            |2:
2222            ]
2223        ]
2224    |3: >get_index_v_eq in ⊢ (??%?)
2225        change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl')
2226        >(split_vector_singleton … bit_one_three_bit_eq)
2227        <arg_nu_nl_eq
2228        whd in hyp:(??%?)
2229        cases (member (BitVector 8) (eq_bv 8) arg m) in hyp
2230        normalize nodelta [*: #ignore @sym_eq ]
2231    ]
2232  |
2233  ].
2234*)
2235(*
2236map_address0 ... (DIRECT arg) = Some .. →
2237  get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) =
2238  get_arg_8 (internal_ram ...) (DIRECT arg)
2239
2240((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) 
2241                     then Some internal_pseudo_address_map M 
2242                     else None internal_pseudo_address_map )
2243                    =Some internal_pseudo_address_map M')
2244*)
2245
2246axiom low_internal_ram_write_at_stack_pointer:
2247 ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma: Word → Word.∀policy: Word → bool.
2248 ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2249  get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP →
2250  low_internal_ram ? cm2 s2 = low_internal_ram T2 cm3 s3 →
2251  sp1 = add ? (get_8051_sfr … cm1 s1 SFR_SP) (bitvector_of_nat 8 1) →
2252  sp2 = add ? sp1 (bitvector_of_nat 8 1) →
2253  bu@@bl = sigma (pbu@@pbl) →
2254   low_internal_ram T1 cm1
2255     (write_at_stack_pointer …
2256       (set_8051_sfr …
2257         (write_at_stack_pointer …
2258           (set_8051_sfr …
2259             (set_low_internal_ram … s1
2260               (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s2)))
2261             SFR_SP sp1)
2262          bl)
2263        SFR_SP sp2)
2264      bu)
2265   = low_internal_ram_of_pseudo_low_internal_ram (sp1::M)
2266      (low_internal_ram …
2267       (write_at_stack_pointer …
2268         (set_8051_sfr …
2269           (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl)
2270          SFR_SP sp2)
2271        pbu)).
2272
2273lemma high_internal_ram_write_at_stack_pointer:
2274 ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma:Word → Word.∀policy: Word → bool.
2275 ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8.
2276  get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP →
2277  high_internal_ram ?? s2 = high_internal_ram T2 cm3 s3 →
2278  sp1 = add ? (get_8051_sfr ? cm1 s1 SFR_SP) (bitvector_of_nat 8 1) →
2279  sp2 = add ? sp1 (bitvector_of_nat 8 1) →
2280  bu@@bl = sigma (pbu@@pbl) →
2281   high_internal_ram T1 cm1
2282     (write_at_stack_pointer …
2283       (set_8051_sfr …
2284         (write_at_stack_pointer …
2285           (set_8051_sfr …
2286             (set_high_internal_ram … s1
2287               (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … s2)))
2288             SFR_SP sp1)
2289          bl)
2290        SFR_SP sp2)
2291      bu)
2292   = high_internal_ram_of_pseudo_high_internal_ram (sp1::M)
2293      (high_internal_ram …
2294       (write_at_stack_pointer …
2295         (set_8051_sfr …
2296           (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl)
2297          SFR_SP sp2)
2298        pbu)).
2299  #T1 #T2 #M #cm1 #s1 #cm2 #s2 #cm3 #s3 #sigma #policy #pbu #pbl #bu #bl #sp1 #sp2
2300  #get_8051_sfr_refl #high_internal_ram_refl #sp1_refl #sp2_refl #sigma_refl
2301  cases daemon (* XXX: !!! *)
2302qed.
2303
2304lemma Some_Some_elim:
2305 ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P.
2306 #T #x #y #P #H #K @H @option_destruct_Some //
2307qed.
2308
2309lemma pair_destruct_right:
2310  ∀A: Type[0].
2311  ∀B: Type[0].
2312  ∀a, c: A.
2313  ∀b, d: B.
2314    〈a, b〉 = 〈c, d〉 → b = d.
2315  #A #B #a #b #c #d //
2316qed.
2317   
2318(*CSC: ???*)
2319lemma snd_assembly_1_pseudoinstruction_ok:
2320  ∀program: pseudo_assembly_program.
2321  ∀sigma: Word → Word.
2322  ∀policy: Word → bool.
2323  ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy.
2324  ∀ppc: Word.
2325  ∀pi.
2326  ∀lookup_labels.
2327  ∀lookup_datalabels.
2328    lookup_labels = (λx. sigma (address_of_word_labels_code_mem (\snd program) x)) →
2329    lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) →
2330    \fst (fetch_pseudo_instruction (\snd program) ppc) = pi →
2331    let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels sigma policy (*(sigma ppc)*) ppc lookup_datalabels  pi) in
2332      sigma (add … ppc (bitvector_of_nat ? 1)) = add … (sigma ppc) (bitvector_of_nat ? len).
2333  #program #sigma #policy #sigma_policy_specification_witness #ppc #pi
2334  #lookup_labels #lookup_datalabels
2335  #lookup_labels_refl #lookup_datalabels_refl #fetch_pseudo_refl
2336  normalize nodelta
2337  generalize in match fetch_pseudo_refl; -fetch_pseudo_refl
2338  #fetch_pseudo_refl
2339  letin assembled ≝ (\fst (assembly program sigma policy))
2340  letin costs ≝ (\snd (assembly program sigma policy))
2341  lapply (assembly_ok program sigma policy sigma_policy_specification_witness assembled costs)
2342  @pair_elim #preamble #instr_list #program_refl
2343  @pair_elim #labels #costs' #create_label_cost_map_refl
2344  <eq_pair_fst_snd #H cases (H (refl …)) -H #costs_refl #H
2345  lapply (H ppc) -H
2346  @pair_elim #pi' #newppc #fetch_pseudo_refl'
2347  @pair_elim #len #assembled #assembly1_refl #H
2348  cases H
2349  #encoding_check_assm #sigma_newppc_refl
2350  >fetch_pseudo_refl' in fetch_pseudo_refl; #pi_refl'
2351  >pi_refl' in assembly1_refl; #assembly1_refl
2352  >lookup_labels_refl >lookup_datalabels_refl
2353  >program_refl normalize nodelta
2354  >assembly1_refl
2355  <sigma_newppc_refl
2356  generalize in match fetch_pseudo_refl';
2357  whd in match (fetch_pseudo_instruction ??);
2358  @pair_elim #lbl #instr #nth_refl normalize nodelta
2359  #G cases (pair_destruct_right ?????? G) %
2360qed.
2361
2362lemma pose: ∀A:Type[0].∀B:A → Type[0].∀a:A. (∀a':A. a'=a → B a') → B a.
2363  /2/
2364qed.
2365
2366(* To be moved in ProofStatus *)
2367lemma program_counter_set_program_counter:
2368  ∀T.
2369  ∀cm.
2370  ∀s.
2371  ∀x.
2372    program_counter T cm (set_program_counter T cm s x) = x.
2373  //
2374qed.
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