source: src/ASM/AssemblyProof.ma @ 943

Last change on this file since 943 was 943, checked in by sacerdot, 9 years ago

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1include "ASM/Assembly.ma".
2include "ASM/Interpret.ma".
3
4(* RUSSEL **)
5
6include "basics/jmeq.ma".
7
8notation > "hvbox(a break ≃ b)"
9  non associative with precedence 45
10for @{ 'jmeq ? $a ? $b }.
11
12notation < "hvbox(term 46 a break maction (≃) (≃\sub(t,u)) term 46 b)"
13  non associative with precedence 45
14for @{ 'jmeq $t $a $u $b }.
15
16interpretation "john major's equality" 'jmeq t x u y = (jmeq t x u y).
17
18lemma eq_to_jmeq:
19  ∀A: Type[0].
20  ∀x, y: A.
21    x = y → x ≃ y.
22  //
23qed.
24
25definition inject : ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ λA,P,a,p. dp … a p.
26definition eject : ∀A.∀P: A → Prop.(Σx:A.P x) → A ≝ λA,P,c.match c with [ dp w p ⇒ w].
27
28coercion inject nocomposites: ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ inject on a:? to Σx:?.?.
29coercion eject nocomposites: ∀A.∀P:A → Prop.∀c:Σx:A.P x.A ≝ eject on _c:Σx:?.? to ?.
30
31axiom VOID: Type[0].
32axiom assert_false: VOID.
33definition bigbang: ∀A:Type[0].False → VOID → A.
34 #A #abs cases abs
35qed.
36
37coercion bigbang nocomposites: ∀A:Type[0].False → ∀v:VOID.A ≝ bigbang on _v:VOID to ?.
38
39lemma sig2: ∀A.∀P:A → Prop. ∀p:Σx:A.P x. P (eject … p).
40 #A #P #p cases p #w #q @q
41qed.
42
43lemma jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. x≃y → x=y.
44 #A #x #y #JMEQ @(jmeq_elim ? x … JMEQ) %
45qed.
46
47coercion jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. ∀p:x≃y.x=y ≝ jmeq_to_eq on _p:?≃? to ?=?.
48
49(* END RUSSELL **)
50
51
52definition bit_elim_prop: ∀P: bool → Prop. Prop ≝
53  λP.
54    P true ∧ P false.
55 
56let rec bitvector_elim_prop_internal
57  (n: nat) (P: BitVector n → Prop) (m: nat) on m: m ≤ n → BitVector (n - m) → Prop ≝
58  match m return λm. m ≤ n → BitVector (n - m) → Prop with
59  [ O    ⇒ λprf1. λprefix. P ?
60  | S n' ⇒ λprf2. λprefix. bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' ? ?)
61  ].
62  [ applyS prefix
63  | letin res ≝ (bit ::: prefix)
64    < (minus_S_S ? ?)
65    > (minus_Sn_m ? ?)
66  [ @ res
67  | @ prf2
68  ]
69  | /2/
70  ].
71qed.
72
73definition bitvector_elim_prop ≝
74  λn: nat.
75  λP: BitVector n → Prop.
76    bitvector_elim_prop_internal n P n ? ?.
77  [ @ (le_n ?)
78  | < (minus_n_n ?)
79    @ [[ ]]
80  ]
81qed.
82
83lemma eq_b_eq:
84  ∀b, c.
85    eq_b b c = true → b = c.
86  #b #c
87  cases b
88  cases c
89  normalize //
90qed.
91
92lemma BitVector_O: ∀v:BitVector 0. v ≃ VEmpty bool.
93 #v generalize in match (refl … 0) cases v in ⊢ (??%? → ?%%??) //
94 #n #hd #tl #abs @⊥ //
95qed.
96
97lemma BitVector_Sn: ∀n.∀v:BitVector (S n).
98 ∃hd.∃tl.v ≃ VCons bool n hd tl.
99 #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??)))
100 [ #abs @⊥ //
101 | #m #hd #tl #EQ <(injective_S … EQ) %[@hd] %[@tl] // ]
102qed.
103
104lemma eq_bv_eq:
105  ∀n, v, q.
106    eq_bv n v q = true → v = q.
107  #n #v #q generalize in match v
108  elim q
109  [ #v #h @BitVector_O
110  | #n #hd #tl #ih #v' #h
111    cases (BitVector_Sn ? v')
112    #hd' * #tl' #jmeq >jmeq in h;
113    #new_h
114    change in new_h with ((andb ? ?) = ?);
115    cases(conjunction_true … new_h)
116    #eq_heads #eq_tails
117    whd in eq_heads:(??(??(%))?);
118    cases(eq_b_eq … eq_heads)
119    whd in eq_tails:(??(?????(%))?);
120    change in eq_tails with (eq_bv ??? = ?);
121    <(ih tl') //
122  ]
123qed.
124
125lemma bool_eq_internal_eq:
126  ∀b, c.
127    (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c.
128  #b #c
129  cases b
130  [ normalize //
131  | normalize
132    cases c
133    [ normalize //
134    | normalize //
135    ]
136  ]
137qed.
138
139lemma eq_bv_refl:
140  ∀n,v. eq_bv n v v = true.
141  #n #v
142  elim v
143  [ //
144  | #n #hd #tl #ih
145    normalize
146    cases hd
147    [ normalize
148      @ ih
149    | normalize
150      @ ih
151    ]
152  ]
153qed.
154
155lemma eq_eq_bv:
156  ∀n, v, q.
157    v = q → eq_bv n v q = true.
158  #n #v
159  elim v
160  [ #q #h <h normalize %
161  | #n #hd #tl #ih #q #h >h //
162  ]
163qed.
164
165let rec foldl_strong_internal
166  (A: Type[0]) (P: list A → Type[0]) (l: list A)
167  (H: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]))
168  (prefix: list A) (suffix: list A) (acc: P prefix) on suffix:
169    l = prefix @ suffix → P(prefix @ suffix) ≝
170  match suffix return λl'. l = prefix @ l' → P (prefix @ l') with
171  [ nil ⇒ λprf. ?
172  | cons hd tl ⇒ λprf. ?
173  ].
174  [ > (append_nil ?)
175    @ acc
176  | applyS (foldl_strong_internal A P l H (prefix @ [hd]) tl ? ?)
177    [ @ (H prefix hd tl prf acc)
178    | applyS prf
179    ]
180  ]
181qed.
182
183definition foldl_strong ≝
184  λA: Type[0].
185  λP: list A → Type[0].
186  λl: list A.
187  λH: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]).
188  λacc: P [ ].
189    foldl_strong_internal A P l H [ ] l acc (refl …).
190
191definition bit_elim: ∀P: bool → bool. bool ≝
192  λP.
193    P true ∧ P false.
194
195let rec bitvector_elim_internal
196  (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝
197  match m return λm. m ≤ n → BitVector (n - m) → bool with
198  [ O    ⇒ λprf1. λprefix. P ?
199  | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?)
200  ].
201  [ applyS prefix
202  | letin res ≝ (bit ::: prefix)
203    < (minus_S_S ? ?)
204    > (minus_Sn_m ? ?)
205    [ @ res
206    | @ prf2
207    ]
208  | /2/
209  ].
210qed.
211
212definition bitvector_elim ≝
213  λn: nat.
214  λP: BitVector n → bool.
215    bitvector_elim_internal n P n ? ?.
216  [ @ (le_n ?)
217  | < (minus_n_n ?)
218    @ [[ ]]
219  ]
220qed.
221
222axiom vector_associative_append:
223  ∀A: Type[0].
224  ∀n, m, o:  nat.
225  ∀v: Vector A n.
226  ∀q: Vector A m.
227  ∀r: Vector A o.
228    ((v @@ q) @@ r)
229    ≃
230    (v @@ (q @@ r)).
231       
232lemma vector_cons_append:
233  ∀A: Type[0].
234  ∀n: nat.
235  ∀e: A.
236  ∀v: Vector A n.
237    e ::: v = [[ e ]] @@ v.
238  # A # N # E # V
239  elim V
240  [ normalize %
241  | # NN # AA # VV # IH
242    normalize
243    %
244  ]
245qed.
246
247lemma super_rewrite2:
248 ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m.
249  ∀P: ∀m. Vector A m → Prop.
250   n=m → v1 ≃ v2 → P n v1 → P m v2.
251 #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ //
252qed.
253
254lemma mem_middle_vector:
255  ∀A: Type[0].
256  ∀m, o: nat.
257  ∀eq: A → A → bool.
258  ∀reflex: ∀a. eq a a = true.
259  ∀p: Vector A m.
260  ∀a: A.
261  ∀r: Vector A o.
262    mem A eq ? (p@@(a:::r)) a = true.
263  # A # M # O # EQ # REFLEX # P # A
264  elim P
265  [ normalize
266    > (REFLEX A)
267    normalize
268    # H
269    %
270  | # NN # AA # PP # IH
271    normalize
272    cases (EQ A AA) //
273     @ IH
274  ]
275qed.
276
277lemma mem_monotonic_wrt_append:
278  ∀A: Type[0].
279  ∀m, o: nat.
280  ∀eq: A → A → bool.
281  ∀reflex: ∀a. eq a a = true.
282  ∀p: Vector A m.
283  ∀a: A.
284  ∀r: Vector A o.
285    mem A eq ? r a = true → mem A eq ? (p @@ r) a = true.
286  # A # M # O # EQ # REFLEX # P # A
287  elim P
288  [ #R #H @H
289  | #NN #AA # PP # IH #R #H
290    normalize
291    cases (EQ A AA)
292    [ normalize %
293    | @ IH @ H
294    ]
295  ]
296qed.
297
298lemma subvector_multiple_append:
299  ∀A: Type[0].
300  ∀o, n: nat.
301  ∀eq: A → A → bool.
302  ∀refl: ∀a. eq a a = true.
303  ∀h: Vector A o.
304  ∀v: Vector A n.
305  ∀m: nat.
306  ∀q: Vector A m.
307    bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)).
308  # A # O # N # EQ # REFLEX # H # V
309  elim V
310  [ normalize
311    # M # V %
312  | # NN # AA # VV # IH # MM # QQ
313    change with (bool_to_Prop (andb ??))
314    cut ((mem A EQ (O + (MM + S NN)) (H@@QQ@@AA:::VV) AA) = true)
315    [
316    | # HH > HH
317      > (vector_cons_append ? ? AA VV)
318      change with (bool_to_Prop (subvector_with ??????))
319      @(super_rewrite2 A ((MM + 1)+ NN) (MM+S NN) ??
320        (λSS.λVS.bool_to_Prop (subvector_with ?? (O+SS) ?? (H@@VS)))
321        ?
322        (vector_associative_append A ? ? ? QQ [[AA]] VV))
323      [ >associative_plus //
324      | @IH ]
325    ]
326    @(mem_monotonic_wrt_append)
327    [ @ REFLEX
328    | @(mem_monotonic_wrt_append)
329      [ @ REFLEX
330      | normalize
331        > REFLEX
332        normalize
333        %
334      ]
335    ]
336qed.
337
338lemma vector_cons_empty:
339  ∀A: Type[0].
340  ∀n: nat.
341  ∀v: Vector A n.
342    [[ ]] @@ v = v.
343  # A # N # V
344  elim V
345  [ normalize %
346  | # NN # HH # VV #H %
347  ]
348qed.
349
350corollary subvector_hd_tl:
351  ∀A: Type[0].
352  ∀o: nat.
353  ∀eq: A → A → bool.
354  ∀refl: ∀a. eq a a = true.
355  ∀h: A.
356  ∀v: Vector A o.
357    bool_to_Prop (subvector_with A ? ? eq v (h ::: v)).
358  # A # O # EQ # REFLEX # H # V
359  > (vector_cons_append A ? H V)
360  < (vector_cons_empty A ? ([[H]] @@ V))
361  @ (subvector_multiple_append A ? ? EQ REFLEX [[]] V ? [[ H ]])
362qed.
363
364lemma eq_a_reflexive:
365  ∀a. eq_a a a = true.
366  # A
367  cases A
368  %
369qed.
370
371lemma is_in_monotonic_wrt_append:
372  ∀m, n: nat.
373  ∀p: Vector addressing_mode_tag m.
374  ∀q: Vector addressing_mode_tag n.
375  ∀to_search: addressing_mode.
376    bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search).
377  # M # N # P # Q # TO_SEARCH
378  # H
379  elim Q
380  [ normalize
381    @ H
382  | # NN # PP # QQ # IH
383    normalize
384    cases (is_a PP TO_SEARCH)
385    [ normalize
386      %
387    | normalize
388      normalize in IH
389      @ IH
390    ]
391  ]
392qed.
393
394corollary is_in_hd_tl:
395  ∀to_search: addressing_mode.
396  ∀hd: addressing_mode_tag.
397  ∀n: nat.
398  ∀v: Vector addressing_mode_tag n.
399    bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search).
400  # TO_SEARCH # HD # N # V
401  elim V
402  [ # H
403    normalize in H;
404    cases H
405  | # NN # HHD # VV # IH # HH
406    > vector_cons_append
407    > (vector_cons_append ? ? HHD VV)
408    @ (is_in_monotonic_wrt_append ? 1 ([[HHD]]@@VV) [[HD]] TO_SEARCH)
409    @ HH
410  ]
411qed.
412 
413let rec list_addressing_mode_tags_elim
414  (n: nat) (l: Vector addressing_mode_tag (S n)) on l: (l → bool) → bool ≝
415  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
416   (l → bool) → bool ] with
417  [ VEmpty      ⇒  true 
418  | VCons len hd tl ⇒ λP.
419    let process_hd ≝
420      match hd return λhd. ∀P: hd:::tl → bool. bool with
421      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
422      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
423      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
424      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
425      | acc_a ⇒ λP.P ACC_A
426      | acc_b ⇒ λP.P ACC_B
427      | dptr ⇒ λP.P DPTR
428      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
429      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
430      | acc_dptr ⇒ λP.P ACC_DPTR
431      | acc_pc ⇒ λP.P ACC_PC
432      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
433      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
434      | carry ⇒ λP.P CARRY
435      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
436      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
437      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
438      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
439      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
440      ]
441    in
442      andb (process_hd P)
443       (match len return λx. x = len → bool with
444         [ O ⇒ λprf. true
445         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
446  ].
447  try %
448  [ 2: cases (sym_eq ??? prf); @tl
449  | generalize in match H; generalize in match tl; cases prf;
450    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
451    #tl
452    normalize in ⊢ (∀_: %. ?)
453    # H
454    whd
455    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
456    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
457qed.
458
459definition product_elim ≝
460  λm, n: nat.
461  λv: Vector addressing_mode_tag (S m).
462  λq: Vector addressing_mode_tag (S n).
463  λP: (v × q) → bool.
464    list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)).
465
466definition union_elim ≝
467  λA, B: Type[0].
468  λelimA: (A → bool) → bool.
469  λelimB: (B → bool) → bool.
470  λelimU: A ⊎ B → bool.
471    elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))).
472
473(*                           
474definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝
475  λP.
476    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧
477    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧
478    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧
479    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧
480    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧
481    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧
482    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧
483    list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
484    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧
485    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧
486    P (DA ? ACC_A) ∧
487    bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧
488    bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧
489    bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧
490    bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧
491    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧
492    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧
493    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧
494    list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
495    P (RL ? ACC_A) ∧
496    P (RLC ? ACC_A) ∧
497    P (RR ? ACC_A) ∧
498    P (RRC ? ACC_A) ∧
499    P (SWAP ? ACC_A) ∧
500    P (RET ?) ∧
501    P (RETI ?) ∧
502    P (NOP ?) ∧
503    bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧
504    list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧
505    bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧
506    bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧
507    union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]])
508                   (product_elim ? ? [[ registr; indirect ]] [[ data ]])
509                   (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧
510    list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧
511    union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]])
512                   (product_elim ? ? [[direct]] [[ acc_a ; data ]])
513                   (λd. P (XRL ? d)) ∧
514    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
515                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
516                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
517                   (λd. P (ANL ? d)) ∧
518    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]])
519                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
520                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
521                   (λd. P (ORL ? d)) ∧
522    union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]])
523                   (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]])
524                   (λd. P (MOVX ? d)) ∧
525    union_elim ? ? (
526      union_elim ? ? (
527        union_elim ? ? (
528          union_elim ? ? (
529            union_elim ? ?  (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
530                            (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]]))
531                            (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]]))
532                            (product_elim ? ? [[dptr]] [[data16]]))
533                            (product_elim ? ? [[carry]] [[bit_addr]]))
534                            (product_elim ? ? [[bit_addr]] [[carry]])
535                            (λd. P (MOV ? d)).
536  %
537qed.
538 
539definition instruction_elim: ∀P: instruction → bool. bool ≝
540  λP. (*
541    bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧
542    bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧
543    bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧
544    bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *)
545    bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (*  ∧
546    P (JMP INDIRECT_DPTR) ∧
547    list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧
548    preinstruction_elim (λp. P (RealInstruction p)). *)
549  %
550qed.
551
552
553axiom instruction_elim_complete:
554 ∀P. instruction_elim P = true → ∀i. P i = true.
555*)
556(*definition eq_instruction ≝
557  λi, j: instruction.
558    true.*)
559axiom eq_instruction: instruction → instruction → bool.
560axiom eq_instruction_refl: ∀i. eq_instruction i i = true.
561
562let rec vect_member
563  (A: Type[0]) (n: nat) (eq: A → A → bool)
564  (v: Vector A n) (a: A) on v: bool ≝
565  match v with
566  [ VEmpty          ⇒ false
567  | VCons len hd tl ⇒
568    eq hd a ∨ (vect_member A ? eq tl a)
569  ].
570
571let rec list_addressing_mode_tags_elim_prop
572  (n: nat)
573  (l: Vector addressing_mode_tag (S n))
574  on l:
575  ∀P: l → Prop.
576  ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a.
577  ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a.
578  ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a.
579  ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a.
580  ∀x: l. P x ≝
581  match l return
582    λy.
583      match y with
584      [ O    ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True
585      | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop.
586               ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True.
587               ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True.
588               ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True.
589               ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True.
590               ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True.
591               ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True.
592               ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True.
593               ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True.
594               ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True.
595               ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True.
596               ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True.
597               ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True.
598               ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True.
599               ∀carry_a: if vect_member … eq_a l carry then P CARRY else True.
600               ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True.
601               ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True.
602               ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True.
603               ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True.
604               ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True.
605               ∀x:l. P x
606      ] with
607  [ VEmpty          ⇒ λAbsurd. ⊥
608  | VCons len hd tl ⇒ λProof. ?
609  ] (refl ? (S n)). cases daemon. qed. (*
610  [ destruct(Absurd)
611  | # A1 # A2 # A3 # A4 # A5 # A6 # A7
612    # A8 # A9 # A10 # A11 # A12 # A13 # A14
613    # A15 # A16 # A17 # A18 # A19 # X
614    cases X
615    # SUB cases daemon ] qed.
616    cases SUB
617    [ # BYTE
618    normalize
619  ].
620 
621 
622(*    let prepare_hd ≝
623      match hd with
624      [ direct ⇒ λdirect_prf. ?
625      | indirect ⇒ λindirect_prf. ?
626      | ext_indirect ⇒ λext_indirect_prf. ?
627      | registr ⇒ λregistr_prf. ?
628      | acc_a ⇒ λacc_a_prf. ?
629      | acc_b ⇒ λacc_b_prf. ?
630      | dptr ⇒ λdptr_prf. ?
631      | data ⇒ λdata_prf. ?
632      | data16 ⇒ λdata16_prf. ?
633      | acc_dptr ⇒ λacc_dptr_prf. ?
634      | acc_pc ⇒ λacc_pc_prf. ?
635      | ext_indirect_dptr ⇒ λext_indirect_prf. ?
636      | indirect_dptr ⇒ λindirect_prf. ?
637      | carry ⇒ λcarry_prf. ?
638      | bit_addr ⇒ λbit_addr_prf. ?
639      | n_bit_addr ⇒ λn_bit_addr_prf. ?
640      | relative ⇒ λrelative_prf. ?
641      | addr11 ⇒ λaddr11_prf. ?
642      | addr16 ⇒ λaddr16_prf. ?
643      ]
644    in ? *)
645  ].
646  [ 1: destruct(absd)
647  | 2: # A1 # A2 # A3 # A4 # A5 # A6
648       # A7 # A8 # A9 # A10 # A11 # A12
649       # A13 # A14 # A15 # A16 # A17 # A18
650       # A19 *
651  ].
652
653
654  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
655   (l → bool) → bool ] with
656  [ VEmpty      ⇒  true 
657  | VCons len hd tl ⇒ λP.
658    let process_hd ≝
659      match hd return λhd. ∀P: hd:::tl → bool. bool with
660      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
661      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
662      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
663      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
664      | acc_a ⇒ λP.P ACC_A
665      | acc_b ⇒ λP.P ACC_B
666      | dptr ⇒ λP.P DPTR
667      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
668      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
669      | acc_dptr ⇒ λP.P ACC_DPTR
670      | acc_pc ⇒ λP.P ACC_PC
671      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
672      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
673      | carry ⇒ λP.P CARRY
674      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
675      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
676      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
677      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
678      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
679      ]
680    in
681      andb (process_hd P)
682       (match len return λx. x = len → bool with
683         [ O ⇒ λprf. true
684         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
685  ].
686  try %
687  [ 2: cases (sym_eq ??? prf); @tl
688  | generalize in match H; generalize in match tl; cases prf;
689    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
690    #tl
691    normalize in ⊢ (∀_: %. ?)
692    # H
693    whd
694    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
695    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
696qed.
697*)
698(*
699lemma test:
700  let i ≝ SJMP (RELATIVE (bitvector_of_nat 8 255)) in
701      (let assembled ≝ assembly1 i in
702      let code_memory ≝ load_code_memory assembled in
703      let fetched ≝ fetch code_memory ? in
704      let 〈instr_pc, ticks〉 ≝ fetched in
705        eq_instruction (\fst instr_pc)) i = true.
706 [2: @ zero
707 | normalize
708 ]*)
709
710lemma BitVectorTrie_O:
711 ∀A:Type[0].∀v:BitVectorTrie A 0.(∃w. v ≃ Leaf A w) ∨ v ≃ Stub A 0.
712 #A #v generalize in match (refl … O) cases v in ⊢ (??%? → (?(??(λ_.?%%??)))(?%%??))
713  [ #w #_ %1 %[@w] %
714  | #n #l #r #abs @⊥ //
715  | #n #EQ %2 >EQ %]
716qed.
717
718lemma BitVectorTrie_Sn:
719 ∀A:Type[0].∀n.∀v:BitVectorTrie A (S n).(∃l,r. v ≃ Node A n l r) ∨ v ≃ Stub A (S n).
720 #A #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → (?(??(λ_.??(λ_.?%%??))))%)
721  [ #m #abs @⊥ //
722  | #m #l #r #EQ %1 <(injective_S … EQ) %[@l] %[@r] //
723  | #m #EQ %2 // ]
724qed.
725
726lemma lookup_prepare_trie_for_insertion_hit:
727 ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.
728  lookup … b (prepare_trie_for_insertion … b v) a = v.
729 #A #a #v #n #b elim b // #m #hd #tl #IH cases hd normalize //
730qed.
731 
732lemma lookup_insert_hit:
733 ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.∀t:BitVectorTrie A n.
734  lookup … b (insert … b v t) a = v.
735 #A #a #v #n #b elim b -b -n //
736 #n #hd #tl #IH #t cases(BitVectorTrie_Sn … t)
737  [ * #l * #r #JMEQ >JMEQ cases hd normalize //
738  | #JMEQ >JMEQ cases hd normalize @lookup_prepare_trie_for_insertion_hit ]
739qed.
740
741coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0].
742
743lemma lookup_prepare_trie_for_insertion_miss:
744 ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.
745  (notb (eq_bv ? b c)) → lookup … b (prepare_trie_for_insertion … c v) a = a.
746 #A #a #v #n #c elim c
747  [ #b >(BitVector_O … b) normalize #abs @⊥ //
748  | #m #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ
749    cases hd cases hd' normalize
750    [2,3: #_ cases tl' //
751    |*: change with (bool_to_Prop (notb (eq_bv ???)) → ?) /2/ ]]
752qed.
753 
754lemma lookup_insert_miss:
755 ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.∀t:BitVectorTrie A n.
756  (notb (eq_bv ? b c)) → lookup … b (insert … c v t) a = lookup … b t a.
757 #A #a #v #n #c elim c -c -n
758  [ #b #t #DIFF @⊥ whd in DIFF; >(BitVector_O … b) in DIFF //
759  | #n #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ
760    #t cases(BitVectorTrie_Sn … t)
761    [ * #l * #r #JMEQ >JMEQ cases hd cases hd' #H normalize in H;
762     [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize // @IH //
763    | #JMEQ >JMEQ cases hd cases hd' #H normalize in H;
764     [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize
765     [3,4: cases tl' // | *: @lookup_prepare_trie_for_insertion_miss //]]]
766qed.
767
768definition load_code_memory_aux ≝
769 fold_left_i_aux … (
770   λi, mem, v.
771     insert … (bitvector_of_nat … i) v mem) (Stub Byte 16).
772
773axiom split_elim:
774 ∀A,l,m,v.∀P: (Vector A l) × (Vector A m) → Prop.
775  (∀vl,vm. v = vl@@vm → P 〈vl,vm〉) → P (split A l m v).
776
777axiom half_add_SO:
778 ∀pc.
779 \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc).
780
781(*
782axiom not_eqvb_S:
783 ∀pc.
784 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))).
785
786axiom not_eqvb_SS:
787 ∀pc.
788 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))).
789 
790axiom bitvector_elim_complete:
791 ∀n,P. bitvector_elim n P = true → ∀bv. P bv.
792
793lemma bitvector_elim_complete':
794 ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true.
795 #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv
796 generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ //
797qed.
798*)
799
800
801
802
803(*
804lemma andb_elim':
805 ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true.
806 #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize //
807qed.
808*)
809
810let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word)
811                       (encoding: list Byte) on encoding: Prop ≝
812  match encoding with
813  [ nil ⇒ final_pc = pc
814  | cons hd tl ⇒
815    let 〈new_pc, byte〉 ≝ next code_memory pc in
816      hd = byte ∧ encoding_check code_memory new_pc final_pc tl
817  ].
818
819lemma encoding_check_append: ∀code_memory,final_pc,l1,pc,l2.
820 encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … final_pc) (l1@l2) →
821  let intermediate_pc ≝ pc + length … l1 in
822   encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … intermediate_pc) l1 ∧
823    encoding_check code_memory (bitvector_of_nat … intermediate_pc) (bitvector_of_nat … final_pc) l2.
824 #code_memory #final_pc #l1 elim l1
825  [ #pc #l2 whd in ⊢ (????% → ?) #H <plus_n_O whd whd in ⊢ (?%?) /2/
826  | #hd #tl #IH #pc #l2 * #H1 #H2 >half_add_SO in H2; #H2 cases (IH … H2) <plus_n_Sm
827    #K1 #K2 % [2:@K2] whd % // >half_add_SO @K1 ]
828qed.
829
830axiom bitvector_3_elim_prop:
831 ∀P: BitVector 3 → Prop.
832  P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] →
833  P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] →
834  P [[true;true;false]] → P [[true;true;true]] → ∀v. P v.
835
836definition ticks_of_instruction ≝
837 λi.
838  let trivial_code_memory ≝ assembly1 i in
839  let trivial_status ≝ load_code_memory trivial_code_memory in
840   \snd (fetch trivial_status (zero ?)).
841
842axiom fetch_assembly:
843  ∀pc,i,code_memory,assembled.
844    assembled = assembly1 i →
845      let len ≝ length … assembled in
846      encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
847      let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in
848      let 〈instr_pc, ticks〉 ≝ fetched in
849      let 〈instr,pc'〉 ≝ instr_pc in
850       (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' (bitvector_of_nat … (pc + len))) = true.
851(* #pc #i #code_memory #assembled cases i [8: *]
852 [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]]
853 [47,48,49:
854 |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg
855  [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,
856   59,60,63,64,65,66,67: #ARG]]
857 [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,
858  56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2
859  [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,
860   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,
861   68,69,70,71: #ARG2]]
862 [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3]
863 normalize in ⊢ (???% → ?)
864 [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)]
865  normalize in ⊢ (???% → ?)]
866 #H >H * #H1 try (change in ⊢ (% → ?) with (? ∧ ?) * #H2)
867 try (change in ⊢ (% → ?) with (? ∧ ?) * #H3) whd in ⊢ (% → ?) #H4
868 change in ⊢ (let fetched ≝ % in ?) with (fetch0 ??)
869 whd in ⊢ (let fetched ≝ ??% in ?) <H1 whd in ⊢ (let fetched ≝ % in ?)
870 [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3]
871 [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,
872  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,
873  69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2]
874 whd >eq_instruction_refl >H4 @eq_bv_refl
875qed. *)
876
877let rec fetch_many code_memory final_pc pc expected on expected: Prop ≝
878 match expected with
879  [ nil ⇒ eq_bv … pc final_pc = true
880  | cons i tl ⇒
881     let fetched ≝ fetch code_memory pc in
882     let 〈instr_pc, ticks〉 ≝ fetched in
883     let 〈instr,pc'〉 ≝ instr_pc in
884      eq_instruction instr i = true ∧
885      ticks = (ticks_of_instruction i) ∧
886      fetch_many code_memory final_pc pc' tl].
887
888lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b.
889 #A #a #b #EQ destruct //
890qed.
891
892lemma pair_destruct: ∀A,B,a1,a2,b1,b2. pair A B a1 a2 = 〈b1,b2〉 → a1=b1 ∧ a2=b2.
893 #A #B #a1 #a2 #b1 #b2 #EQ destruct /2/
894qed.
895
896axiom eq_bv_to_eq: ∀n.∀v1,v2: BitVector n. eq_bv … v1 v2 = true → v1=v2.
897
898axiom eq_instruction_to_eq: ∀i1,i2. eq_instruction i1 i2 = true → i1 = i2.
899
900lemma fetch_assembly_pseudo:
901 ∀program,ppc,lookup_labels,lookup_datalabels.
902  ∀pi,code_memory,len,assembled,instructions,pc.
903   let expansion ≝ jump_expansion ppc program in
904   Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (bitvector_of_nat ? pc) expansion pi →
905    Some … 〈len,assembled〉 = assembly_1_pseudoinstruction program ppc (bitvector_of_nat ? pc) lookup_labels lookup_datalabels pi →
906     encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled →
907      fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions.
908 #program #ppc #lookup_labels #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc
909 #EQ1 whd in ⊢ (???% → ?) <EQ1 whd in ⊢ (???% → ?) #EQ2
910 cases (pair_destruct ?????? (option_destruct_Some … EQ2)) -EQ2; #EQ2a #EQ2b
911 >EQ2a >EQ2b -EQ2a EQ2b;
912  generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc
913  generalize in match pc elim instructions
914  [ #pc whd in ⊢ (% → %) #H >H @eq_bv_refl
915  | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd
916    generalize in match (fetch_assembly pc i code_memory … (refl …) H1)
917    cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %)
918    cases newi_pc #newi #newpc whd in ⊢ (% → %) #K cases (conjunction_true … K) -K; #K1
919    cases (conjunction_true … K1) -K1; #K1 #K2 #K3 % try % //
920    [ @eqb_true_to_eq <(eq_instruction_to_eq … K1) // | >(eq_bv_to_eq … K3) @IH @H2 ]
921qed.
922
923(* This establishes the correspondence between pseudo program counters and
924   program counters. It is at the heart of the proof. *)
925(*CSC: code taken from build_maps *)
926definition sigma0: pseudo_assembly_program → option (nat × (nat × (BitVectorTrie Word 16))) ≝
927 λinstr_list.
928  foldl ??
929    (λt. λi.
930       match t with
931       [ None ⇒ None ?
932       | Some ppc_pc_map ⇒
933         let 〈ppc,pc_map〉 ≝ ppc_pc_map in
934         let 〈program_counter, sigma_map〉 ≝ pc_map in
935         let 〈label, i〉 ≝ i in
936          match construct_costs instr_list ppc program_counter (λx. zero ?) (λx. zero ?) (Stub …) i with
937           [ None ⇒ None ?
938           | Some pc_ignore ⇒
939              let 〈pc,ignore〉 ≝ pc_ignore in
940              Some … 〈S ppc,〈pc, insert ? ? (bitvector_of_nat ? ppc) (bitvector_of_nat ? pc) sigma_map〉〉 ]
941       ]) (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (\snd instr_list).
942       
943definition tech_pc_sigma0: pseudo_assembly_program → option (nat × (BitVectorTrie Word 16)) ≝
944 λinstr_list.
945  match sigma0 instr_list with
946   [ None ⇒ None …
947   | Some result ⇒
948      let 〈ppc,pc_sigma_map〉 ≝ result in
949       Some … pc_sigma_map ].
950
951definition sigma_safe: pseudo_assembly_program → option (Word → Word) ≝       
952 λinstr_list.
953  match sigma0 instr_list with
954  [ None ⇒ None ?
955  | Some result ⇒
956    let 〈ppc,pc_sigma_map〉 ≝ result in
957    let 〈pc, sigma_map〉 ≝ pc_sigma_map in
958      if gtb pc (2^16) then
959        None ?
960      else
961        Some ? (λx.lookup ?? x sigma_map (zero …)) ].
962
963axiom policy_ok: ∀p. sigma_safe p ≠ None ….
964
965definition sigma: pseudo_assembly_program → Word → Word ≝
966 λp.
967  match sigma_safe p return λr:option (Word → Word). r ≠ None … → Word → Word with
968   [ None ⇒ λabs. ⊥
969   | Some r ⇒ λ_.r] (policy_ok p).
970 cases abs //
971qed.
972
973lemma length_append:
974 ∀A.∀l1,l2:list A.
975  |l1 @ l2| = |l1| + |l2|.
976 #A #l1 elim l1
977  [ //
978  | #hd #tl #IH #l2 normalize <IH //]
979qed.
980
981let rec does_not_occur (id:Identifier) (l:list labelled_instruction) on l: bool ≝
982 match l with
983  [ nil ⇒ true
984  | cons hd tl ⇒ notb (instruction_matches_identifier id hd) ∧ does_not_occur id tl].
985
986lemma does_not_occur_None:
987 ∀id,i,list_instr.
988  does_not_occur id (list_instr@[〈None …,i〉]) =
989  does_not_occur id list_instr.
990 #id #i #list_instr elim list_instr
991  [ % | #hd #tl #IH whd in ⊢ (??%%) >IH %]
992qed.
993
994let rec occurs_exactly_once (id:Identifier) (l:list labelled_instruction) on l : bool ≝
995 match l with
996  [ nil ⇒ false
997  | cons hd tl ⇒
998     if instruction_matches_identifier id hd then
999      does_not_occur id tl
1000     else
1001      occurs_exactly_once id tl ].
1002
1003lemma occurs_exactly_once_None:
1004 ∀id,i,list_instr.
1005  occurs_exactly_once id (list_instr@[〈None …,i〉]) =
1006  occurs_exactly_once id list_instr.
1007 #id #i #list_instr elim list_instr
1008  [ % | #hd #tl #IH whd in ⊢ (??%%) >IH >does_not_occur_None %]
1009qed.
1010
1011lemma index_of_internal_None: ∀i,id,instr_list,n.
1012 occurs_exactly_once id (instr_list@[〈None …,i〉]) →
1013  index_of_internal ? (instruction_matches_identifier id) instr_list n =
1014   index_of_internal ? (instruction_matches_identifier id) (instr_list@[〈None …,i〉]) n.
1015 #i #id #instr_list elim instr_list
1016  [ #n #abs whd in abs; cases abs
1017  | #hd #tl #IH #n whd in ⊢ (% → ??%%); whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ?)
1018    cases (instruction_matches_identifier id hd) whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ??%%)
1019    [ #H %
1020    | #H @IH whd in H; cases (occurs_exactly_once ??) in H ⊢ %
1021      [ #_ % | #abs cases abs ]]]
1022qed.
1023
1024lemma address_of_word_labels_code_mem_None: ∀i,id,instr_list.
1025 occurs_exactly_once id (instr_list@[〈None …,i〉]) →
1026  address_of_word_labels_code_mem instr_list id =
1027  address_of_word_labels_code_mem (instr_list@[〈None …,i〉]) id.
1028 #i #id #instr_list #H whd in ⊢ (??%%) whd in ⊢ (??(??%?)(??%?))
1029 >(index_of_internal_None … H) %
1030qed.
1031
1032axiom tech_pc_sigma0_append:
1033 ∀preamble,instr_list,prefix,label,i,pc',code,ppc,pc,costs,costs'.
1034  Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 →
1035   construct_costs 〈preamble,instr_list〉 … ppc pc (λx.zero 16) (λx. zero 16) costs i = Some … 〈pc',code〉 →
1036    tech_pc_sigma0 〈preamble,prefix@[〈label,i〉]〉 = Some … 〈pc',costs'〉.
1037
1038axiom tech_pc_sigma0_append_None:
1039 ∀preamble,instr_list,prefix,i,ppc,pc,costs.
1040  Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 →
1041   construct_costs 〈preamble,instr_list〉 … ppc pc (λx.zero 16) (λx. zero 16) costs i = None …
1042    → False.
1043
1044(*
1045definition build_maps' ≝
1046  λpseudo_program.
1047  let 〈preamble,instr_list〉 ≝ pseudo_program in
1048  let result ≝
1049   foldl_strong
1050    (option Identifier × pseudo_instruction)
1051    (λpre. Σres:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1052      let pre' ≝ 〈preamble,pre〉 in
1053      let 〈labels,pc_costs〉 ≝ res in
1054       tech_pc_sigma0 pre' = Some … pc_costs ∧
1055       ∀id. occurs_exactly_once id pre →
1056        lookup ?? id labels (zero …) = sigma pre' (address_of_word_labels_code_mem pre id))
1057    instr_list
1058    (λprefix,i,tl,prf,t.
1059      let 〈labels, pc_costs〉 ≝ t in
1060      let 〈program_counter, costs〉 ≝ pc_costs in
1061       let 〈label, i'〉 ≝ i in
1062       let labels ≝
1063         match label with
1064         [ None ⇒ labels
1065         | Some label ⇒
1066           let program_counter_bv ≝ bitvector_of_nat ? program_counter in
1067             insert ? ? label program_counter_bv labels
1068         ]
1069       in
1070         match construct_costs 〈preamble,instr_list〉 program_counter (λx. zero ?) (λx. zero ?) costs i' with
1071         [ None ⇒
1072            let dummy ≝ 〈labels,pc_costs〉 in
1073             dummy
1074         | Some construct ⇒ 〈labels, construct〉
1075         ]
1076    ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉
1077  in
1078   let 〈labels, pc_costs〉 ≝ result in
1079   let 〈pc, costs〉 ≝ pc_costs in
1080    〈labels, costs〉.
1081 [3: whd % // #id normalize in ⊢ (% → ?) #abs @⊥ //
1082 | whd cases construct in p3 #PC #CODE #JMEQ %
1083    [ @(tech_pc_sigma0_append ??????????? (jmeq_to_eq ??? JMEQ)) | #id #Hid ]
1084 | (* dummy case *) @⊥
1085   @(tech_pc_sigma0_append_None ?? prefix ???? (jmeq_to_eq ??? p3)) ]
1086 [*: generalize in match (sig2 … t) whd in ⊢ (% → ?)
1087     >p whd in ⊢ (% → ?) >p1 * #IH0 #IH1 >IH0 // ]
1088 whd in ⊢ (??(????%?)?) -labels1;
1089 cases label in Hid
1090  [ #Hid whd in ⊢ (??(????%?)?) >IH1 -IH1
1091     [ >(address_of_word_labels_code_mem_None … Hid)
1092       (* MANCA LEMMA: INDIRIZZO TROVATO NEL PROGRAMMA! *)
1093     | whd in Hid >occurs_exactly_once_None in Hid // ]
1094  | -label #label #Hid whd in ⊢ (??(????%?)?)
1095   
1096  ]
1097qed.
1098
1099lemma build_maps_ok:
1100 ∀p:pseudo_assembly_program.
1101  let 〈labels,costs〉 ≝ build_maps' p in
1102   ∀pc.
1103    (nat_of_bitvector … pc) < length … (\snd p) →
1104     lookup ?? pc labels (zero …) = sigma p (\snd (fetch_pseudo_instruction (\snd p) pc)).
1105 #p cases p #preamble #instr_list
1106  elim instr_list
1107   [ whd #pc #abs normalize in abs; cases (not_le_Sn_O ?) [#H cases (H abs) ]
1108   | #hd #tl #IH
1109    whd in ⊢ (match % with [ _ ⇒ ?])
1110   ]
1111qed.
1112*)
1113
1114(*
1115lemma rev_preserves_length:
1116 ∀A.∀l. length … (rev A l) = length … l.
1117  #A #l elim l
1118   [ %
1119   | #hd #tl #IH normalize >length_append normalize /2/ ]
1120qed.
1121
1122lemma rev_append:
1123 ∀A.∀l1,l2.
1124  rev A (l1@l2) = rev A l2 @ rev A l1.
1125 #A #l1 elim l1 normalize //
1126qed.
1127 
1128lemma rev_rev: ∀A.∀l. rev … (rev A l) = l.
1129 #A #l elim l
1130  [ //
1131  | #hd #tl #IH normalize >rev_append normalize // ]
1132qed.
1133
1134lemma split_len_Sn:
1135 ∀A:Type[0].∀l:list A.∀len.
1136  length … l = S len →
1137   Σl'.Σa. l = l'@[a] ∧ length … l' = len.
1138 #A #l elim l
1139  [ normalize #len #abs destruct
1140  | #hd #tl #IH #len
1141    generalize in match (rev_rev … tl)
1142    cases (rev A tl) in ⊢ (??%? → ?)
1143     [ #H <H normalize #EQ % [@[ ]] % [@hd] normalize /2/ 
1144     | #a #l' #H <H normalize #EQ
1145      %[@(hd::rev … l')] %[@a] % //
1146      >length_append in EQ #EQ normalize in EQ; normalize;
1147      generalize in match (injective_S … EQ) #EQ2 /2/ ]]
1148qed.
1149
1150lemma list_elim_rev:
1151 ∀A:Type[0].∀P:list A → Type[0].
1152  P [ ] → (∀l,a. P l → P (l@[a])) →
1153   ∀l. P l.
1154 #A #P #H1 #H2 #l
1155 generalize in match (refl … (length … l))
1156 generalize in ⊢ (???% → ?) #n generalize in match l
1157 elim n
1158  [ #L cases L [ // | #x #w #abs (normalize in abs) @⊥ // ]
1159  | #m #IH #L #EQ
1160    cases (split_len_Sn … EQ) #l' * #a * /3/ ]
1161qed.
1162
1163axiom is_prefix: ∀A:Type[0]. list A → list A → Prop.
1164axiom prefix_of_append:
1165 ∀A:Type[0].∀l,l1,l2:list A.
1166  is_prefix … l l1 → is_prefix … l (l1@l2).
1167axiom prefix_reflexive: ∀A,l. is_prefix A l l.
1168axiom nil_prefix: ∀A,l. is_prefix A [ ] l.
1169
1170record Propify (A:Type[0]) : Type[0] (*Prop*) ≝ { in_propify: A }.
1171
1172definition Propify_elim: ∀A. ∀P:Prop. (A → P) → (Propify A → P) ≝
1173 λA,P,H,x. match x with [ mk_Propify p ⇒ H p ].
1174
1175definition app ≝
1176 λA:Type[0].λl1:Propify (list A).λl2:list A.
1177  match l1 with
1178   [ mk_Propify l1 ⇒ mk_Propify … (l1@l2) ].
1179
1180lemma app_nil: ∀A,l1. app A l1 [ ] = l1.
1181 #A * /3/
1182qed.
1183
1184lemma app_assoc: ∀A,l1,l2,l3. app A (app A l1 l2) l3 = app A l1 (l2@l3).
1185 #A * #l1 normalize //
1186qed.
1187
1188let rec foldli (A: Type[0]) (B: Propify (list A) → Type[0])
1189 (f: ∀prefix. B prefix → ∀x.B (app … prefix [x]))
1190 (prefix: Propify (list A)) (b: B prefix) (l: list A) on l :
1191 B (app … prefix l) ≝
1192  match l with
1193  [ nil ⇒ ? (* b *)
1194  | cons hd tl ⇒ ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*)
1195  ].
1196 [ applyS b
1197 | <(app_assoc ?? [hd]) @(foldli A B f (app … prefix [hd]) (f prefix b hd) tl) ]
1198qed.
1199
1200(*
1201let rec foldli (A: Type[0]) (B: list A → Type[0]) (f: ∀prefix. B prefix → ∀x. B (prefix@[x]))
1202 (prefix: list A) (b: B prefix) (l: list A) on l : B (prefix@l) ≝
1203  match l with
1204  [ nil ⇒ ? (* b *)
1205  | cons hd tl ⇒
1206     ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*)
1207  ].
1208 [ applyS b
1209 | applyS (foldli A B f (prefix@[hd]) (f prefix b hd) tl) ]
1210qed.
1211*)
1212
1213definition foldll:
1214 ∀A:Type[0].∀B: Propify (list A) → Type[0].
1215  (∀prefix. B prefix → ∀x. B (app … prefix [x])) →
1216   B (mk_Propify … []) → ∀l: list A. B (mk_Propify … l)
1217 ≝ λA,B,f. foldli A B f (mk_Propify … [ ]).
1218
1219axiom is_pprefix: ∀A:Type[0]. Propify (list A) → list A → Prop.
1220axiom pprefix_of_append:
1221 ∀A:Type[0].∀l,l1,l2.
1222  is_pprefix A l l1 → is_pprefix A l (l1@l2).
1223axiom pprefix_reflexive: ∀A,l. is_pprefix A (mk_Propify … l) l.
1224axiom nil_pprefix: ∀A,l. is_pprefix A (mk_Propify … [ ]) l.
1225
1226
1227axiom foldll':
1228 ∀A:Type[0].∀l: list A.
1229  ∀B: ∀prefix:Propify (list A). is_pprefix ? prefix l → Type[0].
1230  (∀prefix,proof. B prefix proof → ∀x,proof'. B (app … prefix [x]) proof') →
1231   B (mk_Propify … [ ]) (nil_pprefix …) → B (mk_Propify … l) (pprefix_reflexive … l).
1232 #A #l #B
1233 generalize in match (foldll A (λprefix. is_pprefix ? prefix l)) #HH
1234 
1235 
1236  #H #acc
1237 @foldll
1238  [
1239  |
1240  ]
1241 
1242 ≝ λA,B,f. foldli A B f (mk_Propify … [ ]).
1243
1244
1245(*
1246record subset (A:Type[0]) (P: A → Prop): Type[0] ≝
1247 { subset_wit:> A;
1248   subset_proof: P subset_wit
1249 }.
1250*)
1251
1252definition build_maps' ≝
1253  λpseudo_program.
1254  let 〈preamble,instr_list〉 ≝ pseudo_program in
1255  let result ≝
1256   foldll
1257    (option Identifier × pseudo_instruction)
1258    (λprefix.
1259      Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1260       match prefix return λ_.Prop with [mk_Propify prefix ⇒ tech_pc_sigma0 〈preamble,prefix〉 ≠ None ?])
1261    (λprefix,t,i.
1262      let 〈labels, pc_costs〉 ≝ t in
1263      let 〈program_counter, costs〉 ≝ pc_costs in
1264       let 〈label, i'〉 ≝ i in
1265       let labels ≝
1266         match label with
1267         [ None ⇒ labels
1268         | Some label ⇒
1269           let program_counter_bv ≝ bitvector_of_nat ? program_counter in
1270             insert ? ? label program_counter_bv labels
1271         ]
1272       in
1273         match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with
1274         [ None ⇒
1275            let dummy ≝ 〈labels,pc_costs〉 in
1276              dummy
1277         | Some construct ⇒ 〈labels, construct〉
1278         ]
1279    ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 instr_list
1280  in
1281   let 〈labels, pc_costs〉 ≝ result in
1282   let 〈pc, costs〉 ≝ pc_costs in
1283    〈labels, costs〉.
1284 [
1285 | @⊥
1286 | normalize % //
1287 ]
1288qed.
1289
1290definition build_maps' ≝
1291  λpseudo_program.
1292  let 〈preamble,instr_list〉 ≝ pseudo_program in
1293  let result ≝
1294   foldl
1295    (Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1296          ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧
1297           tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)))
1298    (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix.
1299          let instr_list_prefix' ≝ instr_list_prefix @ [i] in
1300           is_prefix ? instr_list_prefix' instr_list →
1301           tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?)
1302    (λt: Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1303          ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧
1304           tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)).
1305     λi: Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix.
1306          let instr_list_prefix' ≝ instr_list_prefix @ [i] in
1307           is_prefix ? instr_list_prefix' instr_list →
1308           tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ? .
1309      let 〈labels, pc_costs〉 ≝ t in
1310      let 〈program_counter, costs〉 ≝ pc_costs in
1311       let 〈label, i'〉 ≝ i in
1312       let labels ≝
1313         match label with
1314         [ None ⇒ labels
1315         | Some label ⇒
1316           let program_counter_bv ≝ bitvector_of_nat ? program_counter in
1317             insert ? ? label program_counter_bv labels
1318         ]
1319       in
1320         match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with
1321         [ None ⇒
1322            let dummy ≝ 〈labels,pc_costs〉 in
1323              dummy
1324         | Some construct ⇒ 〈labels, construct〉
1325         ]
1326    ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 ?(*instr_list*)
1327  in
1328   let 〈labels, pc_costs〉 ≝ result in
1329   let 〈pc, costs〉 ≝ pc_costs in
1330    〈labels, costs〉.
1331 [4: @(list_elim_rev ?
1332       (λinstr_list. list (
1333        (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix.
1334          let instr_list_prefix' ≝ instr_list_prefix @ [i] in
1335           is_prefix ? instr_list_prefix' instr_list →
1336           tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?)))
1337       ?? instr_list) (* CSC: BAD ORDER FOR CODE EXTRACTION *)
1338      [ @[ ]
1339      | #l' #a #limage %2
1340        [ %[@a] #PREFIX #PREFIX_OK
1341        | (* CSC: EVEN WORST CODE FOR EXTRACTION: WE SHOULD STRENGTHEN
1342             THE INDUCTION HYPOTHESIS INSTEAD *)
1343          elim limage
1344           [ %1
1345           | #HD #TL #IH @(?::IH) cases HD #ELEM #K1 %[@ELEM] #K2 #K3
1346             @K1 @(prefix_of_append ???? K3)
1347           ] 
1348        ]
1349       
1350       
1351     
1352 
1353  cases t in c2 ⊢ % #t' * #LIST_PREFIX * #H1t' #H2t' #HJMt'
1354     % [@ (LIST_PREFIX @ [i])] %
1355      [ cases (sig2 … i LIST_PREFIX) #K1 #K2 @K1
1356      | (* DOABLE IN PRINCIPLE *)
1357      ]
1358 | (* assert false case *)
1359 |3: % [@ ([ ])] % [2: % | (* DOABLE *)]
1360 |   
1361*)
1362
1363axiom assembly_ok:
1364 ∀program,assembled,costs,labels.
1365  Some … 〈labels,costs〉 = build_maps program →
1366  Some … 〈assembled,costs〉 = assembly program →
1367  let code_memory ≝ load_code_memory assembled in
1368  let preamble ≝ \fst program in
1369  let datalabels ≝ construct_datalabels preamble in
1370  let lookup_labels ≝ λx. sigma program (address_of_word_labels_code_mem (\snd program) x) in
1371  let lookup_datalabels ≝ λx. lookup ?? x datalabels (zero ?) in
1372   ∀ppc,len,assembledi.
1373    let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1374     Some … 〈len,assembledi〉 = assembly_1_pseudoinstruction program ppc (sigma program ppc) lookup_labels lookup_datalabels pi →
1375      encoding_check code_memory (sigma program ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program ppc) + len)) assembledi ∧
1376       sigma program newppc = bitvector_of_nat … (nat_of_bitvector … (sigma program ppc) + len).
1377
1378axiom bitvector_of_nat_nat_of_bitvector:
1379  ∀n,v.
1380    bitvector_of_nat n (nat_of_bitvector n v) = v.
1381
1382axiom assembly_ok_to_expand_pseudo_instruction_ok:
1383 ∀program,assembled,costs.
1384  Some … 〈assembled,costs〉 = assembly program →
1385   ∀ppc.
1386    let code_memory ≝ load_code_memory assembled in
1387    let preamble ≝ \fst program in   
1388    let data_labels ≝ construct_datalabels preamble in
1389    let lookup_labels ≝ λx. sigma program (address_of_word_labels_code_mem (\snd program) x) in
1390    let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in
1391    let expansion ≝ jump_expansion ppc program in
1392    let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1393     ∃instructions.
1394      Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program ppc) expansion pi.
1395
1396lemma fetch_assembly_pseudo2:
1397 ∀program,assembled,costs,labels.
1398  Some … 〈labels,costs〉 = build_maps program →
1399  Some … 〈assembled,costs〉 = assembly program →
1400   ∀ppc.
1401    let code_memory ≝ load_code_memory assembled in
1402    let preamble ≝ \fst program in
1403    let data_labels ≝ construct_datalabels preamble in
1404    let lookup_labels ≝ λx. sigma program (address_of_word_labels_code_mem (\snd program) x) in
1405    let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in
1406    let expansion ≝ jump_expansion ppc program in
1407    let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in
1408     ∃instructions.
1409      Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program ppc) expansion pi ∧
1410       fetch_many code_memory (sigma program newppc) (sigma program ppc) instructions.
1411 #program #assembled #costs #labels #BUILD_MAPS #ASSEMBLY #ppc
1412 generalize in match (assembly_ok_to_expand_pseudo_instruction_ok program assembled costs ASSEMBLY ppc)
1413 letin code_memory ≝ (load_code_memory assembled)
1414 letin preamble ≝ (\fst program)
1415 letin data_labels ≝ (construct_datalabels preamble)
1416 letin lookup_labels ≝ (λx. sigma program (address_of_word_labels_code_mem (\snd program) x))
1417 letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?))
1418 whd in ⊢ (% → %)
1419 generalize in match (assembly_ok … BUILD_MAPS ASSEMBLY ppc)
1420 cases (fetch_pseudo_instruction (\snd program) ppc) #pi #newppc
1421 generalize in match (fetch_assembly_pseudo program ppc
1422  (λx. sigma program (address_of_word_labels_code_mem (\snd program) x)) (λx. lookup ?? x data_labels (zero ?)) pi
1423  (load_code_memory assembled));
1424 whd in ⊢ ((∀_.∀_.∀_.∀_.%) → (∀_.∀_.%) → % → %) #H1 #H2 * #instructions #EXPAND
1425 whd in H1:(∀_.∀_.∀_.∀_.? → ???% → ?) H2:(∀_.∀_.???% → ?);
1426 normalize nodelta in EXPAND; (* HERE *)
1427 generalize in match (λlen, assembled.H1 len assembled instructions (nat_of_bitvector … (sigma program ppc))) -H1; #H1
1428 >bitvector_of_nat_nat_of_bitvector in H1; #H1
1429 <EXPAND in H1 H2; whd in ⊢ ((∀_.∀_.? → ???% → ?) → (∀_.∀_.???% → ?) → ?)
1430 #H1 #H2
1431 cases (H2 ?? (refl …)) -H2; #K1 #K2 >K2
1432 generalize in match (H1 ?? (refl …) (refl …) ?) -H1;
1433  [ #K3 % [2: % [% | @K3]] | @K1 ]
1434qed.
1435
1436(* OLD?
1437definition assembly_specification:
1438  ∀assembly_program: pseudo_assembly_program.
1439  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1440  λpseudo_assembly_program.
1441  λcode_mem.
1442    ∀pc: Word.
1443      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1444      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1445      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1446      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1447      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1448       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1449      match pre_assembled with
1450       [ None ⇒ True
1451       | Some pc_code ⇒
1452          let 〈new_pc,code〉 ≝ pc_code in
1453           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1454
1455axiom assembly_meets_specification:
1456  ∀pseudo_assembly_program.
1457    match assembly pseudo_assembly_program with
1458    [ None ⇒ True
1459    | Some code_mem_cost ⇒
1460      let 〈code_mem, cost〉 ≝ code_mem_cost in
1461        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1462    ].
1463(*
1464  # PROGRAM
1465  [ cases PROGRAM
1466    # PREAMBLE
1467    # INSTR_LIST
1468    elim INSTR_LIST
1469    [ whd
1470      whd in ⊢ (∀_. %)
1471      # PC
1472      whd
1473    | # INSTR
1474      # INSTR_LIST_TL
1475      # H
1476      whd
1477      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1478    ]
1479  | cases not_implemented
1480  ] *)
1481*)
1482
1483definition internal_pseudo_address_map ≝ list (BitVector 8).
1484
1485axiom low_internal_ram_of_pseudo_low_internal_ram:
1486 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1487
1488axiom high_internal_ram_of_pseudo_high_internal_ram:
1489 ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7.
1490
1491axiom low_internal_ram_of_pseudo_internal_ram_hit:
1492 ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀addr:BitVector 7.
1493  member ? (eq_bv 8) (false:::addr) M = true →
1494   let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1495   let pbl ≝ lookup ? 7 addr (low_internal_ram ? s) (zero 8) in
1496   let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram ? s) (zero 8) in
1497   let bl ≝ lookup ? 7 addr ram (zero 8) in
1498   let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in
1499    bu@@bl = sigma (code_memory … s) (pbu@@pbl).
1500
1501axiom low_internal_ram_of_pseudo_internal_ram_miss:
1502 ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀addr:BitVector 7.
1503  let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in
1504  let 〈carr,Saddr〉 ≝ half_add ? addr (bitvector_of_nat 7 1) in
1505  carr = false →
1506  member ? (eq_bv 8) (false:::addr) M = false →
1507   member ? (eq_bv 8) (false:::Saddr) M = false →
1508    lookup ? 7 Saddr ram = lookup ? 7 Saddr (low_internal_ram … s).
1509
1510definition addressing_mode_ok ≝
1511 λM:internal_pseudo_address_map.λs:PseudoStatus.
1512  λaddr:addressing_mode.
1513   match addr with
1514    [ DIRECT d ⇒
1515       ¬(member ? (eq_bv 8) d M) ∧
1516       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1517    | INDIRECT i ⇒
1518       let d ≝ get_register ? s [[false;false;i]] in
1519       ¬(member ? (eq_bv 8) d M) ∧
1520       ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M)
1521    | EXT_INDIRECT _ ⇒ true
1522    | REGISTER _ ⇒ true
1523    | ACC_A ⇒ true
1524    | ACC_B ⇒ true
1525    | DPTR ⇒ true
1526    | DATA _ ⇒ true
1527    | DATA16 _ ⇒ true
1528    | ACC_DPTR ⇒ true
1529    | ACC_PC ⇒ true
1530    | EXT_INDIRECT_DPTR ⇒ true
1531    | INDIRECT_DPTR ⇒ true
1532    | CARRY ⇒ true
1533    | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1534    | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *)
1535    | RELATIVE _ ⇒ true
1536    | ADDR11 _ ⇒ true
1537    | ADDR16 _ ⇒ true ].
1538
1539definition next_internal_pseudo_address_map ≝
1540 λM:internal_pseudo_address_map.
1541  λs:PseudoStatus.
1542   match \fst (fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s)) with
1543    [ Comment _ ⇒ Some ? M
1544    | Cost _ ⇒ Some … M
1545    | Jmp _ ⇒ Some … M
1546    | Call _ ⇒
1547       Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M)
1548    | Mov _ _ ⇒ Some … M
1549    | Instruction instr ⇒
1550       match instr with
1551        [ ADD addr1 addr2 ⇒
1552           if addressing_mode_ok M s addr1 ∧ addressing_mode_ok M s addr2 then
1553            Some ? M
1554           else
1555            None ?
1556        | ADDC addr1 addr2 ⇒
1557           if addressing_mode_ok M s addr1 ∧ addressing_mode_ok M s addr2 then
1558            Some ? M
1559           else
1560            None ?
1561        | SUBB addr1 addr2 ⇒
1562           if addressing_mode_ok M s addr1 ∧ addressing_mode_ok M s addr2 then
1563            Some ? M
1564           else
1565            None ?       
1566        | _ ⇒ (* TO BE COMPLETED *) Some ? M ]].
1567   
1568definition status_of_pseudo_status:
1569 internal_pseudo_address_map → PseudoStatus → option Status ≝
1570 λM,ps.
1571  let pap ≝ code_memory … ps in
1572   match assembly pap with
1573    [ None ⇒ None …
1574    | Some p ⇒
1575       let cm ≝ load_code_memory (\fst p) in
1576       let pc ≝ sigma pap (program_counter ? ps) in
1577       let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in
1578       let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (low_internal_ram … ps) in
1579        Some …
1580         (mk_PreStatus (BitVectorTrie Byte 16)
1581           cm
1582           lir
1583           hir
1584           (external_ram … ps)
1585           pc
1586           (special_function_registers_8051 … ps)
1587           (special_function_registers_8052 … ps)
1588           (p1_latch … ps)
1589           (p3_latch … ps)
1590           (clock … ps)) ].
1591
1592(*
1593definition write_at_stack_pointer':
1594 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1595  λM: Type[0].
1596  λs: PreStatus M.
1597  λv: Byte.
1598    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1599    let bit_zero ≝ get_index_v… nu O ? in
1600    let bit_1 ≝ get_index_v… nu 1 ? in
1601    let bit_2 ≝ get_index_v… nu 2 ? in
1602    let bit_3 ≝ get_index_v… nu 3 ? in
1603      if bit_zero then
1604        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1605                              v (low_internal_ram ? s) in
1606          set_low_internal_ram ? s memory
1607      else
1608        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1609                              v (high_internal_ram ? s) in
1610          set_high_internal_ram ? s memory.
1611  [ cases l0 %
1612  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1613qed.
1614
1615definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1616 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1617
1618  λticks_of.
1619  λs.
1620  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1621  let ticks ≝ ticks_of (program_counter ? s) in
1622  let s ≝ set_clock ? s (clock ? s + ticks) in
1623  let s ≝ set_program_counter ? s pc in
1624    match instr with
1625    [ Instruction instr ⇒
1626       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1627    | Comment cmt ⇒ s
1628    | Cost cst ⇒ s
1629    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1630    | Call call ⇒
1631      let a ≝ address_of_word_labels s call in
1632      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1633      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1634      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1635      let s ≝ write_at_stack_pointer' ? s pc_bl in
1636      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1637      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1638      let s ≝ write_at_stack_pointer' ? s pc_bu in
1639        set_program_counter ? s a
1640    | Mov dptr ident ⇒
1641       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1642    ].
1643 [
1644 |2,3,4: %
1645 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1646 |
1647 | %
1648 ]
1649 cases not_implemented
1650qed.
1651*)
1652
1653axiom execute_1_pseudo_instruction_preserves_code_memory:
1654 ∀ticks_of,ps.
1655  code_memory … (execute_1_pseudo_instruction ticks_of ps) = code_memory … ps.
1656
1657(*
1658lemma execute_code_memory_unchanged:
1659 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1660 #ticks #ps whd in ⊢ (??? (??%))
1661 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1662  (program_counter pseudo_assembly_program ps)) #instr #pc
1663 whd in ⊢ (??? (??%)) cases instr
1664  [ #pre cases pre
1665     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1666       cases (split ????) #z1 #z2 %
1667     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1668       cases (split ????) #z1 #z2 %
1669     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1670       cases (split ????) #z1 #z2 %
1671     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1672       [ #x1 whd in ⊢ (??? (??%))
1673     | *: cases not_implemented
1674     ]
1675  | #comment %
1676  | #cost %
1677  | #label %
1678  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1679    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1680    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1681    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1682    (* CSC: ??? *)
1683  | #dptr #label (* CSC: ??? *)
1684  ]
1685  cases not_implemented
1686qed.
1687*)
1688
1689lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1690 ∀M:internal_pseudo_address_map.
1691 ∀ps,ps': PseudoStatus.
1692  code_memory … ps = code_memory … ps' →
1693   match status_of_pseudo_status M ps with
1694    [ None ⇒ status_of_pseudo_status M ps' = None …
1695    | Some _ ⇒ ∃w. status_of_pseudo_status M ps' = Some … w
1696    ].
1697 #M #ps #ps' #H whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1698 generalize in match (refl … (assembly (code_memory … ps)))
1699 cases (assembly ?) in ⊢ (???% → %)
1700  [ #K whd whd in ⊢ (??%?) <H >K %
1701  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1702qed.
1703
1704definition ticks_of0: pseudo_assembly_program → Word → ? → nat × nat ≝
1705  λprogram: pseudo_assembly_program.
1706  λppc: Word.
1707  λfetched.
1708    match fetched with
1709    [ Instruction instr ⇒
1710      match instr with
1711      [ JC lbl ⇒
1712        match jump_expansion ppc program with
1713        [ short_jump ⇒ 〈2, 2〉
1714        | medium_jump ⇒ ?
1715        | long_jump ⇒ 〈4, 4〉
1716        ]
1717      | JNC lbl ⇒
1718        match jump_expansion ppc program with
1719        [ short_jump ⇒ 〈2, 2〉
1720        | medium_jump ⇒ ?
1721        | long_jump ⇒ 〈4, 4〉
1722        ]
1723      | JB bit lbl ⇒
1724        match jump_expansion ppc program with
1725        [ short_jump ⇒ 〈2, 2〉
1726        | medium_jump ⇒ ?
1727        | long_jump ⇒ 〈4, 4〉
1728        ]
1729      | JNB bit lbl ⇒
1730        match jump_expansion ppc program with
1731        [ short_jump ⇒ 〈2, 2〉
1732        | medium_jump ⇒ ?
1733        | long_jump ⇒ 〈4, 4〉
1734        ]
1735      | JBC bit lbl ⇒
1736        match jump_expansion ppc program with
1737        [ short_jump ⇒ 〈2, 2〉
1738        | medium_jump ⇒ ?
1739        | long_jump ⇒ 〈4, 4〉
1740        ]
1741      | JZ lbl ⇒
1742        match jump_expansion ppc program with
1743        [ short_jump ⇒ 〈2, 2〉
1744        | medium_jump ⇒ ?
1745        | long_jump ⇒ 〈4, 4〉
1746        ]
1747      | JNZ lbl ⇒
1748        match jump_expansion ppc program with
1749        [ short_jump ⇒ 〈2, 2〉
1750        | medium_jump ⇒ ?
1751        | long_jump ⇒ 〈4, 4〉
1752        ]
1753      | CJNE arg lbl ⇒
1754        match jump_expansion ppc program with
1755        [ short_jump ⇒ 〈2, 2〉
1756        | medium_jump ⇒ ?
1757        | long_jump ⇒ 〈4, 4〉
1758        ]
1759      | DJNZ arg lbl ⇒
1760        match jump_expansion ppc program with
1761        [ short_jump ⇒ 〈2, 2〉
1762        | medium_jump ⇒ ?
1763        | long_jump ⇒ 〈4, 4〉
1764        ]
1765      | ADD arg1 arg2 ⇒
1766        let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in
1767         〈ticks, ticks〉
1768      | ADDC arg1 arg2 ⇒
1769        let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in
1770         〈ticks, ticks〉
1771      | SUBB arg1 arg2 ⇒
1772        let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in
1773         〈ticks, ticks〉
1774      | INC arg ⇒
1775        let ticks ≝ ticks_of_instruction (INC ? arg) in
1776         〈ticks, ticks〉
1777      | DEC arg ⇒
1778        let ticks ≝ ticks_of_instruction (DEC ? arg) in
1779         〈ticks, ticks〉
1780      | MUL arg1 arg2 ⇒
1781        let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in
1782         〈ticks, ticks〉
1783      | DIV arg1 arg2 ⇒
1784        let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in
1785         〈ticks, ticks〉
1786      | DA arg ⇒
1787        let ticks ≝ ticks_of_instruction (DA ? arg) in
1788         〈ticks, ticks〉
1789      | ANL arg ⇒
1790        let ticks ≝ ticks_of_instruction (ANL ? arg) in
1791         〈ticks, ticks〉
1792      | ORL arg ⇒
1793        let ticks ≝ ticks_of_instruction (ORL ? arg) in
1794         〈ticks, ticks〉
1795      | XRL arg ⇒
1796        let ticks ≝ ticks_of_instruction (XRL ? arg) in
1797         〈ticks, ticks〉
1798      | CLR arg ⇒
1799        let ticks ≝ ticks_of_instruction (CLR ? arg) in
1800         〈ticks, ticks〉
1801      | CPL arg ⇒
1802        let ticks ≝ ticks_of_instruction (CPL ? arg) in
1803         〈ticks, ticks〉
1804      | RL arg ⇒
1805        let ticks ≝ ticks_of_instruction (RL ? arg) in
1806         〈ticks, ticks〉
1807      | RLC arg ⇒
1808        let ticks ≝ ticks_of_instruction (RLC ? arg) in
1809         〈ticks, ticks〉
1810      | RR arg ⇒
1811        let ticks ≝ ticks_of_instruction (RR ? arg) in
1812         〈ticks, ticks〉
1813      | RRC arg ⇒
1814        let ticks ≝ ticks_of_instruction (RRC ? arg) in
1815         〈ticks, ticks〉
1816      | SWAP arg ⇒
1817        let ticks ≝ ticks_of_instruction (SWAP ? arg) in
1818         〈ticks, ticks〉
1819      | MOV arg ⇒
1820        let ticks ≝ ticks_of_instruction (MOV ? arg) in
1821         〈ticks, ticks〉
1822      | MOVX arg ⇒
1823        let ticks ≝ ticks_of_instruction (MOVX ? arg) in
1824         〈ticks, ticks〉
1825      | SETB arg ⇒
1826        let ticks ≝ ticks_of_instruction (SETB ? arg) in
1827         〈ticks, ticks〉
1828      | PUSH arg ⇒
1829        let ticks ≝ ticks_of_instruction (PUSH ? arg) in
1830         〈ticks, ticks〉
1831      | POP arg ⇒
1832        let ticks ≝ ticks_of_instruction (POP ? arg) in
1833         〈ticks, ticks〉
1834      | XCH arg1 arg2 ⇒
1835        let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in
1836         〈ticks, ticks〉
1837      | XCHD arg1 arg2 ⇒
1838        let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in
1839         〈ticks, ticks〉
1840      | RET ⇒
1841        let ticks ≝ ticks_of_instruction (RET ?) in
1842         〈ticks, ticks〉
1843      | RETI ⇒
1844        let ticks ≝ ticks_of_instruction (RETI ?) in
1845         〈ticks, ticks〉
1846      | NOP ⇒
1847        let ticks ≝ ticks_of_instruction (NOP ?) in
1848         〈ticks, ticks〉
1849      ]
1850    | Comment comment ⇒ 〈0, 0〉
1851    | Cost cost ⇒ 〈0, 0〉
1852    | Jmp jmp ⇒ 〈2, 2〉
1853    | Call call ⇒ 〈2, 2〉
1854    | Mov dptr tgt ⇒ 〈2, 2〉
1855    ].
1856  cases not_implemented (* policy returned medium_jump for conditional jumping = impossible *)
1857qed.
1858
1859definition ticks_of: pseudo_assembly_program → Word → nat × nat ≝
1860  λprogram: pseudo_assembly_program.
1861  λppc: Word.
1862    let 〈preamble, pseudo〉 ≝ program in
1863    let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in
1864     ticks_of0 program ppc fetched.
1865
1866lemma get_register_set_program_counter:
1867 ∀T,s,pc. get_register T (set_program_counter … s pc) = get_register … s.
1868 #T #s #pc %
1869qed.
1870
1871lemma get_8051_sfr_set_program_counter:
1872 ∀T,s,pc. get_8051_sfr T (set_program_counter … s pc) = get_8051_sfr … s.
1873 #T #s #pc %
1874qed.
1875
1876lemma get_bit_addressable_sfr_set_program_counter:
1877 ∀T,s,pc. get_bit_addressable_sfr T (set_program_counter … s pc) = get_bit_addressable_sfr … s.
1878 #T #s #pc %
1879qed.
1880
1881lemma low_internal_ram_set_program_counter:
1882 ∀T,s,pc. low_internal_ram T (set_program_counter … s pc) = low_internal_ram … s.
1883 #T #s #pc %
1884qed.
1885
1886lemma get_arg_8_set_program_counter:
1887 ∀n.∀l:Vector addressing_mode_tag (S n). ¬(is_in ? l ACC_PC) →
1888  ∀T,s,pc,b.∀arg:l.
1889   get_arg_8 T (set_program_counter … s pc) b arg = get_arg_8 … s b arg.
1890 [2,3: cases arg; *; normalize //]
1891 #n #l #H #T #s #pc #b * *; [11: #NH @⊥ //] #X try #Y %
1892qed.
1893
1894lemma set_bit_addressable_sfr_set_code_memory:
1895  ∀T, U: Type[0].
1896  ∀ps: PreStatus ?.
1897  ∀code_mem.
1898  ∀x.
1899  ∀val.
1900  set_bit_addressable_sfr ? (set_code_memory T U ps code_mem) x val =
1901  set_code_memory T U (set_bit_addressable_sfr ? ps x val) code_mem.
1902  #T #U #ps #code_mem #x #val
1903  whd in ⊢ (??%?)
1904  whd in ⊢ (???(???%?))
1905  cases (eqb ? 128) [ normalize nodelta cases not_implemented
1906  | normalize nodelta
1907  cases (eqb (nat_of_bitvector 8 x) ?) [ normalize nodelta cases not_implemented
1908  | normalize nodelta
1909  cases (eqb (nat_of_bitvector 8 x) ?) [ normalize nodelta cases not_implemented
1910  | normalize nodelta
1911  cases (eqb ? 176) [ normalize nodelta %
1912  | normalize nodelta
1913  cases (eqb ? 153) [ normalize nodelta %
1914  | normalize nodelta
1915  cases (eqb ? 138) [ normalize nodelta %
1916  | normalize nodelta
1917  cases (eqb ? 139) [ normalize nodelta %
1918  | normalize nodelta
1919  cases (eqb ? 140) [ normalize nodelta %
1920  | normalize nodelta
1921  cases (eqb ? 141) [ normalize nodelta %
1922  | normalize nodelta
1923  cases (eqb ? 200) [ normalize nodelta %
1924  | normalize nodelta
1925  cases (eqb ? 202) [ normalize nodelta %
1926  | normalize nodelta
1927  cases (eqb ? 203) [ normalize nodelta %
1928  | normalize nodelta
1929  cases (eqb ? 204) [ normalize nodelta %
1930  | normalize nodelta
1931  cases (eqb ? 205) [ normalize nodelta %
1932  | normalize nodelta
1933  cases (eqb ? 135) [ normalize nodelta %
1934  | normalize nodelta
1935  cases (eqb ? 136) [ normalize nodelta %
1936  | normalize nodelta
1937  cases (eqb ? 137) [ normalize nodelta %
1938  | normalize nodelta
1939  cases (eqb ? 152) [ normalize nodelta %
1940  | normalize nodelta
1941  cases (eqb ? 168) [ normalize nodelta %
1942  | normalize nodelta
1943  cases (eqb ? 184) [ normalize nodelta %
1944  | normalize nodelta
1945  cases (eqb ? 129) [ normalize nodelta %
1946  | normalize nodelta
1947  cases (eqb ? 130) [ normalize nodelta %
1948  | normalize nodelta
1949  cases (eqb ? 131) [ normalize nodelta %
1950  | normalize nodelta
1951  cases (eqb ? 208) [ normalize nodelta %
1952  | normalize nodelta
1953  cases (eqb ? 224) [ normalize nodelta %
1954  | normalize nodelta
1955  cases (eqb ? 240) [ normalize nodelta %
1956  | normalize nodelta
1957    cases not_implemented
1958  ]]]]]]]]]]]]]]]]]]]]]]]]]]
1959qed.
1960
1961lemma set_arg_8_set_code_memory:
1962  ∀n:nat.
1963  ∀l:Vector addressing_mode_tag (S n).
1964    ¬(is_in ? l ACC_PC) →
1965    ∀T: Type[0].
1966    ∀U: Type[0].
1967    ∀ps: PreStatus ?.
1968    ∀code_mem.
1969    ∀val.
1970    ∀b: l.
1971  set_arg_8 ? (set_code_memory T U ps code_mem) b val =
1972  set_code_memory T U (set_arg_8 ? ps b val) code_mem.
1973  [2,3: cases b; *; normalize //]
1974  #n #l #prf #T #U #ps #code_mem #val * *;
1975  [*:
1976    [1,2,3,4,8,9,15,16,17,18,19: #x]
1977    try #y whd in ⊢ (??(%)?)
1978    whd in ⊢ (???(???(%)?))
1979    [1,2:
1980      cases (split bool 4 4 ?)
1981      #nu' #nl'
1982      normalize nodelta
1983      cases (split bool 1 3 nu')
1984      #bit1' #ignore'
1985      normalize nodelta
1986      cases (get_index_v bool 4 nu' 1 ?)
1987      [1,2,3,4:
1988        normalize nodelta
1989        try %
1990        try (@(set_bit_addressable_sfr_set_code_memory T U ps code_mem x val))
1991        try (normalize in ⊢ (???(???%?)))
1992      ]
1993    |3,4: %
1994    |*:
1995       try normalize nodelta
1996       normalize cases (not_implemented)
1997    ]
1998 ]
1999qed.
2000
2001axiom set_arg_8_set_program_counter:
2002  ∀n:nat.
2003  ∀l:Vector addressing_mode_tag (S n).
2004    ¬(is_in ? l ACC_PC) →
2005    ∀T: Type[0].
2006    ∀ps: PreStatus ?.
2007    ∀pc.
2008    ∀val.
2009    ∀b: l.
2010  set_arg_8 ? (set_program_counter T ps pc) b val =
2011  set_program_counter T (set_arg_8 ? ps b val) pc.
2012  [1,2: cases b; *; normalize //]
2013qed.
2014 
2015
2016lemma get_arg_8_set_code_memory:
2017 ∀T1,T2,s,code_mem,b,arg.
2018   get_arg_8 T1 (set_code_memory T2 T1 s code_mem) b arg = get_arg_8 … s b arg.
2019 #T1 #T2 #s #code_mem #b #arg %
2020qed.
2021
2022lemma set_code_memory_set_flags:
2023 ∀T1,T2,s,f1,f2,f3,code_mem.
2024  set_code_memory T1 T2 (set_flags T1 s f1 f2 f3) code_mem =
2025   set_flags … (set_code_memory … s code_mem) f1 f2 f3.
2026 #T1 #T2 #s #f1 #f2 #f3 #code_mem %
2027qed.
2028
2029lemma set_program_counter_set_flags:
2030 ∀T1,s,f1,f2,f3,pc.
2031  set_program_counter T1 (set_flags T1 s f1 f2 f3) pc =
2032   set_flags … (set_program_counter … s pc) f1 f2 f3.
2033 #T1 #s #f1 #f2 #f3 #pc  %
2034qed.
2035
2036lemma program_counter_set_flags:
2037 ∀T1,s,f1,f2,f3.
2038  program_counter T1 (set_flags T1 s f1 f2 f3) = program_counter … s.
2039 #T1 #s #f1 #f2 #f3 %
2040qed.
2041
2042lemma eq_rect_Type1_r:
2043  ∀A: Type[1].
2044  ∀a:A.
2045  ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p.
2046  #A #a #P #H #x #p
2047  generalize in match H
2048  generalize in match P
2049  cases p
2050  //
2051qed.
2052
2053lemma split_eq_status:
2054 ∀T.
2055 ∀A1,A2,A3,A4,A5,A6,A7,A8,A9,A10.
2056 ∀B1,B2,B3,B4,B5,B6,B7,B8,B9,B10.
2057  A1=B1 → A2=B2 → A3=B3 → A4=B4 → A5=B5 → A6=B6 → A7=B7 → A8=B8 → A9=B9 → A10=B10 →
2058   mk_PreStatus T A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 =
2059   mk_PreStatus T B1 B2 B3 B4 B5 B6 B7 B8 B9 B10.
2060 //
2061qed.
2062
2063lemma main_thm:
2064 ∀M,M',ps,s,s''.
2065  next_internal_pseudo_address_map M ps = Some … M' →
2066  status_of_pseudo_status M ps = Some … s →
2067  status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps)) ps) = Some … s'' →
2068   ∃n. execute n s = s''.
2069 #M #M' #ps #s #s''
2070 generalize in match (fetch_assembly_pseudo2 (code_memory … ps))
2071 whd in ⊢ (? → ? → ??%? → ??%? → ?)
2072 >execute_1_pseudo_instruction_preserves_code_memory
2073 generalize in match (refl … (assembly (code_memory … ps)))
2074 generalize in match (assembly (code_memory … ps)) in ⊢ (??%? → %) #ASS whd in ⊢ (???% → ?)
2075 cases (build_maps (code_memory … ps))
2076  [ cases (code_memory … ps) #preamble #instr_list whd in ⊢ (???% → ?) #EQ >EQ #H #MAP
2077    #abs @⊥ normalize in abs; destruct (abs) ]
2078 * #labels #costs whd in ⊢ (? → ? → ??%? → ?) generalize in match (refl … (code_memory … ps))
2079 cases (code_memory … ps) in ⊢ (???% → %) #preamble #instr_list #EQ0 normalize nodelta;
2080 generalize in ⊢ (???(match % with [_ ⇒ ? | _ ⇒ ?]) → ?) *; normalize nodelta;
2081  [ #EQ >EQ #_ #_ #abs @⊥ normalize in abs; destruct (abs) ]
2082 * #final_ppc * #final_pc #assembled #EQ >EQ -EQ ASS; normalize nodelta;
2083 #H generalize in match (H ??? (refl …) (refl …)) -H; #H;
2084 #MAP
2085 #H1 generalize in match (option_destruct_Some ??? H1) -H1; #H1 <H1 -H1;
2086 #H2 generalize in match (option_destruct_Some ??? H2) -H2; #H2 <H2 -H2;
2087 change with
2088  (∃n.
2089    execute n
2090     (set_low_internal_ram ?
2091       (set_high_internal_ram ?
2092         (set_program_counter ?
2093           (set_code_memory ?? ps (load_code_memory assembled))
2094          (sigma 〈preamble,instr_list〉 (program_counter ? ps)))
2095        (high_internal_ram_of_pseudo_high_internal_ram M ?))
2096      (low_internal_ram_of_pseudo_low_internal_ram M ?))
2097   = set_low_internal_ram ?
2098      (set_high_internal_ram ?
2099        (set_program_counter ?
2100          (set_code_memory ?? (execute_1_pseudo_instruction ? ps) (load_code_memory assembled))
2101         (sigma ??))
2102       ?)
2103     ?)
2104 whd in match (\snd 〈preamble,instr_list〉) in H;
2105 whd in match (\fst 〈preamble,instr_list〉) in H;
2106 whd in match (\snd 〈final_pc,assembled〉) in H;
2107 whd in match (\snd 〈preamble,instr_list〉) in MAP;
2108 -s s'' labels costs final_ppc final_pc;
2109 letin ps' ≝ (execute_1_pseudo_instruction (ticks_of 〈preamble,instr_list〉) ps)
2110 (* NICE STATEMENT HERE *)
2111 generalize in match (refl … ps') generalize in ⊢ (??%? → ?) normalize nodelta; -ps'; #ps'
2112 #K <K generalize in match K; -K;
2113 (* STATEMENT WITH EQUALITY HERE *)
2114 whd in ⊢ (???(?%?) → ?)
2115 whd in ⊢ (???% → ?) generalize in match (H (program_counter … ps)) -H; >EQ0 normalize nodelta;
2116 cases (fetch_pseudo_instruction instr_list (program_counter … ps)) in MAP ⊢ %
2117 #pi #newppc normalize nodelta; #MAP * #instructions *;
2118 cases pi in MAP; normalize nodelta;
2119  [2,3: (* Comment, Cost *) #ARG #MAP #H1 #H2 #EQ %[1,3:@0]
2120    generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP M';
2121    normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2122    #H2 >(eq_bv_to_eq … H2) >EQ %
2123(*  |6: (* Mov *) #arg1 #arg2
2124       #H1 #H2 #EQ %[@1]
2125       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2126       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2127       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2128       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2129       >H2b >(eq_instruction_to_eq … H2a)
2130       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2131       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?)
2132       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2133       normalize nodelta;
2134       [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2135       #result #flags
2136       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) % *)
2137(*  |4,5: (* Jmp, Call *) #label
2138      whd in ⊢ (???% → ?) cases (jump_expansion ??) normalize nodelta;
2139       [3,6: (* long *) #H1 #H2 #EQ %[1,3:@1]
2140           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2141           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2142           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2143           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2144           >H2b >(eq_instruction_to_eq … H2a)
2145           generalize in match EQ; -EQ;
2146           [2: whd in ⊢ (???% → ??%?);
2147               cases (half_add ???) #carry #new_sp normalize nodelta;
2148               >(eq_bv_to_eq … H2c)
2149               change with
2150                ((?=let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 newppc in ?) →
2151                    (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?)
2152               (* MISMATCH ON WHAT IS ON THE STACK!!!! *)
2153               (* HOW TO PROVE THIS?? *)
2154           ]
2155           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c)
2156           cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX
2157            [2: % | ]
2158       |4: (* short Call *) #abs @⊥ destruct (abs)
2159       |1: (* short Jmp *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2160           generalize in match
2161            (refl ?
2162             (sub_16_with_carry
2163              (sigma 〈preamble,instr_list〉 (program_counter … ps))
2164              (sigma 〈preamble,instr_list〉 (address_of_word_labels_code_mem instr_list label))
2165              false))
2166           cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta;
2167           generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta;
2168           generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta;
2169           #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)]
2170           generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2171           change in ⊢ (? → ??%?) with (execute_1_0 ??)
2172           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2173           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2174           >H2b >(eq_instruction_to_eq … H2a)
2175           generalize in match EQ; -EQ;
2176           whd in ⊢ (???% → ?);
2177           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c)
2178           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ??) ? in ?) = ?)
2179           generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 newppc) (sign_extension lower)))
2180           cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4
2181           @split_eq_status try % change with (newpc = sigma ? (address_of_word_labels ps label))
2182           (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2183       | (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
2184         generalize in match
2185          (refl …
2186            (split … 5 11 (sigma 〈preamble,instr_list〉 (address_of_word_labels_code_mem instr_list label))))
2187         cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1
2188         generalize in match
2189          (refl …
2190            (split … 5 11 (sigma 〈preamble,instr_list〉 (program_counter … ps))))
2191         cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2
2192         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
2193         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
2194         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
2195         change in ⊢ (? →??%?) with (execute_1_0 ??)
2196           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2197           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2198           >H2b >(eq_instruction_to_eq … H2a)
2199           generalize in match EQ; -EQ;
2200           whd in ⊢ (???% → ?);
2201           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) whd in ⊢ (??%?)
2202           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?)
2203           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc)))
2204           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
2205           generalize in match (refl … (split bool 4 4 pc_bu))
2206           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
2207           generalize in match (refl … (split bool 3 8 rest_addr))
2208           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
2209           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
2210           generalize in match
2211            (refl …
2212             (half_add 16 (sigma 〈preamble,instr_list〉 newppc)
2213             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
2214           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7
2215           @split_eq_status try %
2216            [ change with (? = sigma ? (address_of_word_labels ps label))
2217              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
2218            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
2219              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]]
2220  (*|5: (* Call *)*) *)
2221  | (* Instruction *) -pi; *; normalize nodelta;
2222    [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1]
2223       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2224       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2225       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2226       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2227       >H2b >(eq_instruction_to_eq … H2a)
2228       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP;
2229       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1;
2230       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
2231       normalize nodelta; #MAP;
2232       [1: change in ⊢ (? → %) with
2233        ((let 〈result,flags〉 ≝
2234          add_8_with_carry
2235           (get_arg_8 ? ps false ACC_A)
2236           (get_arg_8 ?
2237             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
2238             false (DIRECT ARG2))
2239           ? in ?) = ?)
2240       [1,2: cut (addressing_mode_ok M ps (DIRECT ARG2) = true) [2:
2241        [1,2:whd in MAP:(??(match % with [ _ ⇒ ? | _ ⇒ ?])?)]
2242       [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
2243       #result #flags
2244       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) %
2245    | (* INC *) #arg1 #H1 #H2 #EQ %[@1]
2246       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
2247       change in ⊢ (? → ??%?) with (execute_1_0 ??)
2248       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
2249       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
2250       >H2b >(eq_instruction_to_eq … H2a)
2251       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
2252       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG]
2253       [1,2,3,4: cases (half_add ???) #carry #result
2254       | cases (half_add ???) #carry #bl normalize nodelta;
2255         cases (full_add ????) #carry' #bu normalize nodelta ]
2256        #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) -newppc';
2257        [5: %
2258        |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program
2259      (set_program_counter pseudo_assembly_program ps newppc)
2260      (\fst  (ticks_of0 〈preamble,instr_list〉
2261                   (program_counter pseudo_assembly_program ps)
2262                   (Instruction (INC Identifier (DIRECT ARG))))
2263       +clock pseudo_assembly_program
2264        (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG))
2265        [2,3: // ]
2266            <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://]
2267            whd in ⊢ (??%%)
2268            cases (split bool 4 4 ARG)
2269            #nu' #nl'
2270            normalize nodelta
2271            cases (split bool 1 3 nu')
2272            #bit_1' #ignore'
2273            normalize nodelta
2274            cases (get_index_v bool 4 nu' ? ?)
2275            [ normalize nodelta (* HERE *) whd in ⊢ (??%%) %
2276            |
2277            ]
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