source: src/ASM/AssemblyProof.ma @ 877

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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
51let rec foldl_strong_internal
52  (A: Type[0]) (P: list A → Type[0]) (l: list A)
53  (H: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]))
54  (prefix: list A) (suffix: list A) (acc: P prefix) on suffix:
55    l = prefix @ suffix → P(prefix @ suffix) ≝
56  match suffix return λl'. l = prefix @ l' → P (prefix @ l') with
57  [ nil ⇒ λprf. ?
58  | cons hd tl ⇒ λprf. ?
59  ].
60  [ > (append_nil ?)
61    @ acc
62  | applyS (foldl_strong_internal A P l H (prefix @ [hd]) tl ? ?)
63    [ @ (H prefix hd tl prf acc)
64    | applyS prf
65    ]
66  ]
67qed.
68
69definition foldl_strong ≝
70  λA: Type[0].
71  λP: list A → Type[0].
72  λl: list A.
73  λH: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]).
74  λacc: P [ ].
75    foldl_strong_internal A P l H [ ] l acc (refl …).
76
77definition bit_elim: ∀P: bool → bool. bool ≝
78  λP.
79    P true ∧ P false.
80
81let rec bitvector_elim_internal
82  (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝
83  match m return λm. m ≤ n → BitVector (n - m) → bool with
84  [ O    ⇒ λprf1. λprefix. P ?
85  | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?)
86  ].
87  [ applyS prefix
88  | letin res ≝ (bit ::: prefix)
89    < (minus_S_S ? ?)
90    > (minus_Sn_m ? ?)
91    [ @ res
92    | @ prf2
93    ]
94  | /2/
95  ].
96qed.
97
98definition bitvector_elim ≝
99  λn: nat.
100  λP: BitVector n → bool.
101    bitvector_elim_internal n P n ? ?.
102  [ @ (le_n ?)
103  | < (minus_n_n ?)
104    @ [[ ]]
105  ]
106qed.
107
108axiom vector_associative_append:
109  ∀A: Type[0].
110  ∀n, m, o:  nat.
111  ∀v: Vector A n.
112  ∀q: Vector A m.
113  ∀r: Vector A o.
114    ((v @@ q) @@ r)
115    ≃
116    (v @@ (q @@ r)).
117       
118lemma vector_cons_append:
119  ∀A: Type[0].
120  ∀n: nat.
121  ∀e: A.
122  ∀v: Vector A n.
123    e ::: v = [[ e ]] @@ v.
124  # A # N # E # V
125  elim V
126  [ normalize %
127  | # NN # AA # VV # IH
128    normalize
129    %
130  ]
131qed.
132
133lemma super_rewrite2:
134 ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m.
135  ∀P: ∀m. Vector A m → Prop.
136   n=m → v1 ≃ v2 → P n v1 → P m v2.
137 #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ //
138qed.
139
140lemma mem_middle_vector:
141  ∀A: Type[0].
142  ∀m, o: nat.
143  ∀eq: A → A → bool.
144  ∀reflex: ∀a. eq a a = true.
145  ∀p: Vector A m.
146  ∀a: A.
147  ∀r: Vector A o.
148    mem A eq ? (p@@(a:::r)) a = true.
149  # A # M # O # EQ # REFLEX # P # A
150  elim P
151  [ normalize
152    > (REFLEX A)
153    normalize
154    # H
155    %
156  | # NN # AA # PP # IH
157    normalize
158    cases (EQ A AA) //
159     @ IH
160  ]
161qed.
162
163lemma mem_monotonic_wrt_append:
164  ∀A: Type[0].
165  ∀m, o: nat.
166  ∀eq: A → A → bool.
167  ∀reflex: ∀a. eq a a = true.
168  ∀p: Vector A m.
169  ∀a: A.
170  ∀r: Vector A o.
171    mem A eq ? r a = true → mem A eq ? (p @@ r) a = true.
172  # A # M # O # EQ # REFLEX # P # A
173  elim P
174  [ #R #H @H
175  | #NN #AA # PP # IH #R #H
176    normalize
177    cases (EQ A AA)
178    [ normalize %
179    | @ IH @ H
180    ]
181  ]
182qed.
183
184lemma subvector_multiple_append:
185  ∀A: Type[0].
186  ∀o, n: nat.
187  ∀eq: A → A → bool.
188  ∀refl: ∀a. eq a a = true.
189  ∀h: Vector A o.
190  ∀v: Vector A n.
191  ∀m: nat.
192  ∀q: Vector A m.
193    bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)).
194  # A # O # N # EQ # REFLEX # H # V
195  elim V
196  [ normalize
197    # M # V %
198  | # NN # AA # VV # IH # MM # QQ
199    change with (bool_to_Prop (andb ??))
200    cut ((mem A EQ (O + (MM + S NN)) (H@@QQ@@AA:::VV) AA) = true)
201    [
202    | # HH > HH
203      > (vector_cons_append ? ? AA VV)
204      change with (bool_to_Prop (subvector_with ??????))
205      @(super_rewrite2 A ((MM + 1)+ NN) (MM+S NN) ??
206        (λSS.λVS.bool_to_Prop (subvector_with ?? (O+SS) ?? (H@@VS)))
207        ?
208        (vector_associative_append A ? ? ? QQ [[AA]] VV))
209      [ >associative_plus //
210      | @IH ]
211    ]
212    @(mem_monotonic_wrt_append)
213    [ @ REFLEX
214    | @(mem_monotonic_wrt_append)
215      [ @ REFLEX
216      | normalize
217        > REFLEX
218        normalize
219        %
220      ]
221    ]
222qed.
223
224lemma vector_cons_empty:
225  ∀A: Type[0].
226  ∀n: nat.
227  ∀v: Vector A n.
228    [[ ]] @@ v = v.
229  # A # N # V
230  elim V
231  [ normalize %
232  | # NN # HH # VV #H %
233  ]
234qed.
235
236corollary subvector_hd_tl:
237  ∀A: Type[0].
238  ∀o: nat.
239  ∀eq: A → A → bool.
240  ∀refl: ∀a. eq a a = true.
241  ∀h: A.
242  ∀v: Vector A o.
243    bool_to_Prop (subvector_with A ? ? eq v (h ::: v)).
244  # A # O # EQ # REFLEX # H # V
245  > (vector_cons_append A ? H V)
246  < (vector_cons_empty A ? ([[H]] @@ V))
247  @ (subvector_multiple_append A ? ? EQ REFLEX [[]] V ? [[ H ]])
248qed.
249
250lemma eq_a_reflexive:
251  ∀a. eq_a a a = true.
252  # A
253  cases A
254  %
255qed.
256
257lemma is_in_monotonic_wrt_append:
258  ∀m, n: nat.
259  ∀p: Vector addressing_mode_tag m.
260  ∀q: Vector addressing_mode_tag n.
261  ∀to_search: addressing_mode.
262    bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search).
263  # M # N # P # Q # TO_SEARCH
264  # H
265  elim Q
266  [ normalize
267    @ H
268  | # NN # PP # QQ # IH
269    normalize
270    cases (is_a PP TO_SEARCH)
271    [ normalize
272      %
273    | normalize
274      normalize in IH
275      @ IH
276    ]
277  ]
278qed.
279
280corollary is_in_hd_tl:
281  ∀to_search: addressing_mode.
282  ∀hd: addressing_mode_tag.
283  ∀n: nat.
284  ∀v: Vector addressing_mode_tag n.
285    bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search).
286  # TO_SEARCH # HD # N # V
287  elim V
288  [ # H
289    normalize in H;
290    cases H
291  | # NN # HHD # VV # IH # HH
292    > vector_cons_append
293    > (vector_cons_append ? ? HHD VV)
294    @ (is_in_monotonic_wrt_append ? 1 ([[HHD]]@@VV) [[HD]] TO_SEARCH)
295    @ HH
296  ]
297qed.
298 
299let rec list_addressing_mode_tags_elim
300  (n: nat) (l: Vector addressing_mode_tag (S n)) on l: (l → bool) → bool ≝
301  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
302   (l → bool) → bool ] with
303  [ VEmpty      ⇒  true 
304  | VCons len hd tl ⇒ λP.
305    let process_hd ≝
306      match hd return λhd. ∀P: hd:::tl → bool. bool with
307      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
308      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
309      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
310      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
311      | acc_a ⇒ λP.P ACC_A
312      | acc_b ⇒ λP.P ACC_B
313      | dptr ⇒ λP.P DPTR
314      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
315      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
316      | acc_dptr ⇒ λP.P ACC_DPTR
317      | acc_pc ⇒ λP.P ACC_PC
318      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
319      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
320      | carry ⇒ λP.P CARRY
321      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
322      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
323      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
324      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
325      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
326      ]
327    in
328      andb (process_hd P)
329       (match len return λx. x = len → bool with
330         [ O ⇒ λprf. true
331         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
332  ].
333  try %
334  [ 2: cases (sym_eq ??? prf); @tl
335  | generalize in match H; generalize in match tl; cases prf;
336    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
337    #tl
338    normalize in ⊢ (∀_: %. ?)
339    # H
340    whd
341    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
342    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
343qed.
344
345definition product_elim ≝
346  λm, n: nat.
347  λv: Vector addressing_mode_tag (S m).
348  λq: Vector addressing_mode_tag (S n).
349  λP: (v × q) → bool.
350    list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)).
351
352definition union_elim ≝
353  λA, B: Type[0].
354  λelimA: (A → bool) → bool.
355  λelimB: (B → bool) → bool.
356  λelimU: A ⊎ B → bool.
357    elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))).
358                           
359definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝
360  λP.
361    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧
362    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧
363    list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧
364    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧
365    list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧
366    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧
367    list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧
368    list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
369    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧
370    list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧
371    P (DA ? ACC_A) ∧
372    bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧
373    bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧
374    bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧
375    bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧
376    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧
377    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧
378    bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧
379    list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧
380    P (RL ? ACC_A) ∧
381    P (RLC ? ACC_A) ∧
382    P (RR ? ACC_A) ∧
383    P (RRC ? ACC_A) ∧
384    P (SWAP ? ACC_A) ∧
385    P (RET ?) ∧
386    P (RETI ?) ∧
387    P (NOP ?) ∧
388    bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧
389    list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧
390    bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧
391    bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧
392    union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]])
393                   (product_elim ? ? [[ registr; indirect ]] [[ data ]])
394                   (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧
395    list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧
396    union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]])
397                   (product_elim ? ? [[direct]] [[ acc_a ; data ]])
398                   (λd. P (XRL ? d)) ∧
399    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
400                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
401                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
402                   (λd. P (ANL ? d)) ∧
403    union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]])
404                                   (product_elim ? ? [[direct]] [[ acc_a ; data ]]))
405                   (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]])
406                   (λd. P (ORL ? d)) ∧
407    union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]])
408                   (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]])
409                   (λd. P (MOVX ? d)) ∧
410    union_elim ? ? (
411      union_elim ? ? (
412        union_elim ? ? (
413          union_elim ? ? (
414            union_elim ? ?  (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]])
415                            (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]]))
416                            (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]]))
417                            (product_elim ? ? [[dptr]] [[data16]]))
418                            (product_elim ? ? [[carry]] [[bit_addr]]))
419                            (product_elim ? ? [[bit_addr]] [[carry]])
420                            (λd. P (MOV ? d)).
421  %
422qed.
423 
424definition instruction_elim: ∀P: instruction → bool. bool ≝
425  λP. (*
426    bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧
427    bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧
428    bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧
429    bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *)
430    bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (*  ∧
431    P (JMP INDIRECT_DPTR) ∧
432    list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧
433    preinstruction_elim (λp. P (RealInstruction p)). *)
434  %
435qed.
436
437
438axiom instruction_elim_complete:
439 ∀P. instruction_elim P = true → ∀i. P i = true.
440
441definition eq_instruction ≝
442  λi, j: instruction.
443    true.
444   
445let rec vect_member
446  (A: Type[0]) (n: nat) (eq: A → A → bool)
447  (v: Vector A n) (a: A) on v: bool ≝
448  match v with
449  [ VEmpty          ⇒ false
450  | VCons len hd tl ⇒
451    eq hd a ∨ (vect_member A ? eq tl a)
452  ].
453   
454let rec list_addressing_mode_tags_elim_prop
455  (n: nat)
456  (l: Vector addressing_mode_tag (S n))
457  (P: addressing_mode → Prop) on l:
458  ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a.
459  ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a.
460  ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a.
461  ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a.
462  ∀x: l. P x ≝
463  match l return
464    λy.
465      match y with
466      [ O    ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True
467      | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.
468               ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True.
469               ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True.
470               ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True.
471               ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True.
472               ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True.
473               ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True.
474               ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True.
475               ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True.
476               ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True.
477               ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True.
478               ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True.
479               ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True.
480               ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True.
481               ∀carry_a: if vect_member … eq_a l carry then P CARRY else True.
482               ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True.
483               ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True.
484               ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True.
485               ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True.
486               ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True.
487               ∀x:l. P x
488      ] with
489  [ VEmpty          ⇒ λAbsurd. ⊥
490  | VCons len hd tl ⇒ λProof. ?
491  ] (refl ? (S n)).
492  [ destruct(Absurd)
493  | # A1 # A2 # A3 # A4 # A5 # A6 # A7
494    # A8 # A9 # A10 # A11 # A12 # A13 # A14
495    # A15 # A16 # A17 # A18 # A19 # X
496    cases X
497    # SUB
498    cases SUB
499    [ # BYTE
500    normalize
501  ].
502 
503 
504(*    let prepare_hd ≝
505      match hd with
506      [ direct ⇒ λdirect_prf. ?
507      | indirect ⇒ λindirect_prf. ?
508      | ext_indirect ⇒ λext_indirect_prf. ?
509      | registr ⇒ λregistr_prf. ?
510      | acc_a ⇒ λacc_a_prf. ?
511      | acc_b ⇒ λacc_b_prf. ?
512      | dptr ⇒ λdptr_prf. ?
513      | data ⇒ λdata_prf. ?
514      | data16 ⇒ λdata16_prf. ?
515      | acc_dptr ⇒ λacc_dptr_prf. ?
516      | acc_pc ⇒ λacc_pc_prf. ?
517      | ext_indirect_dptr ⇒ λext_indirect_prf. ?
518      | indirect_dptr ⇒ λindirect_prf. ?
519      | carry ⇒ λcarry_prf. ?
520      | bit_addr ⇒ λbit_addr_prf. ?
521      | n_bit_addr ⇒ λn_bit_addr_prf. ?
522      | relative ⇒ λrelative_prf. ?
523      | addr11 ⇒ λaddr11_prf. ?
524      | addr16 ⇒ λaddr16_prf. ?
525      ]
526    in ? *)
527  ].
528  [ 1: destruct(absd)
529  | 2: # A1 # A2 # A3 # A4 # A5 # A6
530       # A7 # A8 # A9 # A10 # A11 # A12
531       # A13 # A14 # A15 # A16 # A17 # A18
532       # A19 *
533  ].
534
535
536  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
537   (l → bool) → bool ] with
538  [ VEmpty      ⇒  true 
539  | VCons len hd tl ⇒ λP.
540    let process_hd ≝
541      match hd return λhd. ∀P: hd:::tl → bool. bool with
542      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
543      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
544      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
545      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
546      | acc_a ⇒ λP.P ACC_A
547      | acc_b ⇒ λP.P ACC_B
548      | dptr ⇒ λP.P DPTR
549      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
550      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
551      | acc_dptr ⇒ λP.P ACC_DPTR
552      | acc_pc ⇒ λP.P ACC_PC
553      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
554      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
555      | carry ⇒ λP.P CARRY
556      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
557      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
558      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
559      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
560      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
561      ]
562    in
563      andb (process_hd P)
564       (match len return λx. x = len → bool with
565         [ O ⇒ λprf. true
566         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
567  ].
568  try %
569  [ 2: cases (sym_eq ??? prf); @tl
570  | generalize in match H; generalize in match tl; cases prf;
571    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
572    #tl
573    normalize in ⊢ (∀_: %. ?)
574    # H
575    whd
576    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
577    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
578qed.
579
580(*
581lemma test:
582  let i ≝ SJMP (RELATIVE (bitvector_of_nat 8 255)) in
583      (let assembled ≝ assembly1 i in
584      let code_memory ≝ load_code_memory assembled in
585      let fetched ≝ fetch code_memory ? in
586      let 〈instr_pc, ticks〉 ≝ fetched in
587        eq_instruction (\fst instr_pc)) i = true.
588 [2: @ zero
589 | normalize
590 ]*)
591
592lemma test:
593  ∀i.
594      (let assembled ≝ assembly1 i in
595      let code_memory ≝ load_code_memory assembled in
596      let fetched ≝ fetch code_memory ? in
597      let 〈instr_pc, ticks〉 ≝ fetched in
598        eq_instruction (\fst instr_pc)) i = true.
599 [ #i cases i #arg try #arg2 whd in ⊢ (??%?)
600   [2: whd in ⊢ (??(match ? (? %) ?with [ _ ⇒ ?] ?)?)
601       cases arg #sam cases sam #XX try #PP normalize in PP; try cases PP;
602       whd in ⊢ (??(match ? (? %) ? with [ _ ⇒ ?] ?)?)
603 
604   [2: #addr whd in ⊢ (??%?)
605 
606   @ (instruction_elim_complete )
607 | @ zero
608 ]
609 normalize
610   
611 
612(* This establishes the correspondence between pseudo program counters and
613   program counters. It is at the heart of the proof. *)
614(*CSC: code taken from build_maps *)
615definition sigma0: pseudo_assembly_program → option (nat × (nat × (BitVectorTrie Word 16))) ≝
616 λinstr_list.
617  foldl ??
618    (λt. λi.
619       match t with
620       [ None ⇒ None ?
621       | Some ppc_pc_map ⇒
622         let 〈ppc,pc_map〉 ≝ ppc_pc_map in
623         let 〈program_counter, sigma_map〉 ≝ pc_map in
624         let 〈label, i〉 ≝ i in
625          match construct_costs instr_list program_counter (λx. zero ?) (λx. zero ?) (Stub …) i with
626           [ None ⇒ None ?
627           | Some pc_ignore ⇒
628              let 〈pc,ignore〉 ≝ pc_ignore in
629              Some … 〈S ppc,〈pc, insert ? ? (bitvector_of_nat ? ppc) (bitvector_of_nat ? pc) sigma_map〉〉 ]
630       ]) (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (\snd instr_list).
631       
632definition tech_pc_sigma0: pseudo_assembly_program → option (nat × (BitVectorTrie Word 16)) ≝
633 λinstr_list.
634  match sigma0 instr_list with
635   [ None ⇒ None …
636   | Some result ⇒
637      let 〈ppc,pc_sigma_map〉 ≝ result in
638       Some … pc_sigma_map ].
639
640definition sigma_safe: pseudo_assembly_program → option (Word → Word) ≝       
641 λinstr_list.
642  match sigma0 instr_list with
643  [ None ⇒ None ?
644  | Some result ⇒
645    let 〈ppc,pc_sigma_map〉 ≝ result in
646    let 〈pc, sigma_map〉 ≝ pc_sigma_map in
647      if gtb pc (2^16) then
648        None ?
649      else
650        Some ? (λx.lookup ?? x sigma_map (zero …)) ].
651
652axiom policy_ok: ∀p. sigma_safe p ≠ None ….
653
654definition sigma: pseudo_assembly_program → Word → Word ≝
655 λp.
656  match sigma_safe p return λr:option (Word → Word). r ≠ None … → Word → Word with
657   [ None ⇒ λabs. ⊥
658   | Some r ⇒ λ_.r] (policy_ok p).
659 cases abs //
660qed.
661
662lemma length_append:
663 ∀A.∀l1,l2:list A.
664  |l1 @ l2| = |l1| + |l2|.
665 #A #l1 elim l1
666  [ //
667  | #hd #tl #IH #l2 normalize <IH //]
668qed.
669
670let rec does_not_occur (id:Identifier) (l:list labelled_instruction) on l: bool ≝
671 match l with
672  [ nil ⇒ true
673  | cons hd tl ⇒ notb (instruction_matches_identifier id hd) ∧ does_not_occur id tl].
674
675lemma does_not_occur_None:
676 ∀id,i,list_instr.
677  does_not_occur id (list_instr@[〈None …,i〉]) =
678  does_not_occur id list_instr.
679 #id #i #list_instr elim list_instr
680  [ % | #hd #tl #IH whd in ⊢ (??%%) >IH %]
681qed.
682
683let rec occurs_exactly_once (id:Identifier) (l:list labelled_instruction) on l : bool ≝
684 match l with
685  [ nil ⇒ false
686  | cons hd tl ⇒
687     if instruction_matches_identifier id hd then
688      does_not_occur id tl
689     else
690      occurs_exactly_once id tl ].
691
692lemma occurs_exactly_once_None:
693 ∀id,i,list_instr.
694  occurs_exactly_once id (list_instr@[〈None …,i〉]) =
695  occurs_exactly_once id list_instr.
696 #id #i #list_instr elim list_instr
697  [ % | #hd #tl #IH whd in ⊢ (??%%) >IH >does_not_occur_None %]
698qed.
699
700coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0].
701
702lemma index_of_internal_None: ∀i,id,instr_list,n.
703 occurs_exactly_once id (instr_list@[〈None …,i〉]) →
704  index_of_internal ? (instruction_matches_identifier id) instr_list n =
705   index_of_internal ? (instruction_matches_identifier id) (instr_list@[〈None …,i〉]) n.
706 #i #id #instr_list elim instr_list
707  [ #n #abs whd in abs; cases abs
708  | #hd #tl #IH #n whd in ⊢ (% → ??%%); whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ?)
709    cases (instruction_matches_identifier id hd) whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ??%%)
710    [ #H %
711    | #H @IH whd in H; cases (occurs_exactly_once ??) in H ⊢ %
712      [ #_ % | #abs cases abs ]]]
713qed.
714
715lemma address_of_word_labels_code_mem_None: ∀i,id,instr_list.
716 occurs_exactly_once id (instr_list@[〈None …,i〉]) →
717  address_of_word_labels_code_mem instr_list id =
718  address_of_word_labels_code_mem (instr_list@[〈None …,i〉]) id.
719 #i #id #instr_list #H whd in ⊢ (??%%) whd in ⊢ (??(??%?)(??%?))
720 >(index_of_internal_None … H) %
721qed.
722
723axiom tech_pc_sigma0_append:
724 ∀preamble,instr_list,prefix,label,i,pc',code,pc,costs,costs'.
725  Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 →
726   construct_costs 〈preamble,instr_list〉 … pc (λx.zero 16) (λx. zero 16) costs i = Some … 〈pc',code〉 →
727    tech_pc_sigma0 〈preamble,prefix@[〈label,i〉]〉 = Some … 〈pc',costs'〉.
728
729axiom tech_pc_sigma0_append_None:
730 ∀preamble,instr_list,prefix,i,pc,costs.
731  Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 →
732   construct_costs 〈preamble,instr_list〉 … pc (λx.zero 16) (λx. zero 16) costs i = None …
733    → False.
734
735lemma BitVectorTrie_O:
736 ∀A:Type[0].∀v:BitVectorTrie A 0.(∃w. v ≃ Leaf A w) ∨ v ≃ Stub A 0.
737 #A #v generalize in match (refl … O) cases v in ⊢ (??%? → (?(??(λ_.?%%??)))(?%%??))
738  [ #w #_ %1 %[@w] %
739  | #n #l #r #abs @⊥ //
740  | #n #EQ %2 >EQ %]
741qed.
742
743lemma BitVectorTrie_Sn:
744 ∀A:Type[0].∀n.∀v:BitVectorTrie A (S n).(∃l,r. v ≃ Node A n l r) ∨ v ≃ Stub A (S n).
745 #A #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → (?(??(λ_.??(λ_.?%%??))))%)
746  [ #m #abs @⊥ //
747  | #m #l #r #EQ %1 <(injective_S … EQ) %[@l] %[@r] //
748  | #m #EQ %2 // ]
749qed.
750
751lemma lookup_prepare_trie_for_insertion_hit:
752 ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.
753  lookup … b (prepare_trie_for_insertion … b v) a = v.
754 #A #a #v #n #b elim b // #m #hd #tl #IH cases hd normalize //
755qed.
756 
757lemma lookup_insert_hit:
758 ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.∀t:BitVectorTrie A n.
759  lookup … b (insert … b v t) a = v.
760 #A #a #v #n #b elim b -b -n //
761 #n #hd #tl #IH #t cases(BitVectorTrie_Sn … t)
762  [ * #l * #r #JMEQ >JMEQ cases hd normalize //
763  | #JMEQ >JMEQ cases hd normalize @lookup_prepare_trie_for_insertion_hit ]
764qed.
765
766lemma BitVector_O: ∀v:BitVector 0. v ≃ VEmpty bool.
767 #v generalize in match (refl … 0) cases v in ⊢ (??%? → ?%%??) //
768 #n #hd #tl #abs @⊥ //
769qed.
770
771lemma BitVector_Sn: ∀n.∀v:BitVector (S n).
772 ∃hd.∃tl.v ≃ VCons bool n hd tl.
773 #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??)))
774 [ #abs @⊥ //
775 | #m #hd #tl #EQ <(injective_S … EQ) %[@hd] %[@tl] // ]
776qed.
777
778lemma lookup_prepare_trie_for_insertion_miss:
779 ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.
780  (notb (eq_bv ? b c)) → lookup … b (prepare_trie_for_insertion … c v) a = a.
781 #A #a #v #n #c elim c
782  [ #b >(BitVector_O … b) normalize #abs @⊥ //
783  | #m #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ
784    cases hd cases hd' normalize
785    [2,3: #_ cases tl' //
786    |*: change with (bool_to_Prop (notb (eq_bv ???)) → ?) /2/ ]]
787qed.
788 
789lemma lookup_insert_miss:
790 ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.∀t:BitVectorTrie A n.
791  (notb (eq_bv ? b c)) → lookup … b (insert … c v t) a = lookup … b t a.
792 #A #a #v #n #c elim c -c -n
793  [ #b #t #DIFF @⊥ whd in DIFF; >(BitVector_O … b) in DIFF //
794  | #n #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ
795    #t cases(BitVectorTrie_Sn … t)
796    [ * #l * #r #JMEQ >JMEQ cases hd cases hd' #H normalize in H;
797     [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize // @IH //
798    | #JMEQ >JMEQ cases hd cases hd' #H normalize in H;
799     [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize
800     [3,4: cases tl' // | *: @lookup_prepare_trie_for_insertion_miss //]]]
801qed.
802
803definition build_maps' ≝
804  λpseudo_program.
805  let 〈preamble,instr_list〉 ≝ pseudo_program in
806  let result ≝
807   foldl_strong
808    (option Identifier × pseudo_instruction)
809    (λpre. Σres:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
810      let pre' ≝ 〈preamble,pre〉 in
811      let 〈labels,pc_costs〉 ≝ res in
812       tech_pc_sigma0 pre' = Some … pc_costs ∧
813       ∀id. occurs_exactly_once id pre →
814        lookup ?? id labels (zero …) = sigma pre' (address_of_word_labels_code_mem pre id))
815    instr_list
816    (λprefix,i,tl,prf,t.
817      let 〈labels, pc_costs〉 ≝ t in
818      let 〈program_counter, costs〉 ≝ pc_costs in
819       let 〈label, i'〉 ≝ i in
820       let labels ≝
821         match label with
822         [ None ⇒ labels
823         | Some label ⇒
824           let program_counter_bv ≝ bitvector_of_nat ? program_counter in
825             insert ? ? label program_counter_bv labels
826         ]
827       in
828         match construct_costs 〈preamble,instr_list〉 program_counter (λx. zero ?) (λx. zero ?) costs i' with
829         [ None ⇒
830            let dummy ≝ 〈labels,pc_costs〉 in
831             dummy
832         | Some construct ⇒ 〈labels, construct〉
833         ]
834    ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉
835  in
836   let 〈labels, pc_costs〉 ≝ result in
837   let 〈pc, costs〉 ≝ pc_costs in
838    〈labels, costs〉.
839 [3: whd % // #id normalize in ⊢ (% → ?) #abs @⊥ //
840 | whd cases construct in p3 #PC #CODE #JMEQ %
841    [ @(tech_pc_sigma0_append ??????????? (jmeq_to_eq ??? JMEQ)) | #id #Hid ]
842 | (* dummy case *) @⊥
843   @(tech_pc_sigma0_append_None ?? prefix ???? (jmeq_to_eq ??? p3)) ]
844 [*: generalize in match (sig2 … t) whd in ⊢ (% → ?)
845     >p whd in ⊢ (% → ?) >p1 * #IH0 #IH1 >IH0 // ]
846 whd in ⊢ (??(????%?)?) -labels1;
847 cases label in Hid
848  [ #Hid whd in ⊢ (??(????%?)?) >IH1 -IH1
849     [ >(address_of_word_labels_code_mem_None … Hid)
850       (* MANCA LEMMA: INDIRIZZO TROVATO NEL PROGRAMMA! *)
851     | whd in Hid >occurs_exactly_once_None in Hid // ]
852  | -label #label #Hid whd in ⊢ (??(????%?)?)
853   
854  ]
855qed.
856
857(*
858(*
859notation < "hvbox('let' 〈ident x,ident y〉 ≝ t 'in' s)"
860 with precedence 10
861for @{ match $t with [ pair ${ident x} ${ident y} ⇒ $s ] }.
862*)
863
864lemma build_maps_ok:
865 ∀p:pseudo_assembly_program.
866  let 〈labels,costs〉 ≝ build_maps' p in
867   ∀pc.
868    (nat_of_bitvector … pc) < length … (\snd p) →
869     lookup ?? pc labels (zero …) = sigma p (\snd (fetch_pseudo_instruction (\snd p) pc)).
870 #p cases p #preamble #instr_list
871  elim instr_list
872   [ whd #pc #abs normalize in abs; cases (not_le_Sn_O ?) [#H cases (H abs) ]
873   | #hd #tl #IH
874    whd in ⊢ (match % with [ _ ⇒ ?])
875   ]
876qed.
877*)
878
879(*
880lemma list_elim_rev:
881 ∀A:Type[0].∀P:list A → Prop.
882  P [ ] → (∀n,l. length l = n → P l → 
883  P [ ] → (∀l,a. P l → P (l@[a])) →
884   ∀l. P l.
885 #A #P
886qed.*)
887
888lemma rev_preserves_length:
889 ∀A.∀l. length … (rev A l) = length … l.
890  #A #l elim l
891   [ %
892   | #hd #tl #IH normalize >length_append normalize /2/ ]
893qed.
894
895lemma rev_append:
896 ∀A.∀l1,l2.
897  rev A (l1@l2) = rev A l2 @ rev A l1.
898 #A #l1 elim l1 normalize //
899qed.
900 
901lemma rev_rev: ∀A.∀l. rev … (rev A l) = l.
902 #A #l elim l
903  [ //
904  | #hd #tl #IH normalize >rev_append normalize // ]
905qed.
906
907lemma split_len_Sn:
908 ∀A:Type[0].∀l:list A.∀len.
909  length … l = S len →
910   Σl'.Σa. l = l'@[a] ∧ length … l' = len.
911 #A #l elim l
912  [ normalize #len #abs destruct
913  | #hd #tl #IH #len
914    generalize in match (rev_rev … tl)
915    cases (rev A tl) in ⊢ (??%? → ?)
916     [ #H <H normalize #EQ % [@[ ]] % [@hd] normalize /2/ 
917     | #a #l' #H <H normalize #EQ
918      %[@(hd::rev … l')] %[@a] % //
919      >length_append in EQ #EQ normalize in EQ; normalize;
920      generalize in match (injective_S … EQ) #EQ2 /2/ ]]
921qed.
922
923lemma list_elim_rev:
924 ∀A:Type[0].∀P:list A → Type[0].
925  P [ ] → (∀l,a. P l → P (l@[a])) →
926   ∀l. P l.
927 #A #P #H1 #H2 #l
928 generalize in match (refl … (length … l))
929 generalize in ⊢ (???% → ?) #n generalize in match l
930 elim n
931  [ #L cases L [ // | #x #w #abs (normalize in abs) @⊥ // ]
932  | #m #IH #L #EQ
933    cases (split_len_Sn … EQ) #l' * #a * /3/ ]
934qed.
935
936axiom is_prefix: ∀A:Type[0]. list A → list A → Prop.
937axiom prefix_of_append:
938 ∀A:Type[0].∀l,l1,l2:list A.
939  is_prefix … l l1 → is_prefix … l (l1@l2).
940axiom prefix_reflexive: ∀A,l. is_prefix A l l.
941axiom nil_prefix: ∀A,l. is_prefix A [ ] l.
942
943record Propify (A:Type[0]) : Type[0] (*Prop*) ≝ { in_propify: A }.
944
945definition Propify_elim: ∀A. ∀P:Prop. (A → P) → (Propify A → P) ≝
946 λA,P,H,x. match x with [ mk_Propify p ⇒ H p ].
947
948definition app ≝
949 λA:Type[0].λl1:Propify (list A).λl2:list A.
950  match l1 with
951   [ mk_Propify l1 ⇒ mk_Propify … (l1@l2) ].
952
953lemma app_nil: ∀A,l1. app A l1 [ ] = l1.
954 #A * /3/
955qed.
956
957lemma app_assoc: ∀A,l1,l2,l3. app A (app A l1 l2) l3 = app A l1 (l2@l3).
958 #A * #l1 normalize //
959qed.
960
961let rec foldli (A: Type[0]) (B: Propify (list A) → Type[0])
962 (f: ∀prefix. B prefix → ∀x.B (app … prefix [x]))
963 (prefix: Propify (list A)) (b: B prefix) (l: list A) on l :
964 B (app … prefix l) ≝
965  match l with
966  [ nil ⇒ ? (* b *)
967  | cons hd tl ⇒ ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*)
968  ].
969 [ applyS b
970 | <(app_assoc ?? [hd]) @(foldli A B f (app … prefix [hd]) (f prefix b hd) tl) ]
971qed.
972
973(*
974let rec foldli (A: Type[0]) (B: list A → Type[0]) (f: ∀prefix. B prefix → ∀x. B (prefix@[x]))
975 (prefix: list A) (b: B prefix) (l: list A) on l : B (prefix@l) ≝
976  match l with
977  [ nil ⇒ ? (* b *)
978  | cons hd tl ⇒
979     ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*)
980  ].
981 [ applyS b
982 | applyS (foldli A B f (prefix@[hd]) (f prefix b hd) tl) ]
983qed.
984*)
985
986definition foldll:
987 ∀A:Type[0].∀B: Propify (list A) → Type[0].
988  (∀prefix. B prefix → ∀x. B (app … prefix [x])) →
989   B (mk_Propify … []) → ∀l: list A. B (mk_Propify … l)
990 ≝ λA,B,f. foldli A B f (mk_Propify … [ ]).
991
992axiom is_pprefix: ∀A:Type[0]. Propify (list A) → list A → Prop.
993axiom pprefix_of_append:
994 ∀A:Type[0].∀l,l1,l2.
995  is_pprefix A l l1 → is_pprefix A l (l1@l2).
996axiom pprefix_reflexive: ∀A,l. is_pprefix A (mk_Propify … l) l.
997axiom nil_pprefix: ∀A,l. is_pprefix A (mk_Propify … [ ]) l.
998
999
1000axiom foldll':
1001 ∀A:Type[0].∀l: list A.
1002  ∀B: ∀prefix:Propify (list A). is_pprefix ? prefix l → Type[0].
1003  (∀prefix,proof. B prefix proof → ∀x,proof'. B (app … prefix [x]) proof') →
1004   B (mk_Propify … [ ]) (nil_pprefix …) → B (mk_Propify … l) (pprefix_reflexive … l).
1005 #A #l #B
1006 generalize in match (foldll A (λprefix. is_pprefix ? prefix l)) #HH
1007 
1008 
1009  #H #acc
1010 @foldll
1011  [
1012  |
1013  ]
1014 
1015 ≝ λA,B,f. foldli A B f (mk_Propify … [ ]).
1016
1017
1018(*
1019record subset (A:Type[0]) (P: A → Prop): Type[0] ≝
1020 { subset_wit:> A;
1021   subset_proof: P subset_wit
1022 }.
1023*)
1024
1025definition build_maps' ≝
1026  λpseudo_program.
1027  let 〈preamble,instr_list〉 ≝ pseudo_program in
1028  let result ≝
1029   foldll
1030    (option Identifier × pseudo_instruction)
1031    (λprefix.
1032      Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1033       match prefix return λ_.Prop with [mk_Propify prefix ⇒ tech_pc_sigma0 〈preamble,prefix〉 ≠ None ?])
1034    (λprefix,t,i.
1035      let 〈labels, pc_costs〉 ≝ t in
1036      let 〈program_counter, costs〉 ≝ pc_costs in
1037       let 〈label, i'〉 ≝ i in
1038       let labels ≝
1039         match label with
1040         [ None ⇒ labels
1041         | Some label ⇒
1042           let program_counter_bv ≝ bitvector_of_nat ? program_counter in
1043             insert ? ? label program_counter_bv labels
1044         ]
1045       in
1046         match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with
1047         [ None ⇒
1048            let dummy ≝ 〈labels,pc_costs〉 in
1049              dummy
1050         | Some construct ⇒ 〈labels, construct〉
1051         ]
1052    ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 instr_list
1053  in
1054   let 〈labels, pc_costs〉 ≝ result in
1055   let 〈pc, costs〉 ≝ pc_costs in
1056    〈labels, costs〉.
1057 [
1058 | @⊥
1059 | normalize % //
1060 ]
1061qed.
1062
1063definition build_maps' ≝
1064  λpseudo_program.
1065  let 〈preamble,instr_list〉 ≝ pseudo_program in
1066  let result ≝
1067   foldl
1068    (Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1069          ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧
1070           tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)))
1071    (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix.
1072          let instr_list_prefix' ≝ instr_list_prefix @ [i] in
1073           is_prefix ? instr_list_prefix' instr_list →
1074           tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?)
1075    (λt: Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1076          ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧
1077           tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)).
1078     λi: Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix.
1079          let instr_list_prefix' ≝ instr_list_prefix @ [i] in
1080           is_prefix ? instr_list_prefix' instr_list →
1081           tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ? .
1082      let 〈labels, pc_costs〉 ≝ t in
1083      let 〈program_counter, costs〉 ≝ pc_costs in
1084       let 〈label, i'〉 ≝ i in
1085       let labels ≝
1086         match label with
1087         [ None ⇒ labels
1088         | Some label ⇒
1089           let program_counter_bv ≝ bitvector_of_nat ? program_counter in
1090             insert ? ? label program_counter_bv labels
1091         ]
1092       in
1093         match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with
1094         [ None ⇒
1095            let dummy ≝ 〈labels,pc_costs〉 in
1096              dummy
1097         | Some construct ⇒ 〈labels, construct〉
1098         ]
1099    ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 ?(*instr_list*)
1100  in
1101   let 〈labels, pc_costs〉 ≝ result in
1102   let 〈pc, costs〉 ≝ pc_costs in
1103    〈labels, costs〉.
1104 [4: @(list_elim_rev ?
1105       (λinstr_list. list (
1106        (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix.
1107          let instr_list_prefix' ≝ instr_list_prefix @ [i] in
1108           is_prefix ? instr_list_prefix' instr_list →
1109           tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?)))
1110       ?? instr_list) (* CSC: BAD ORDER FOR CODE EXTRACTION *)
1111      [ @[ ]
1112      | #l' #a #limage %2
1113        [ %[@a] #PREFIX #PREFIX_OK
1114        | (* CSC: EVEN WORST CODE FOR EXTRACTION: WE SHOULD STRENGTHEN
1115             THE INDUCTION HYPOTHESIS INSTEAD *)
1116          elim limage
1117           [ %1
1118           | #HD #TL #IH @(?::IH) cases HD #ELEM #K1 %[@ELEM] #K2 #K3
1119             @K1 @(prefix_of_append ???? K3)
1120           ] 
1121        ]
1122       
1123       
1124     
1125 
1126  cases t in c2 ⊢ % #t' * #LIST_PREFIX * #H1t' #H2t' #HJMt'
1127     % [@ (LIST_PREFIX @ [i])] %
1128      [ cases (sig2 … i LIST_PREFIX) #K1 #K2 @K1
1129      | (* DOABLE IN PRINCIPLE *)
1130      ]
1131 | (* assert false case *)
1132 |3: % [@ ([ ])] % [2: % | (* DOABLE *)]
1133 |   
1134
1135let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word)
1136                       (encoding: list Byte) on encoding: Prop ≝
1137  match encoding with
1138  [ nil ⇒ final_pc = pc
1139  | cons hd tl ⇒
1140    let 〈new_pc, byte〉 ≝ next code_memory pc in
1141      hd = byte ∧ encoding_check code_memory new_pc final_pc tl
1142  ].
1143
1144definition assembly_specification:
1145  ∀assembly_program: pseudo_assembly_program.
1146  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1147  λpseudo_assembly_program.
1148  λcode_mem.
1149    ∀pc: Word.
1150      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1151      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1152      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1153      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1154      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1155       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1156      match pre_assembled with
1157       [ None ⇒ True
1158       | Some pc_code ⇒
1159          let 〈new_pc,code〉 ≝ pc_code in
1160           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1161
1162axiom assembly_meets_specification:
1163  ∀pseudo_assembly_program.
1164    match assembly pseudo_assembly_program with
1165    [ None ⇒ True
1166    | Some code_mem_cost ⇒
1167      let 〈code_mem, cost〉 ≝ code_mem_cost in
1168        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1169    ].
1170(*
1171  # PROGRAM
1172  [ cases PROGRAM
1173    # PREAMBLE
1174    # INSTR_LIST
1175    elim INSTR_LIST
1176    [ whd
1177      whd in ⊢ (∀_. %)
1178      # PC
1179      whd
1180    | # INSTR
1181      # INSTR_LIST_TL
1182      # H
1183      whd
1184      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1185    ]
1186  | cases not_implemented
1187  ] *)
1188
1189definition status_of_pseudo_status: PseudoStatus → option Status ≝
1190 λps.
1191  let pap ≝ code_memory … ps in
1192   match assembly pap with
1193    [ None ⇒ None …
1194    | Some p ⇒
1195       let cm ≝ load_code_memory (\fst p) in
1196       let pc ≝ sigma' pap (program_counter ? ps) in
1197        Some …
1198         (mk_PreStatus (BitVectorTrie Byte 16)
1199           cm
1200           (low_internal_ram … ps)
1201           (high_internal_ram … ps)
1202           (external_ram … ps)
1203           pc
1204           (special_function_registers_8051 … ps)
1205           (special_function_registers_8052 … ps)
1206           (p1_latch … ps)
1207           (p3_latch … ps)
1208           (clock … ps)) ].
1209
1210definition write_at_stack_pointer':
1211 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1212  λM: Type[0].
1213  λs: PreStatus M.
1214  λv: Byte.
1215    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1216    let bit_zero ≝ get_index_v… nu O ? in
1217    let bit_1 ≝ get_index_v… nu 1 ? in
1218    let bit_2 ≝ get_index_v… nu 2 ? in
1219    let bit_3 ≝ get_index_v… nu 3 ? in
1220      if bit_zero then
1221        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1222                              v (low_internal_ram ? s) in
1223          set_low_internal_ram ? s memory
1224      else
1225        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1226                              v (high_internal_ram ? s) in
1227          set_high_internal_ram ? s memory.
1228  [ cases l0 %
1229  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1230qed.
1231
1232definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1233 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1234
1235  λticks_of.
1236  λs.
1237  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1238  let ticks ≝ ticks_of (program_counter ? s) in
1239  let s ≝ set_clock ? s (clock ? s + ticks) in
1240  let s ≝ set_program_counter ? s pc in
1241    match instr with
1242    [ Instruction instr ⇒
1243       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1244    | Comment cmt ⇒ s
1245    | Cost cst ⇒ s
1246    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1247    | Call call ⇒
1248      let a ≝ address_of_word_labels s call in
1249      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1250      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1251      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1252      let s ≝ write_at_stack_pointer' ? s pc_bl in
1253      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1254      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1255      let s ≝ write_at_stack_pointer' ? s pc_bu in
1256        set_program_counter ? s a
1257    | Mov dptr ident ⇒
1258       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1259    ].
1260 [
1261 |2,3,4: %
1262 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1263 |
1264 | %
1265 ]
1266 cases not_implemented
1267qed.
1268
1269(*
1270lemma execute_code_memory_unchanged:
1271 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1272 #ticks #ps whd in ⊢ (??? (??%))
1273 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1274  (program_counter pseudo_assembly_program ps)) #instr #pc
1275 whd in ⊢ (??? (??%)) cases instr
1276  [ #pre cases pre
1277     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1278       cases (split ????) #z1 #z2 %
1279     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1280       cases (split ????) #z1 #z2 %
1281     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1282       cases (split ????) #z1 #z2 %
1283     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1284       [ #x1 whd in ⊢ (??? (??%))
1285     | *: cases not_implemented
1286     ]
1287  | #comment %
1288  | #cost %
1289  | #label %
1290  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1291    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1292    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1293    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1294    (* CSC: ??? *)
1295  | #dptr #label (* CSC: ??? *)
1296  ]
1297  cases not_implemented
1298qed.
1299*)
1300
1301lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1302 ∀ps,ps': PseudoStatus.
1303  code_memory … ps = code_memory … ps' →
1304   match status_of_pseudo_status ps with
1305    [ None ⇒ status_of_pseudo_status ps' = None …
1306    | Some _ ⇒ ∃w. status_of_pseudo_status ps' = Some … w
1307    ].
1308 #ps #ps' #H whd in ⊢ (mat
1309 ch % with [ _ ⇒ ? | _ ⇒ ? ])
1310 generalize in match (refl … (assembly (code_memory … ps)))
1311 cases (assembly ?) in ⊢ (???% → %)
1312  [ #K whd whd in ⊢ (??%?) <H >K %
1313  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1314qed.*)
1315
1316let rec encoding_check' (code_memory: BitVectorTrie Byte 16) (pc: Word) (encoding: list Byte) on encoding: Prop ≝
1317  match encoding with
1318  [ nil ⇒ True
1319  | cons hd tl ⇒
1320    let 〈new_pc, byte〉 ≝ next code_memory pc in
1321      hd = byte ∧ encoding_check' code_memory new_pc tl
1322  ].
1323
1324(* prove later *)
1325axiom test:
1326  ∀pc: Word.
1327  ∀code_memory: BitVectorTrie Byte 16.
1328  ∀i: instruction.
1329    let assembled ≝ assembly1 i in
1330      encoding_check' code_memory pc assembled →
1331        let 〈instr_pc, ignore〉 ≝ fetch code_memory pc in
1332        let 〈instr, pc〉 ≝ instr_pc in
1333          instr = i.
1334 
1335lemma main_thm:
1336 ∀ticks_of.
1337 ∀ps: PseudoStatus.
1338  match status_of_pseudo_status ps with [ None ⇒ True | Some s ⇒
1339  let ps' ≝ execute_1_pseudo_instruction ticks_of ps in
1340  match status_of_pseudo_status ps' with [ None ⇒ True | Some s'' ⇒
1341  let s' ≝ execute_1 s in
1342   s = s'']].
1343 #ticks_of #ps
1344 whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1345 cases (assembly (code_memory pseudo_assembly_program ps)) [%] * #cm #costs whd
1346 whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1347 generalize in match (sig2 … (execute_1_pseudo_instruction' ticks_of ps))
1348 
1349 cases (status_of_pseudo_status (execute_1_pseudo_instruction ticks_of ps)) [%] #s'' whd
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