source: src/ASM/AssemblyProof.ma @ 884

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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
441(*definition eq_instruction ≝
442  λi, j: instruction.
443    true.*)
444axiom eq_instruction: instruction → instruction → bool.
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  on l:
458  ∀P: l → Prop.
459  ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a.
460  ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a.
461  ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a.
462  ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a.
463  ∀x: l. P x ≝
464  match l return
465    λy.
466      match y with
467      [ O    ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True
468      | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop.
469               ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True.
470               ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True.
471               ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True.
472               ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True.
473               ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True.
474               ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True.
475               ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True.
476               ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True.
477               ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True.
478               ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True.
479               ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True.
480               ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True.
481               ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True.
482               ∀carry_a: if vect_member … eq_a l carry then P CARRY else True.
483               ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True.
484               ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True.
485               ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True.
486               ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True.
487               ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True.
488               ∀x:l. P x
489      ] with
490  [ VEmpty          ⇒ λAbsurd. ⊥
491  | VCons len hd tl ⇒ λProof. ?
492  ] (refl ? (S n)). cases daemon. qed. (*
493  [ destruct(Absurd)
494  | # A1 # A2 # A3 # A4 # A5 # A6 # A7
495    # A8 # A9 # A10 # A11 # A12 # A13 # A14
496    # A15 # A16 # A17 # A18 # A19 # X
497    cases X
498    # SUB cases daemon ] qed.
499    cases SUB
500    [ # BYTE
501    normalize
502  ].
503 
504 
505(*    let prepare_hd ≝
506      match hd with
507      [ direct ⇒ λdirect_prf. ?
508      | indirect ⇒ λindirect_prf. ?
509      | ext_indirect ⇒ λext_indirect_prf. ?
510      | registr ⇒ λregistr_prf. ?
511      | acc_a ⇒ λacc_a_prf. ?
512      | acc_b ⇒ λacc_b_prf. ?
513      | dptr ⇒ λdptr_prf. ?
514      | data ⇒ λdata_prf. ?
515      | data16 ⇒ λdata16_prf. ?
516      | acc_dptr ⇒ λacc_dptr_prf. ?
517      | acc_pc ⇒ λacc_pc_prf. ?
518      | ext_indirect_dptr ⇒ λext_indirect_prf. ?
519      | indirect_dptr ⇒ λindirect_prf. ?
520      | carry ⇒ λcarry_prf. ?
521      | bit_addr ⇒ λbit_addr_prf. ?
522      | n_bit_addr ⇒ λn_bit_addr_prf. ?
523      | relative ⇒ λrelative_prf. ?
524      | addr11 ⇒ λaddr11_prf. ?
525      | addr16 ⇒ λaddr16_prf. ?
526      ]
527    in ? *)
528  ].
529  [ 1: destruct(absd)
530  | 2: # A1 # A2 # A3 # A4 # A5 # A6
531       # A7 # A8 # A9 # A10 # A11 # A12
532       # A13 # A14 # A15 # A16 # A17 # A18
533       # A19 *
534  ].
535
536
537  match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x').
538   (l → bool) → bool ] with
539  [ VEmpty      ⇒  true 
540  | VCons len hd tl ⇒ λP.
541    let process_hd ≝
542      match hd return λhd. ∀P: hd:::tl → bool. bool with
543      [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x))
544      | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x))
545      | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x))
546      | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x))
547      | acc_a ⇒ λP.P ACC_A
548      | acc_b ⇒ λP.P ACC_B
549      | dptr ⇒ λP.P DPTR
550      | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x))
551      | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x))
552      | acc_dptr ⇒ λP.P ACC_DPTR
553      | acc_pc ⇒ λP.P ACC_PC
554      | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR
555      | indirect_dptr ⇒ λP.P INDIRECT_DPTR
556      | carry ⇒ λP.P CARRY
557      | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x))
558      | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x))
559      | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x))
560      | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x))
561      | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x))
562      ]
563    in
564      andb (process_hd P)
565       (match len return λx. x = len → bool with
566         [ O ⇒ λprf. true
567         | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len))
568  ].
569  try %
570  [ 2: cases (sym_eq ??? prf); @tl
571  | generalize in match H; generalize in match tl; cases prf;
572    (* cases prf in tl H; : ??? WAS WORKING BEFORE *)
573    #tl
574    normalize in ⊢ (∀_: %. ?)
575    # H
576    whd
577    normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
578    cases (is_a hd (subaddressing_modeel y tl H)) whd // ]
579qed.
580*)
581(*
582lemma test:
583  let i ≝ SJMP (RELATIVE (bitvector_of_nat 8 255)) in
584      (let assembled ≝ assembly1 i in
585      let code_memory ≝ load_code_memory assembled in
586      let fetched ≝ fetch code_memory ? in
587      let 〈instr_pc, ticks〉 ≝ fetched in
588        eq_instruction (\fst instr_pc)) i = true.
589 [2: @ zero
590 | normalize
591 ]*)
592
593lemma BitVectorTrie_O:
594 ∀A:Type[0].∀v:BitVectorTrie A 0.(∃w. v ≃ Leaf A w) ∨ v ≃ Stub A 0.
595 #A #v generalize in match (refl … O) cases v in ⊢ (??%? → (?(??(λ_.?%%??)))(?%%??))
596  [ #w #_ %1 %[@w] %
597  | #n #l #r #abs @⊥ //
598  | #n #EQ %2 >EQ %]
599qed.
600
601lemma BitVectorTrie_Sn:
602 ∀A:Type[0].∀n.∀v:BitVectorTrie A (S n).(∃l,r. v ≃ Node A n l r) ∨ v ≃ Stub A (S n).
603 #A #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → (?(??(λ_.??(λ_.?%%??))))%)
604  [ #m #abs @⊥ //
605  | #m #l #r #EQ %1 <(injective_S … EQ) %[@l] %[@r] //
606  | #m #EQ %2 // ]
607qed.
608
609lemma lookup_prepare_trie_for_insertion_hit:
610 ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.
611  lookup … b (prepare_trie_for_insertion … b v) a = v.
612 #A #a #v #n #b elim b // #m #hd #tl #IH cases hd normalize //
613qed.
614 
615lemma lookup_insert_hit:
616 ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.∀t:BitVectorTrie A n.
617  lookup … b (insert … b v t) a = v.
618 #A #a #v #n #b elim b -b -n //
619 #n #hd #tl #IH #t cases(BitVectorTrie_Sn … t)
620  [ * #l * #r #JMEQ >JMEQ cases hd normalize //
621  | #JMEQ >JMEQ cases hd normalize @lookup_prepare_trie_for_insertion_hit ]
622qed.
623
624lemma BitVector_O: ∀v:BitVector 0. v ≃ VEmpty bool.
625 #v generalize in match (refl … 0) cases v in ⊢ (??%? → ?%%??) //
626 #n #hd #tl #abs @⊥ //
627qed.
628
629lemma BitVector_Sn: ∀n.∀v:BitVector (S n).
630 ∃hd.∃tl.v ≃ VCons bool n hd tl.
631 #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??)))
632 [ #abs @⊥ //
633 | #m #hd #tl #EQ <(injective_S … EQ) %[@hd] %[@tl] // ]
634qed.
635
636coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0].
637
638lemma lookup_prepare_trie_for_insertion_miss:
639 ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.
640  (notb (eq_bv ? b c)) → lookup … b (prepare_trie_for_insertion … c v) a = a.
641 #A #a #v #n #c elim c
642  [ #b >(BitVector_O … b) normalize #abs @⊥ //
643  | #m #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ
644    cases hd cases hd' normalize
645    [2,3: #_ cases tl' //
646    |*: change with (bool_to_Prop (notb (eq_bv ???)) → ?) /2/ ]]
647qed.
648 
649lemma lookup_insert_miss:
650 ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.∀t:BitVectorTrie A n.
651  (notb (eq_bv ? b c)) → lookup … b (insert … c v t) a = lookup … b t a.
652 #A #a #v #n #c elim c -c -n
653  [ #b #t #DIFF @⊥ whd in DIFF; >(BitVector_O … b) in DIFF //
654  | #n #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ
655    #t cases(BitVectorTrie_Sn … t)
656    [ * #l * #r #JMEQ >JMEQ cases hd cases hd' #H normalize in H;
657     [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize // @IH //
658    | #JMEQ >JMEQ cases hd cases hd' #H normalize in H;
659     [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize
660     [3,4: cases tl' // | *: @lookup_prepare_trie_for_insertion_miss //]]]
661qed.
662
663definition load_code_memory_aux ≝
664 fold_left_i_aux … (
665   λi, mem, v.
666     insert … (bitvector_of_nat … i) v mem) (Stub Byte 16).
667
668axiom split_elim:
669 ∀A,l,m,v.∀P: (Vector A l) × (Vector A m) → Prop.
670  (∀vl,vm. v = vl@@vm → P 〈vl,vm〉) → P (split A l m v).
671
672axiom half_add_SO:
673 ∀pc.
674 \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc).
675
676axiom not_eqvb_S:
677 ∀pc.
678 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))).
679
680axiom not_eqvb_SS:
681 ∀pc.
682 (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))).
683
684axiom bitvector_elim_complete:
685 ∀n,P. bitvector_elim n P = true → ∀bv. P bv.
686
687lemma bitvector_elim_complete':
688 ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true.
689 #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv
690 generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ //
691qed.
692
693lemma andb_elim':
694 ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true.
695 #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize //
696qed.
697
698lemma test:
699  ∀pc,i.
700     (let assembled ≝ assembly1 i in
701      let code_memory ≝ load_code_memory_aux pc assembled in
702      let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in
703      let 〈instr_pc, ticks〉 ≝ fetched in
704        eq_instruction (\fst instr_pc)) i = true.
705 #pc #i cases i #arg try #arg2 whd in ⊢ (??%?)
706   [2,4: @(list_addressing_mode_tags_elim_prop … arg) whd try % #XX
707         whd in ⊢ (??(match ? (? ? %) ? with [ _ ⇒ ?] ?)?)
708         @split_elim #b1 #b2 #EQ >EQ -EQ;
709        change in ⊢ (??(match % with [ _ ⇒ ?] ?)?) with (fetch0 ??)
710        whd in ⊢ (??(match ?%% with [ _ ⇒ ?] ?)?)
711        >lookup_insert_miss try @not_eqvb_SS
712        >lookup_insert_miss //
713        >lookup_insert_hit
714        whd in ⊢ (??%?) whd in ⊢ (??(?%?)?)
715        >half_add_SO >half_add_SO
716        >lookup_insert_miss try @not_eqvb_S
717        >lookup_insert_hit
718        >lookup_insert_hit
719        (* REFL MISSING *)
720        cases daemon
721   |1,3: @(list_addressing_mode_tags_elim_prop … arg) whd try % #XX
722     whd in ⊢ (??(match ? (? ? %) ? with [ _ ⇒ ?] ?)?)
723     @split_elim #b1 #b2 #EQ >EQ -EQ;
724     change in ⊢ (??(match % with [ _ ⇒ ?] ?)?) with (fetch0 ??)
725     whd in ⊢ (??(match ?%% with [ _ ⇒ ?] ?)?)
726     >lookup_insert_miss //
727     >lookup_insert_hit
728     generalize in match b1 @(bitvector_elim_complete')
729     @andb_elim' @andb_elim' @andb_elim' whd in ⊢ (??%?)
730     >half_add_SO >lookup_insert_hit normalize
731     (* FALSO!!! AJMP vs ACALL *)
732   
733     
734     whd in ⊢ (??%?) whd in ⊢ (??(match % with [ _ ⇒ % | _ ⇒ ?])?)
735     check andb_elim.
736      (λV.(let 〈instr_pc,ticks〉 ≝
737           fetch0
738           (insert Byte 16 (bitvector_of_nat 16 (S pc)) b2
739            (insert Byte 16 (bitvector_of_nat 16 pc)
740             (V@@[[true; false; false; false; true]]) (Stub Byte 16)))
741           〈\snd  (half_add 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 1)),
742           V@@[[true; false; false; false; true]]〉 in 
743    eq_instruction (\fst  instr_pc))
744   (ACALL (ADDR11 (V@@b2)))) ?)
745     whd in ⊢ (??%?) whd in ⊢ (??(?%?)?)
746     >half_add_SO
747     
748       
749         
750 
751   [2: #addr whd in ⊢ (??%?)
752 
753   @ (instruction_elim_complete )
754 | @ zero
755 ]
756 normalize
757   
758 
759(* This establishes the correspondence between pseudo program counters and
760   program counters. It is at the heart of the proof. *)
761(*CSC: code taken from build_maps *)
762definition sigma0: pseudo_assembly_program → option (nat × (nat × (BitVectorTrie Word 16))) ≝
763 λinstr_list.
764  foldl ??
765    (λt. λi.
766       match t with
767       [ None ⇒ None ?
768       | Some ppc_pc_map ⇒
769         let 〈ppc,pc_map〉 ≝ ppc_pc_map in
770         let 〈program_counter, sigma_map〉 ≝ pc_map in
771         let 〈label, i〉 ≝ i in
772          match construct_costs instr_list program_counter (λx. zero ?) (λx. zero ?) (Stub …) i with
773           [ None ⇒ None ?
774           | Some pc_ignore ⇒
775              let 〈pc,ignore〉 ≝ pc_ignore in
776              Some … 〈S ppc,〈pc, insert ? ? (bitvector_of_nat ? ppc) (bitvector_of_nat ? pc) sigma_map〉〉 ]
777       ]) (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (\snd instr_list).
778       
779definition tech_pc_sigma0: pseudo_assembly_program → option (nat × (BitVectorTrie Word 16)) ≝
780 λinstr_list.
781  match sigma0 instr_list with
782   [ None ⇒ None …
783   | Some result ⇒
784      let 〈ppc,pc_sigma_map〉 ≝ result in
785       Some … pc_sigma_map ].
786
787definition sigma_safe: pseudo_assembly_program → option (Word → Word) ≝       
788 λinstr_list.
789  match sigma0 instr_list with
790  [ None ⇒ None ?
791  | Some result ⇒
792    let 〈ppc,pc_sigma_map〉 ≝ result in
793    let 〈pc, sigma_map〉 ≝ pc_sigma_map in
794      if gtb pc (2^16) then
795        None ?
796      else
797        Some ? (λx.lookup ?? x sigma_map (zero …)) ].
798
799axiom policy_ok: ∀p. sigma_safe p ≠ None ….
800
801definition sigma: pseudo_assembly_program → Word → Word ≝
802 λp.
803  match sigma_safe p return λr:option (Word → Word). r ≠ None … → Word → Word with
804   [ None ⇒ λabs. ⊥
805   | Some r ⇒ λ_.r] (policy_ok p).
806 cases abs //
807qed.
808
809lemma length_append:
810 ∀A.∀l1,l2:list A.
811  |l1 @ l2| = |l1| + |l2|.
812 #A #l1 elim l1
813  [ //
814  | #hd #tl #IH #l2 normalize <IH //]
815qed.
816
817let rec does_not_occur (id:Identifier) (l:list labelled_instruction) on l: bool ≝
818 match l with
819  [ nil ⇒ true
820  | cons hd tl ⇒ notb (instruction_matches_identifier id hd) ∧ does_not_occur id tl].
821
822lemma does_not_occur_None:
823 ∀id,i,list_instr.
824  does_not_occur id (list_instr@[〈None …,i〉]) =
825  does_not_occur id list_instr.
826 #id #i #list_instr elim list_instr
827  [ % | #hd #tl #IH whd in ⊢ (??%%) >IH %]
828qed.
829
830let rec occurs_exactly_once (id:Identifier) (l:list labelled_instruction) on l : bool ≝
831 match l with
832  [ nil ⇒ false
833  | cons hd tl ⇒
834     if instruction_matches_identifier id hd then
835      does_not_occur id tl
836     else
837      occurs_exactly_once id tl ].
838
839lemma occurs_exactly_once_None:
840 ∀id,i,list_instr.
841  occurs_exactly_once id (list_instr@[〈None …,i〉]) =
842  occurs_exactly_once id list_instr.
843 #id #i #list_instr elim list_instr
844  [ % | #hd #tl #IH whd in ⊢ (??%%) >IH >does_not_occur_None %]
845qed.
846
847lemma index_of_internal_None: ∀i,id,instr_list,n.
848 occurs_exactly_once id (instr_list@[〈None …,i〉]) →
849  index_of_internal ? (instruction_matches_identifier id) instr_list n =
850   index_of_internal ? (instruction_matches_identifier id) (instr_list@[〈None …,i〉]) n.
851 #i #id #instr_list elim instr_list
852  [ #n #abs whd in abs; cases abs
853  | #hd #tl #IH #n whd in ⊢ (% → ??%%); whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ?)
854    cases (instruction_matches_identifier id hd) whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ??%%)
855    [ #H %
856    | #H @IH whd in H; cases (occurs_exactly_once ??) in H ⊢ %
857      [ #_ % | #abs cases abs ]]]
858qed.
859
860lemma address_of_word_labels_code_mem_None: ∀i,id,instr_list.
861 occurs_exactly_once id (instr_list@[〈None …,i〉]) →
862  address_of_word_labels_code_mem instr_list id =
863  address_of_word_labels_code_mem (instr_list@[〈None …,i〉]) id.
864 #i #id #instr_list #H whd in ⊢ (??%%) whd in ⊢ (??(??%?)(??%?))
865 >(index_of_internal_None … H) %
866qed.
867
868axiom tech_pc_sigma0_append:
869 ∀preamble,instr_list,prefix,label,i,pc',code,pc,costs,costs'.
870  Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 →
871   construct_costs 〈preamble,instr_list〉 … pc (λx.zero 16) (λx. zero 16) costs i = Some … 〈pc',code〉 →
872    tech_pc_sigma0 〈preamble,prefix@[〈label,i〉]〉 = Some … 〈pc',costs'〉.
873
874axiom tech_pc_sigma0_append_None:
875 ∀preamble,instr_list,prefix,i,pc,costs.
876  Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 →
877   construct_costs 〈preamble,instr_list〉 … pc (λx.zero 16) (λx. zero 16) costs i = None …
878    → False.
879
880
881definition build_maps' ≝
882  λpseudo_program.
883  let 〈preamble,instr_list〉 ≝ pseudo_program in
884  let result ≝
885   foldl_strong
886    (option Identifier × pseudo_instruction)
887    (λpre. Σres:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
888      let pre' ≝ 〈preamble,pre〉 in
889      let 〈labels,pc_costs〉 ≝ res in
890       tech_pc_sigma0 pre' = Some … pc_costs ∧
891       ∀id. occurs_exactly_once id pre →
892        lookup ?? id labels (zero …) = sigma pre' (address_of_word_labels_code_mem pre id))
893    instr_list
894    (λprefix,i,tl,prf,t.
895      let 〈labels, pc_costs〉 ≝ t in
896      let 〈program_counter, costs〉 ≝ pc_costs in
897       let 〈label, i'〉 ≝ i in
898       let labels ≝
899         match label with
900         [ None ⇒ labels
901         | Some label ⇒
902           let program_counter_bv ≝ bitvector_of_nat ? program_counter in
903             insert ? ? label program_counter_bv labels
904         ]
905       in
906         match construct_costs 〈preamble,instr_list〉 program_counter (λx. zero ?) (λx. zero ?) costs i' with
907         [ None ⇒
908            let dummy ≝ 〈labels,pc_costs〉 in
909             dummy
910         | Some construct ⇒ 〈labels, construct〉
911         ]
912    ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉
913  in
914   let 〈labels, pc_costs〉 ≝ result in
915   let 〈pc, costs〉 ≝ pc_costs in
916    〈labels, costs〉.
917 [3: whd % // #id normalize in ⊢ (% → ?) #abs @⊥ //
918 | whd cases construct in p3 #PC #CODE #JMEQ %
919    [ @(tech_pc_sigma0_append ??????????? (jmeq_to_eq ??? JMEQ)) | #id #Hid ]
920 | (* dummy case *) @⊥
921   @(tech_pc_sigma0_append_None ?? prefix ???? (jmeq_to_eq ??? p3)) ]
922 [*: generalize in match (sig2 … t) whd in ⊢ (% → ?)
923     >p whd in ⊢ (% → ?) >p1 * #IH0 #IH1 >IH0 // ]
924 whd in ⊢ (??(????%?)?) -labels1;
925 cases label in Hid
926  [ #Hid whd in ⊢ (??(????%?)?) >IH1 -IH1
927     [ >(address_of_word_labels_code_mem_None … Hid)
928       (* MANCA LEMMA: INDIRIZZO TROVATO NEL PROGRAMMA! *)
929     | whd in Hid >occurs_exactly_once_None in Hid // ]
930  | -label #label #Hid whd in ⊢ (??(????%?)?)
931   
932  ]
933qed.
934
935(*
936(*
937notation < "hvbox('let' 〈ident x,ident y〉 ≝ t 'in' s)"
938 with precedence 10
939for @{ match $t with [ pair ${ident x} ${ident y} ⇒ $s ] }.
940*)
941
942lemma build_maps_ok:
943 ∀p:pseudo_assembly_program.
944  let 〈labels,costs〉 ≝ build_maps' p in
945   ∀pc.
946    (nat_of_bitvector … pc) < length … (\snd p) →
947     lookup ?? pc labels (zero …) = sigma p (\snd (fetch_pseudo_instruction (\snd p) pc)).
948 #p cases p #preamble #instr_list
949  elim instr_list
950   [ whd #pc #abs normalize in abs; cases (not_le_Sn_O ?) [#H cases (H abs) ]
951   | #hd #tl #IH
952    whd in ⊢ (match % with [ _ ⇒ ?])
953   ]
954qed.
955*)
956
957(*
958lemma list_elim_rev:
959 ∀A:Type[0].∀P:list A → Prop.
960  P [ ] → (∀n,l. length l = n → P l → 
961  P [ ] → (∀l,a. P l → P (l@[a])) →
962   ∀l. P l.
963 #A #P
964qed.*)
965
966lemma rev_preserves_length:
967 ∀A.∀l. length … (rev A l) = length … l.
968  #A #l elim l
969   [ %
970   | #hd #tl #IH normalize >length_append normalize /2/ ]
971qed.
972
973lemma rev_append:
974 ∀A.∀l1,l2.
975  rev A (l1@l2) = rev A l2 @ rev A l1.
976 #A #l1 elim l1 normalize //
977qed.
978 
979lemma rev_rev: ∀A.∀l. rev … (rev A l) = l.
980 #A #l elim l
981  [ //
982  | #hd #tl #IH normalize >rev_append normalize // ]
983qed.
984
985lemma split_len_Sn:
986 ∀A:Type[0].∀l:list A.∀len.
987  length … l = S len →
988   Σl'.Σa. l = l'@[a] ∧ length … l' = len.
989 #A #l elim l
990  [ normalize #len #abs destruct
991  | #hd #tl #IH #len
992    generalize in match (rev_rev … tl)
993    cases (rev A tl) in ⊢ (??%? → ?)
994     [ #H <H normalize #EQ % [@[ ]] % [@hd] normalize /2/ 
995     | #a #l' #H <H normalize #EQ
996      %[@(hd::rev … l')] %[@a] % //
997      >length_append in EQ #EQ normalize in EQ; normalize;
998      generalize in match (injective_S … EQ) #EQ2 /2/ ]]
999qed.
1000
1001lemma list_elim_rev:
1002 ∀A:Type[0].∀P:list A → Type[0].
1003  P [ ] → (∀l,a. P l → P (l@[a])) →
1004   ∀l. P l.
1005 #A #P #H1 #H2 #l
1006 generalize in match (refl … (length … l))
1007 generalize in ⊢ (???% → ?) #n generalize in match l
1008 elim n
1009  [ #L cases L [ // | #x #w #abs (normalize in abs) @⊥ // ]
1010  | #m #IH #L #EQ
1011    cases (split_len_Sn … EQ) #l' * #a * /3/ ]
1012qed.
1013
1014axiom is_prefix: ∀A:Type[0]. list A → list A → Prop.
1015axiom prefix_of_append:
1016 ∀A:Type[0].∀l,l1,l2:list A.
1017  is_prefix … l l1 → is_prefix … l (l1@l2).
1018axiom prefix_reflexive: ∀A,l. is_prefix A l l.
1019axiom nil_prefix: ∀A,l. is_prefix A [ ] l.
1020
1021record Propify (A:Type[0]) : Type[0] (*Prop*) ≝ { in_propify: A }.
1022
1023definition Propify_elim: ∀A. ∀P:Prop. (A → P) → (Propify A → P) ≝
1024 λA,P,H,x. match x with [ mk_Propify p ⇒ H p ].
1025
1026definition app ≝
1027 λA:Type[0].λl1:Propify (list A).λl2:list A.
1028  match l1 with
1029   [ mk_Propify l1 ⇒ mk_Propify … (l1@l2) ].
1030
1031lemma app_nil: ∀A,l1. app A l1 [ ] = l1.
1032 #A * /3/
1033qed.
1034
1035lemma app_assoc: ∀A,l1,l2,l3. app A (app A l1 l2) l3 = app A l1 (l2@l3).
1036 #A * #l1 normalize //
1037qed.
1038
1039let rec foldli (A: Type[0]) (B: Propify (list A) → Type[0])
1040 (f: ∀prefix. B prefix → ∀x.B (app … prefix [x]))
1041 (prefix: Propify (list A)) (b: B prefix) (l: list A) on l :
1042 B (app … prefix l) ≝
1043  match l with
1044  [ nil ⇒ ? (* b *)
1045  | cons hd tl ⇒ ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*)
1046  ].
1047 [ applyS b
1048 | <(app_assoc ?? [hd]) @(foldli A B f (app … prefix [hd]) (f prefix b hd) tl) ]
1049qed.
1050
1051(*
1052let rec foldli (A: Type[0]) (B: list A → Type[0]) (f: ∀prefix. B prefix → ∀x. B (prefix@[x]))
1053 (prefix: list A) (b: B prefix) (l: list A) on l : B (prefix@l) ≝
1054  match l with
1055  [ nil ⇒ ? (* b *)
1056  | cons hd tl ⇒
1057     ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*)
1058  ].
1059 [ applyS b
1060 | applyS (foldli A B f (prefix@[hd]) (f prefix b hd) tl) ]
1061qed.
1062*)
1063
1064definition foldll:
1065 ∀A:Type[0].∀B: Propify (list A) → Type[0].
1066  (∀prefix. B prefix → ∀x. B (app … prefix [x])) →
1067   B (mk_Propify … []) → ∀l: list A. B (mk_Propify … l)
1068 ≝ λA,B,f. foldli A B f (mk_Propify … [ ]).
1069
1070axiom is_pprefix: ∀A:Type[0]. Propify (list A) → list A → Prop.
1071axiom pprefix_of_append:
1072 ∀A:Type[0].∀l,l1,l2.
1073  is_pprefix A l l1 → is_pprefix A l (l1@l2).
1074axiom pprefix_reflexive: ∀A,l. is_pprefix A (mk_Propify … l) l.
1075axiom nil_pprefix: ∀A,l. is_pprefix A (mk_Propify … [ ]) l.
1076
1077
1078axiom foldll':
1079 ∀A:Type[0].∀l: list A.
1080  ∀B: ∀prefix:Propify (list A). is_pprefix ? prefix l → Type[0].
1081  (∀prefix,proof. B prefix proof → ∀x,proof'. B (app … prefix [x]) proof') →
1082   B (mk_Propify … [ ]) (nil_pprefix …) → B (mk_Propify … l) (pprefix_reflexive … l).
1083 #A #l #B
1084 generalize in match (foldll A (λprefix. is_pprefix ? prefix l)) #HH
1085 
1086 
1087  #H #acc
1088 @foldll
1089  [
1090  |
1091  ]
1092 
1093 ≝ λA,B,f. foldli A B f (mk_Propify … [ ]).
1094
1095
1096(*
1097record subset (A:Type[0]) (P: A → Prop): Type[0] ≝
1098 { subset_wit:> A;
1099   subset_proof: P subset_wit
1100 }.
1101*)
1102
1103definition build_maps' ≝
1104  λpseudo_program.
1105  let 〈preamble,instr_list〉 ≝ pseudo_program in
1106  let result ≝
1107   foldll
1108    (option Identifier × pseudo_instruction)
1109    (λprefix.
1110      Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1111       match prefix return λ_.Prop with [mk_Propify prefix ⇒ tech_pc_sigma0 〈preamble,prefix〉 ≠ None ?])
1112    (λprefix,t,i.
1113      let 〈labels, pc_costs〉 ≝ t in
1114      let 〈program_counter, costs〉 ≝ pc_costs in
1115       let 〈label, i'〉 ≝ i in
1116       let labels ≝
1117         match label with
1118         [ None ⇒ labels
1119         | Some label ⇒
1120           let program_counter_bv ≝ bitvector_of_nat ? program_counter in
1121             insert ? ? label program_counter_bv labels
1122         ]
1123       in
1124         match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with
1125         [ None ⇒
1126            let dummy ≝ 〈labels,pc_costs〉 in
1127              dummy
1128         | Some construct ⇒ 〈labels, construct〉
1129         ]
1130    ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 instr_list
1131  in
1132   let 〈labels, pc_costs〉 ≝ result in
1133   let 〈pc, costs〉 ≝ pc_costs in
1134    〈labels, costs〉.
1135 [
1136 | @⊥
1137 | normalize % //
1138 ]
1139qed.
1140
1141definition build_maps' ≝
1142  λpseudo_program.
1143  let 〈preamble,instr_list〉 ≝ pseudo_program in
1144  let result ≝
1145   foldl
1146    (Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1147          ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧
1148           tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)))
1149    (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix.
1150          let instr_list_prefix' ≝ instr_list_prefix @ [i] in
1151           is_prefix ? instr_list_prefix' instr_list →
1152           tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?)
1153    (λt: Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))).
1154          ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧
1155           tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)).
1156     λi: Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix.
1157          let instr_list_prefix' ≝ instr_list_prefix @ [i] in
1158           is_prefix ? instr_list_prefix' instr_list →
1159           tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ? .
1160      let 〈labels, pc_costs〉 ≝ t in
1161      let 〈program_counter, costs〉 ≝ pc_costs in
1162       let 〈label, i'〉 ≝ i in
1163       let labels ≝
1164         match label with
1165         [ None ⇒ labels
1166         | Some label ⇒
1167           let program_counter_bv ≝ bitvector_of_nat ? program_counter in
1168             insert ? ? label program_counter_bv labels
1169         ]
1170       in
1171         match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with
1172         [ None ⇒
1173            let dummy ≝ 〈labels,pc_costs〉 in
1174              dummy
1175         | Some construct ⇒ 〈labels, construct〉
1176         ]
1177    ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 ?(*instr_list*)
1178  in
1179   let 〈labels, pc_costs〉 ≝ result in
1180   let 〈pc, costs〉 ≝ pc_costs in
1181    〈labels, costs〉.
1182 [4: @(list_elim_rev ?
1183       (λinstr_list. list (
1184        (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix.
1185          let instr_list_prefix' ≝ instr_list_prefix @ [i] in
1186           is_prefix ? instr_list_prefix' instr_list →
1187           tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?)))
1188       ?? instr_list) (* CSC: BAD ORDER FOR CODE EXTRACTION *)
1189      [ @[ ]
1190      | #l' #a #limage %2
1191        [ %[@a] #PREFIX #PREFIX_OK
1192        | (* CSC: EVEN WORST CODE FOR EXTRACTION: WE SHOULD STRENGTHEN
1193             THE INDUCTION HYPOTHESIS INSTEAD *)
1194          elim limage
1195           [ %1
1196           | #HD #TL #IH @(?::IH) cases HD #ELEM #K1 %[@ELEM] #K2 #K3
1197             @K1 @(prefix_of_append ???? K3)
1198           ] 
1199        ]
1200       
1201       
1202     
1203 
1204  cases t in c2 ⊢ % #t' * #LIST_PREFIX * #H1t' #H2t' #HJMt'
1205     % [@ (LIST_PREFIX @ [i])] %
1206      [ cases (sig2 … i LIST_PREFIX) #K1 #K2 @K1
1207      | (* DOABLE IN PRINCIPLE *)
1208      ]
1209 | (* assert false case *)
1210 |3: % [@ ([ ])] % [2: % | (* DOABLE *)]
1211 |   
1212
1213let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word)
1214                       (encoding: list Byte) on encoding: Prop ≝
1215  match encoding with
1216  [ nil ⇒ final_pc = pc
1217  | cons hd tl ⇒
1218    let 〈new_pc, byte〉 ≝ next code_memory pc in
1219      hd = byte ∧ encoding_check code_memory new_pc final_pc tl
1220  ].
1221
1222definition assembly_specification:
1223  ∀assembly_program: pseudo_assembly_program.
1224  ∀code_mem: BitVectorTrie Byte 16. Prop ≝
1225  λpseudo_assembly_program.
1226  λcode_mem.
1227    ∀pc: Word.
1228      let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in
1229      let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in
1230      let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in
1231      let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in
1232      let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program
1233       (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in
1234      match pre_assembled with
1235       [ None ⇒ True
1236       | Some pc_code ⇒
1237          let 〈new_pc,code〉 ≝ pc_code in
1238           encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ].
1239
1240axiom assembly_meets_specification:
1241  ∀pseudo_assembly_program.
1242    match assembly pseudo_assembly_program with
1243    [ None ⇒ True
1244    | Some code_mem_cost ⇒
1245      let 〈code_mem, cost〉 ≝ code_mem_cost in
1246        assembly_specification pseudo_assembly_program (load_code_memory code_mem)
1247    ].
1248(*
1249  # PROGRAM
1250  [ cases PROGRAM
1251    # PREAMBLE
1252    # INSTR_LIST
1253    elim INSTR_LIST
1254    [ whd
1255      whd in ⊢ (∀_. %)
1256      # PC
1257      whd
1258    | # INSTR
1259      # INSTR_LIST_TL
1260      # H
1261      whd
1262      whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?])
1263    ]
1264  | cases not_implemented
1265  ] *)
1266
1267definition status_of_pseudo_status: PseudoStatus → option Status ≝
1268 λps.
1269  let pap ≝ code_memory … ps in
1270   match assembly pap with
1271    [ None ⇒ None …
1272    | Some p ⇒
1273       let cm ≝ load_code_memory (\fst p) in
1274       let pc ≝ sigma' pap (program_counter ? ps) in
1275        Some …
1276         (mk_PreStatus (BitVectorTrie Byte 16)
1277           cm
1278           (low_internal_ram … ps)
1279           (high_internal_ram … ps)
1280           (external_ram … ps)
1281           pc
1282           (special_function_registers_8051 … ps)
1283           (special_function_registers_8052 … ps)
1284           (p1_latch … ps)
1285           (p3_latch … ps)
1286           (clock … ps)) ].
1287
1288definition write_at_stack_pointer':
1289 ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝
1290  λM: Type[0].
1291  λs: PreStatus M.
1292  λv: Byte.
1293    let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in
1294    let bit_zero ≝ get_index_v… nu O ? in
1295    let bit_1 ≝ get_index_v… nu 1 ? in
1296    let bit_2 ≝ get_index_v… nu 2 ? in
1297    let bit_3 ≝ get_index_v… nu 3 ? in
1298      if bit_zero then
1299        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1300                              v (low_internal_ram ? s) in
1301          set_low_internal_ram ? s memory
1302      else
1303        let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl)
1304                              v (high_internal_ram ? s) in
1305          set_high_internal_ram ? s memory.
1306  [ cases l0 %
1307  |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ]
1308qed.
1309
1310definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus.
1311 Σps':PseudoStatus.(code_memory … ps = code_memory … ps')
1312
1313  λticks_of.
1314  λs.
1315  let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in
1316  let ticks ≝ ticks_of (program_counter ? s) in
1317  let s ≝ set_clock ? s (clock ? s + ticks) in
1318  let s ≝ set_program_counter ? s pc in
1319    match instr with
1320    [ Instruction instr ⇒
1321       execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s
1322    | Comment cmt ⇒ s
1323    | Cost cst ⇒ s
1324    | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp)
1325    | Call call ⇒
1326      let a ≝ address_of_word_labels s call in
1327      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1328      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1329      let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in
1330      let s ≝ write_at_stack_pointer' ? s pc_bl in
1331      let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in
1332      let s ≝ set_8051_sfr ? s SFR_SP new_sp in
1333      let s ≝ write_at_stack_pointer' ? s pc_bu in
1334        set_program_counter ? s a
1335    | Mov dptr ident ⇒
1336       set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr
1337    ].
1338 [
1339 |2,3,4: %
1340 | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) %
1341 |
1342 | %
1343 ]
1344 cases not_implemented
1345qed.
1346
1347(*
1348lemma execute_code_memory_unchanged:
1349 ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps).
1350 #ticks #ps whd in ⊢ (??? (??%))
1351 cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps))
1352  (program_counter pseudo_assembly_program ps)) #instr #pc
1353 whd in ⊢ (??? (??%)) cases instr
1354  [ #pre cases pre
1355     [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1356       cases (split ????) #z1 #z2 %
1357     | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1358       cases (split ????) #z1 #z2 %
1359     | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%))
1360       cases (split ????) #z1 #z2 %
1361     | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x
1362       [ #x1 whd in ⊢ (??? (??%))
1363     | *: cases not_implemented
1364     ]
1365  | #comment %
1366  | #cost %
1367  | #label %
1368  | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%))
1369    cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2
1370    whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2
1371    whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%))
1372    (* CSC: ??? *)
1373  | #dptr #label (* CSC: ??? *)
1374  ]
1375  cases not_implemented
1376qed.
1377*)
1378
1379lemma status_of_pseudo_status_failure_depends_only_on_code_memory:
1380 ∀ps,ps': PseudoStatus.
1381  code_memory … ps = code_memory … ps' →
1382   match status_of_pseudo_status ps with
1383    [ None ⇒ status_of_pseudo_status ps' = None …
1384    | Some _ ⇒ ∃w. status_of_pseudo_status ps' = Some … w
1385    ].
1386 #ps #ps' #H whd in ⊢ (mat
1387 ch % with [ _ ⇒ ? | _ ⇒ ? ])
1388 generalize in match (refl … (assembly (code_memory … ps)))
1389 cases (assembly ?) in ⊢ (???% → %)
1390  [ #K whd whd in ⊢ (??%?) <H >K %
1391  | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ]
1392qed.*)
1393
1394let rec encoding_check' (code_memory: BitVectorTrie Byte 16) (pc: Word) (encoding: list Byte) on encoding: Prop ≝
1395  match encoding with
1396  [ nil ⇒ True
1397  | cons hd tl ⇒
1398    let 〈new_pc, byte〉 ≝ next code_memory pc in
1399      hd = byte ∧ encoding_check' code_memory new_pc tl
1400  ].
1401
1402(* prove later *)
1403axiom test:
1404  ∀pc: Word.
1405  ∀code_memory: BitVectorTrie Byte 16.
1406  ∀i: instruction.
1407    let assembled ≝ assembly1 i in
1408      encoding_check' code_memory pc assembled →
1409        let 〈instr_pc, ignore〉 ≝ fetch code_memory pc in
1410        let 〈instr, pc〉 ≝ instr_pc in
1411          instr = i.
1412 
1413lemma main_thm:
1414 ∀ticks_of.
1415 ∀ps: PseudoStatus.
1416  match status_of_pseudo_status ps with [ None ⇒ True | Some s ⇒
1417  let ps' ≝ execute_1_pseudo_instruction ticks_of ps in
1418  match status_of_pseudo_status ps' with [ None ⇒ True | Some s'' ⇒
1419  let s' ≝ execute_1 s in
1420   s = s'']].
1421 #ticks_of #ps
1422 whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1423 cases (assembly (code_memory pseudo_assembly_program ps)) [%] * #cm #costs whd
1424 whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ])
1425 generalize in match (sig2 … (execute_1_pseudo_instruction' ticks_of ps))
1426 
1427 cases (status_of_pseudo_status (execute_1_pseudo_instruction ticks_of ps)) [%] #s'' whd
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