source: LTS/Vm.ma @ 3505

Last change on this file since 3505 was 3505, checked in by piccolo, 6 years ago
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1include "costs.ma".
2include "basics/lists/list.ma".
3include "../src/utilities/option.ma".
4include "basics/jmeq.ma".
5
6lemma bind_inversion : ∀A,B : Type[0].∀m : option A.
7∀f : A → option B.∀y : B.
8! x ← m; f x = return y →
9∃ x.(m = return x) ∧ (f x = return y).
10#A #B * [| #a] #f #y normalize #EQ [destruct]
11%{a} %{(refl …)} //
12qed.
13
14record assembler_params : Type[1] ≝
15{ seq_instr : Type[0]
16; jump_condition : Type[0]
17; io_instr : Type[0] }.
18
19inductive AssemblerInstr (p : assembler_params) : Type[0] ≝
20| Seq : seq_instr p → option (NonFunctionalLabel) →  AssemblerInstr p
21| Ijmp: ℕ → AssemblerInstr p
22| CJump : jump_condition p → ℕ → NonFunctionalLabel → NonFunctionalLabel → AssemblerInstr p
23| Iio : NonFunctionalLabel → io_instr p → NonFunctionalLabel → AssemblerInstr p
24| Icall: FunctionName → AssemblerInstr p
25| Iret: AssemblerInstr p.
26
27definition labels_pc_of_instr : ∀p.AssemblerInstr p → ℕ → list (CostLabel × ℕ) ≝
28λp,i,program_counter.
29match i with
30  [ Seq _ opt_l ⇒ match opt_l with
31                  [ Some lbl ⇒ [〈(a_non_functional_label lbl),S program_counter〉]
32                  | None ⇒ [ ]
33                  ]
34  | Ijmp _ ⇒ [ ]
35  | CJump _ newpc ltrue lfalse ⇒ [〈(a_non_functional_label ltrue),newpc〉;
36                                  〈(a_non_functional_label lfalse),S program_counter〉]
37  | Iio lin _ lout ⇒ [〈(a_non_functional_label lin),program_counter〉;
38                      〈(a_non_functional_label lout),S program_counter〉]
39  | Icall f ⇒ [ ]
40  | Iret ⇒ [ ]
41  ].
42
43let rec labels_pc (p : assembler_params)
44(prog : list (AssemblerInstr p)) (call_label_fun : list (ℕ × CallCostLabel))
45              (return_label_fun : list (ℕ × ReturnPostCostLabel)) (i_act : NonFunctionalLabel)
46              (program_counter : ℕ) on prog : list (CostLabel × ℕ) ≝
47match prog with
48[ nil ⇒ [〈a_non_functional_label (i_act),O〉] @
49        map … (λx.let〈y,z〉 ≝ x in 〈(a_call z),y〉) (call_label_fun) @
50        map … (λx.let〈y,z〉 ≝ x in 〈(a_return_post z),y〉) (return_label_fun)
51| cons i is ⇒ (labels_pc_of_instr … i program_counter)@labels_pc p is call_label_fun return_label_fun i_act (S program_counter)
52].
53
54include "basics/lists/listb.ma".
55
56(*doppione da mettere a posto*)
57let rec no_duplicates (A : DeqSet) (l : list A) on l : Prop ≝
58match l with
59[ nil ⇒ True
60| cons x xs ⇒ ¬ (bool_to_Prop (x ∈ xs)) ∧ no_duplicates … xs
61].
62
63
64record AssemblerProgram (p : assembler_params) : Type[0] ≝
65{ instructions : list (AssemblerInstr p)
66; endmain : ℕ
67; endmain_ok : endmain < |instructions|
68; entry_of_function : FunctionName → ℕ
69; call_label_fun : list (ℕ × CallCostLabel)
70; return_label_fun : list (ℕ × ReturnPostCostLabel)
71; in_act : NonFunctionalLabel
72; asm_no_duplicates : no_duplicates … (map ?? \fst … (labels_pc … instructions call_label_fun return_label_fun in_act O))
73}.
74
75
76definition fetch: ∀p.AssemblerProgram p → ℕ → option (AssemblerInstr p) ≝
77 λp,l,n. nth_opt ? n (instructions … l).
78
79definition stackT: Type[0] ≝ list (nat).
80
81record sem_params (p : assembler_params) : Type[1] ≝
82{ m : monoid
83; asm_store_type : Type[0]
84; eval_asm_seq : seq_instr p → asm_store_type → option asm_store_type
85; eval_asm_cond : jump_condition p → asm_store_type → option bool
86; eval_asm_io : io_instr p → asm_store_type → option asm_store_type
87; cost_of_io : io_instr p → asm_store_type → m
88; cost_of : AssemblerInstr p → asm_store_type →  m
89}.
90
91record vm_state (p : assembler_params) (p' : sem_params p) : Type[0] ≝
92{ pc : ℕ
93; asm_stack : stackT
94; asm_store : asm_store_type … p'
95; asm_is_io : bool
96; cost : m … p'
97}.
98
99definition label_of_pc ≝ λL.λl.λpc.find …
100   (λp.let 〈x,y〉 ≝ p in if eqb x pc then Some L y else None ? ) l.
101   
102definition option_pop ≝
103  λA.λl:list A. match l with
104  [ nil ⇒ None ?
105  | cons a tl ⇒ Some ? (〈a,tl〉) ].
106
107
108inductive vmstep (p : assembler_params) (p' : sem_params p)
109   (prog : AssemblerProgram p)  :
110      ActionLabel → relation (vm_state p p') ≝
111| vm_seq : ∀st1,st2 : vm_state p p'.∀i,l.
112           fetch … prog (pc … st1) = return (Seq p i l) →
113           asm_is_io … st1 = false →
114           eval_asm_seq p p' i (asm_store … st1) = return asm_store … st2 → 
115           asm_stack … st1 = asm_stack … st2 →
116           asm_is_io … st1 = asm_is_io … st2 →
117           S (pc … st1) = pc … st2 →
118           op … (cost … st1) (cost_of p p' (Seq p i l) (asm_store … st1)) = cost … st2 →
119           vmstep … (cost_act l) st1 st2
120| vm_ijump : ∀st1,st2 : vm_state p p'.∀new_pc : ℕ.
121           fetch … prog (pc … st1) = return (Ijmp p new_pc) →
122           asm_is_io … st1 = false →
123           asm_store … st1 = asm_store … st2 →
124           asm_stack … st1 = asm_stack … st2 →
125           asm_is_io … st1 = asm_is_io … st2 →
126           new_pc = pc … st2 →
127           op … (cost … st1) (cost_of p p' (Ijmp … new_pc) (asm_store … st1)) = cost … st2 →
128           vmstep … (cost_act (None ?)) st1 st2
129| vm_cjump_true :
130           ∀st1,st2 : vm_state p p'.∀cond,new_pc,ltrue,lfalse.
131           eval_asm_cond p p' cond (asm_store … st1) = return true→
132           fetch … prog (pc … st1) = return (CJump p cond new_pc ltrue lfalse) →
133           asm_is_io … st1 = false →
134           asm_store … st1 = asm_store … st2 →
135           asm_stack … st1 = asm_stack … st2 →
136           asm_is_io … st1 = asm_is_io … st2 →
137           pc … st2 = new_pc →
138           op … (cost … st1) (cost_of p p' (CJump p cond new_pc ltrue lfalse) (asm_store … st1)) = cost … st2 →
139           vmstep … (cost_act (Some ? ltrue)) st1 st2
140| vm_cjump_false :
141           ∀st1,st2 : vm_state p p'.∀cond,new_pc,ltrue,lfalse.
142           eval_asm_cond p p' cond (asm_store … st1) = return false→
143           fetch … prog (pc … st1) = return (CJump p cond new_pc ltrue lfalse) →
144           asm_is_io … st1 = false →
145           asm_store … st1 = asm_store … st2 →
146           asm_stack … st1 = asm_stack … st2 →
147           asm_is_io … st1 = asm_is_io … st2 →
148           S (pc … st1) = pc … st2 →
149           op … (cost … st1) (cost_of p p' (CJump … cond new_pc ltrue lfalse) (asm_store … st1)) = cost … st2 →
150           vmstep … (cost_act (Some ? lfalse)) st1 st2
151| vm_io_in : 
152           ∀st1,st2 : vm_state p p'.∀lin,io,lout.
153           fetch … prog (pc … st1) = return (Iio p lin io lout) →
154           asm_is_io … st1 = false →
155           asm_store … st1 = asm_store … st2 →
156           asm_stack … st1 = asm_stack … st2 →
157           true = asm_is_io … st2 →
158           pc … st1 = pc … st2 →
159           cost … st1 = cost … st2 →
160           vmstep … (cost_act (Some ? lin)) st1 st2
161| vm_io_out :
162           ∀st1,st2 : vm_state p p'.∀lin,io,lout.
163           fetch … prog (pc … st1) = return (Iio p lin io lout) →
164           asm_is_io … st1 = true →
165           eval_asm_io … io (asm_store … st1) = return asm_store … st2 →
166           asm_stack … st1 = asm_stack … st2 →
167           false = asm_is_io … st2 →
168           S (pc … st1) = pc … st2 →
169           op … (cost … st1) (cost_of_io p p' io (asm_store … st1)) = cost … st2 →
170           vmstep … (cost_act (Some ? lout)) st1 st2
171| vm_call :
172           ∀st1,st2 : vm_state p p'.∀f,lbl.
173           fetch … prog (pc … st1) = return (Icall p f) →
174           asm_is_io … st1 = false →
175           asm_store … st1 = asm_store … st2 →
176           S (pc … st1) ::  asm_stack … st1 = asm_stack … st2 →
177           asm_is_io … st1 = asm_is_io … st2 →
178           entry_of_function … prog f = pc … st2 →
179           op … (cost … st1) (cost_of p p' (Icall p f) (asm_store … st1)) = cost … st2 →
180           label_of_pc ? (call_label_fun … prog) (entry_of_function … prog f) = return lbl →
181           vmstep … (call_act f lbl) st1 st2
182| vm_ret :
183          ∀st1,st2 : vm_state p p'.∀newpc,lbl.
184           fetch … prog (pc … st1) = return (Iret p) →
185           asm_is_io … st1 = false →
186           asm_store … st1 = asm_store … st2 →
187           asm_stack … st1 = newpc ::  asm_stack … st2  →
188           asm_is_io … st1 = asm_is_io … st2 →
189           newpc = pc … st2 →
190           label_of_pc ? (return_label_fun … prog) newpc = return lbl →
191           op … (cost … st1) (cost_of p p' (Iret p) (asm_store … st1)) = cost … st2 →
192           vmstep … (ret_act (Some ? lbl)) st1 st2.
193
194definition eval_vmstate : ∀p : assembler_params.∀p' : sem_params p.
195AssemblerProgram p → vm_state p p' → option (ActionLabel × (vm_state p p')) ≝
196λp,p',prog,st.
197! i ← fetch … prog (pc … st);
198match i with
199[ Seq x opt_l ⇒
200   if asm_is_io … st then
201     None ?
202   else
203     ! new_store ← eval_asm_seq p p' x (asm_store … st);
204     return 〈cost_act opt_l,
205             mk_vm_state ?? (S (pc … st)) (asm_stack … st) new_store false
206                (op … (cost … st) (cost_of p p' (Seq p x opt_l) (asm_store … st)))〉
207| Ijmp newpc ⇒
208   if asm_is_io … st then
209     None ?
210   else
211     return 〈cost_act (None ?),
212             mk_vm_state ?? newpc (asm_stack … st) (asm_store … st) false
213                (op … (cost … st) (cost_of p p' (Ijmp … newpc) (asm_store … st)))〉
214| CJump cond newpc ltrue lfalse ⇒
215   if asm_is_io … st then
216     None ?
217   else
218     ! b ← eval_asm_cond p p' cond (asm_store … st);
219     if b then
220       return 〈cost_act (Some ? ltrue),
221               mk_vm_state ?? newpc (asm_stack … st) (asm_store … st) false
222                (op … (cost … st) (cost_of p p' (CJump … cond newpc ltrue lfalse) (asm_store … st)))〉
223     else
224       return 〈cost_act (Some ? lfalse),
225               mk_vm_state ?? (S (pc … st)) (asm_stack … st) (asm_store … st) false
226                (op … (cost … st) (cost_of p p' (CJump … cond newpc ltrue lfalse) (asm_store … st)))〉
227| Iio lin io lout ⇒
228              if asm_is_io … st then
229                 ! new_store ← eval_asm_io … io (asm_store … st);
230                 return 〈cost_act (Some ? lout),
231                         mk_vm_state ?? (S (pc … st)) (asm_stack … st)
232                         new_store false
233                         (op … (cost … st)
234                               (cost_of_io p p' io (asm_store … st)))〉   
235              else
236                return 〈cost_act (Some ? lin),
237                        mk_vm_state ?? (pc … st) (asm_stack … st) (asm_store … st)
238                                    true (cost … st)〉
239| Icall f ⇒
240    if asm_is_io … st then
241      None ?
242    else
243      ! lbl ← label_of_pc ? (call_label_fun … prog) (entry_of_function … prog f);
244      return 〈call_act f lbl,
245              mk_vm_state ?? (entry_of_function … prog f)
246                             ((S (pc … st)) :: (asm_stack … st))
247                             (asm_store … st) false
248                             (op … (cost … st) (cost_of p p' (Icall p f) (asm_store … st)))〉
249| Iret ⇒ if asm_is_io … st then
250            None ?
251         else
252            ! 〈newpc,tl〉 ← option_pop … (asm_stack … st);
253            ! lbl ← label_of_pc ? (return_label_fun … prog) newpc;
254            return 〈ret_act (Some ? lbl),
255                    mk_vm_state ?? newpc tl (asm_store … st) false   
256                     (op … (cost … st) (cost_of p p' (Iret p) (asm_store … st)))〉
257].
258
259lemma eval_vmstate_to_Prop : ∀p,p',prog,st1,st2,l.
260eval_vmstate p p' prog st1 = return 〈l,st2〉 → vmstep … prog l st1 st2.
261#p #p' #prog #st1 #st2 #l whd in match eval_vmstate; normalize nodelta
262#H cases(bind_inversion ????? H) -H * normalize nodelta
263[ #seq #opt_l * #EQfetch inversion(asm_is_io ???) normalize nodelta
264  [ #_ whd in ⊢ (??%% → ?); #EQ destruct] #EQio #H cases(bind_inversion ????? H)
265  #newstore * #EQnewstore whd in ⊢ (??%% → ?); #EQ destruct
266  @vm_seq //
267| #newpc * #EQfetch inversion(asm_is_io ???) normalize nodelta #EQio
268  [ whd in ⊢ (??%% → ?); #EQ destruct] whd in ⊢ (??%% → ?); #EQ destruct
269  @vm_ijump //
270| #cond #new_pc #ltrue #lfase * #EQfetch inversion(asm_is_io ???) normalize nodelta
271  [ #_ whd in ⊢ (??%% → ?); #EQ destruct] #EQio #H cases(bind_inversion ????? H) -H
272  * normalize nodelta * #EQcond whd in ⊢ (??%% → ?); #EQ destruct
273  [ @(vm_cjump_true … EQfetch) // | @(vm_cjump_false … EQfetch) //]
274| #lin #io #lout * #EQfetch inversion(asm_is_io ???) normalize nodelta #EQio
275  [ #H cases(bind_inversion ????? H) -H #newstore * #EQnewstore ]
276  whd in ⊢ (??%% → ?); #EQ destruct
277  [ @(vm_io_out … EQfetch) // | @(vm_io_in … EQfetch) // ]
278| #f * #EQfetch inversion(asm_is_io ???) #EQio normalize nodelta
279  [ whd in ⊢ (??%% → ?); #EQ destruct ] #H cases (bind_inversion ????? H) -H
280  #lbl * #EQlb whd in ⊢ (??%% → ?); #EQ destruct @(vm_call … EQfetch) //
281| * #EQfetch inversion(asm_is_io ???) normalize nodelta #EQio
282  [ whd in ⊢ (??%% → ?); #EQ destruct] #H cases(bind_inversion ????? H) -H
283  * #newpc #tl * whd in match option_pop; normalize nodelta inversion(asm_stack ???)
284  normalize nodelta [#_ whd in ⊢ (??%% → ?); #EQ destruct] #newpc1 #tl1 #_
285  #EQstack whd in ⊢ (??%% → ?); #EQ destruct #H cases(bind_inversion ????? H) #lbl
286  * #EQlbl whd in ⊢ (??%% → ?); #EQ destruct @vm_ret //
287]
288qed.
289
290 
291lemma vm_step_to_eval : ∀p,p',prog,st1,st2,l.vmstep … prog l st1 st2 →
292eval_vmstate p p' prog st1 = return 〈l,st2〉.
293#p #p' #prog * #pc1 #stack1 #store1 #io1 #cost1
294* #pc2 #stack2 #store2 #io2 #cost2 #l #H inversion H
295[ #s1 #s2 #i #opt_l #EQfetch #EQio #EQstore #EQstack #EQio1 #EQpc #EQcost
296  #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in match eval_vmstate; normalize nodelta   
297  >EQfetch >m_return_bind normalize nodelta >EQio normalize nodelta >EQstore
298  >m_return_bind <EQio1 >EQio <EQpc >EQstack >EQcost %
299| #s1 #s2 #newpc #EQfetch #EQio1 #EQstore #EQstack #EQio2 #EQnewpc #EQcost
300  #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in match eval_vmstate; normalize nodelta
301  >EQfetch >m_return_bind normalize nodelta >EQio1 normalize nodelta <EQio2 >EQio1
302  >EQstore >EQstack <EQcost >EQstore %
303|3,4: #s1 #s2 #cond #newoc #ltrue #lfalse #EQev_cond #EQfetch #EQio1 #EQstore
304  #EQstack #EQio2 #EQnewoc #EQcost #EQ1 #EQ2 #EQ3 #EQ4 destruct
305  whd in match eval_vmstate; normalize nodelta >EQfetch >m_return_bind
306  normalize nodelta >EQio1 normalize nodelta >EQev_cond >m_return_bind
307  normalize nodelta <EQio1 >EQio2 >EQstore >EQstack <EQcost >EQstore [%] >EQnewoc %
308|5,6: #s1 #s2 #lin #io #lout #EQfetch #EQio1 #EQstore #EQstack #EQio2 #EQpc
309   #EQcost #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in match eval_vmstate;
310   normalize nodelta >EQfetch >m_return_bind normalize nodelta >EQio1
311   normalize nodelta >EQstack <EQpc >EQcost [ >EQstore <EQio2 %]
312   >EQstore >m_return_bind <EQpc <EQio2 %
313| #s1 #s2 #f #lbl #EQfetch #EQio1 #EQstore #EQstack #EQio2 #EQentry
314  #EQcost #EQlab_pc #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in match eval_vmstate;
315  normalize nodelta >EQfetch >m_return_bind normalize nodelta >EQio1
316  normalize nodelta >EQlab_pc >m_return_bind >EQentry >EQcost <EQio2 >EQio1
317  <EQstack >EQstore %
318| #s1 #s2 #newpc #lbl #EQfetch #EQio1 #EQstore #EQstack #EQio2 #EQnewpc
319  #EQlab_pc #EQcosts #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in match eval_vmstate;
320  normalize nodelta >EQfetch >m_return_bind normalize nodelta >EQio1 normalize nodelta
321  >EQstack whd in match option_pop; normalize nodelta >m_return_bind
322  >EQlab_pc >m_return_bind >EQcosts >EQstore  <EQio2 >EQio1 %
323]
324qed.
325
326coercion vm_step_to_eval.
327
328include "../src/utilities/hide.ma".
329
330discriminator option.
331
332inductive vm_ass_state  (p : assembler_params) (p' : sem_params p) : Type[0] ≝
333| INITIAL : vm_ass_state p p'
334| FINAL : vm_ass_state p p'
335| STATE : vm_state p p' → vm_ass_state p p'.
336
337definition ass_vmstep ≝
338 λp,p',prog.
339  λl.λs1,s2 : vm_ass_state p p'.
340                    match s1 with
341                    [ STATE st1 ⇒
342                        match s2 with
343                        [ STATE st2 ⇒ 
344                            (eqb (pc ?? st1) (endmain … prog)) = false ∧ vmstep p p' prog l st1 st2
345                        | INITIAL ⇒ False
346                        | FINAL ⇒ eqb (pc … st1) (endmain … prog) = true ∧
347                           l = cost_act (Some … (in_act … prog))
348                        ]
349                    | INITIAL ⇒ match s2 with
350                                [ STATE st2 ⇒ eqb (pc … st2) O = true ∧
351                                   l = cost_act (Some … (in_act … prog))
352                                | _ ⇒ False
353                                ]
354                    | FINAL ⇒ False
355                    ].
356
357definition asm_operational_semantics : ∀p.sem_params p → AssemblerProgram p →  abstract_status ≝
358λp,p',prog.let init_act ≝ cost_act (Some ? (in_act … prog)) in
359           let end_act ≝ cost_act (Some ? (in_act … prog)) in
360    mk_abstract_status
361                (vm_ass_state p p')
362                (ass_vmstep … prog)
363                (λ_.λ_.True)
364                (λst.match st with
365                    [ INITIAL ⇒ cl_other | FINAL ⇒ cl_other |
366                     STATE s ⇒
367                      match fetch … prog (pc … s) with
368                      [ Some i ⇒ match i with
369                               [ Seq _ _ ⇒ cl_other
370                               | Ijmp _ ⇒ cl_other
371                               | CJump _ _ _ _ ⇒ cl_jump
372                               | Iio _ _ _ ⇒ if asm_is_io … s then cl_io else cl_other
373                               | Icall _ ⇒ cl_other
374                               | Iret ⇒ cl_other
375                               ]
376                     | None ⇒ cl_other
377                     ]
378                    ]
379                )
380                (λ_.true)
381                (λs.match s with [ INITIAL ⇒ true | _ ⇒ false])
382                (λs.match s with [ FINAL ⇒ true | _ ⇒ false])
383                ???.
384@hide_prf
385[ #s1 #s2 #l
386 cases s1 normalize nodelta [1,2: #abs destruct] -s1 #s1
387 cases s2 normalize nodelta [1: #_ * |2: #_ * #_ #EQ >EQ normalize /2/ ] #s2 
388 inversion(fetch ???) normalize nodelta
389  [ #_ #EQ destruct] * normalize nodelta
390  [ #seq #lbl #_
391  | #n #_
392  | #cond #newpc #ltrue #lfalse #EQfetch
393  | #lin #io #lout #_ cases (asm_is_io ??) normalize nodelta
394  | #f #_
395  | #_
396  ]
397  #EQ destruct * #_ #H lapply(vm_step_to_eval … H) whd in match eval_vmstate;
398  normalize nodelta >EQfetch >m_return_bind normalize nodelta cases(asm_is_io ??)
399  normalize nodelta [ whd in ⊢ (??%% → ?); #EQ destruct] #H cases(bind_inversion ????? H) -H
400  * * #_ normalize nodelta whd in ⊢ (??%% → ?); #EQ destruct % //
401| #s1 #s2 #l
402  cases s1 normalize nodelta [2: #_ * |3: #s1 ]
403  cases s2 normalize nodelta [1,2,4,5: #abs destruct ] #s2 [2: #_ * #_ /2/ ]
404  inversion(fetch ???) normalize nodelta
405  [ #_ #EQ destruct] * normalize nodelta
406  [ #seq #lbl #_
407  | #n #_
408  | #cond #newpc #ltrue #lfalse #_
409  | #lin #io #lout #EQfetch inversion (asm_is_io ??) #EQio normalize nodelta
410  | #f #_
411  | #_
412  ]
413  #EQ destruct * #_ #H lapply(vm_step_to_eval … H) whd in match eval_vmstate;
414  normalize nodelta #H cases(bind_inversion ????? H) -H *
415  [ #seq1 #lbl1
416  | #n1
417  | #cond1 #newpc1 #ltrue1 #lfalse1
418  | #lin1 #io1 #lout1
419  | #f
420  |
421  ]
422  normalize nodelta * #_ cases(asm_is_io ??) normalize nodelta
423  [1,3,5,9,11: whd in ⊢ (??%% → ?); #EQ destruct
424  |2,6,7,10,12: #H cases(bind_inversion ????? H) -H #x * #_
425    [2: cases x normalize nodelta
426    |5: #H cases(bind_inversion ????? H) -H #y * #_
427    ]
428  ]
429  whd in ⊢ (??%% → ?); #EQ destruct destruct % //
430| #s1 #s2 #l
431  cases s1 normalize nodelta [1,2: #abs destruct ] #s1
432  cases s2 normalize nodelta [ #_ * | #_ * /2/ ] #s2
433  inversion(fetch ???) normalize nodelta
434  [ #_ #EQ destruct] * normalize nodelta
435  [ #seq #lbl #_
436  | #n #_
437  | #cond #newpc #ltrue #lfalse #_
438  | #lin #io #lout #EQfetch inversion(asm_is_io ??) normalize nodelta #EQio
439  | #f #_
440  | #_
441  ]
442  #EQ destruct * #_ #H lapply(vm_step_to_eval … H) whd in match eval_vmstate;
443  normalize nodelta  >EQfetch >m_return_bind normalize nodelta >EQio
444  normalize nodelta #H cases(bind_inversion ????? H) -H #x * #_
445  whd in ⊢ (??%% → ?); #EQ destruct % //
446qed.
447
448definition asm_concrete_trans_sys ≝
449λp,p',prog.mk_concrete_transition_sys …
450             (asm_operational_semantics p p' prog) (m … p')
451             (λs.match s with [STATE st ⇒ cost … st | _ ⇒ e …] ).
452
453definition emits_labels ≝
454λp.λinstr : AssemblerInstr p.match instr with
455        [ Seq _ opt_l ⇒ match opt_l with
456                        [ None ⇒ Some ? (λpc.S pc)
457                        | Some _ ⇒ None ?
458                        ]
459        | Ijmp newpc ⇒ Some ? (λ_.newpc)
460        | _ ⇒ None ?
461        ].
462
463definition fetch_state : ∀p,p'.AssemblerProgram p → vm_state p p' → option (AssemblerInstr p) ≝
464λp,p',prog,st.fetch … prog (pc … st).
465
466record asm_galois_connection (p: assembler_params) (p': sem_params p) (prog: AssemblerProgram p) : Type[2] ≝
467{ aabs_d : abstract_transition_sys (m … p')
468; agalois_rel:> galois_rel (asm_concrete_trans_sys p p' prog) aabs_d
469}.
470
471definition galois_connection_of_asm_galois_connection:
472 ∀p,p',prog. asm_galois_connection p p' prog → galois_connection ≝
473 λp,p',prog,agc.
474  mk_galois_connection
475   (asm_concrete_trans_sys p p' prog)
476   (aabs_d … agc)
477   (agalois_rel … agc).
478
479coercion galois_connection_of_asm_galois_connection.
480
481definition ass_fetch ≝
482 λp,p',prog.
483    λs.match s with [ STATE st ⇒ if eqb (pc … st) (endmain … prog) then
484                                   Some ? (None ?)
485                               else ! x ← fetch_state p p' prog st; Some ? (Some ? x)
486                    | INITIAL ⇒ Some ? (None ?)
487                    | FINAL  ⇒ None ? ].
488
489definition ass_instr_map ≝
490 λp,p',prog.λasm_galois_conn: asm_galois_connection p p' prog.
491  λinstr_map: AssemblerInstr p → (*option*) (abs_instr … (abs_d asm_galois_conn)).
492  (λi.match i with [None ⇒ (*Some …*) (e …) |Some x ⇒ instr_map … x]).
493
494record asm_abstract_interpretation (p: assembler_params) (p': sem_params p) (prog: AssemblerProgram p) : Type[2] ≝
495{ asm_galois_conn: asm_galois_connection p p' prog
496; instr_map : AssemblerInstr p → (*option*) (abs_instr … (abs_d asm_galois_conn))
497; instr_map_ok :
498   ∀s,s': concr … asm_galois_conn. ∀a: abs_d … asm_galois_conn.∀l,i.
499    as_execute … l s s' →
500     ass_fetch … prog s = Some ? i →
501      ∀I. ass_instr_map … instr_map i = (*Some ?*) I →
502       asm_galois_conn s a → asm_galois_conn s' (〚I〛 a)
503}.
504
505definition abstract_interpretation_of_asm_abstract_interpretation:
506 ∀p,p',prog. asm_abstract_interpretation p p' prog → abstract_interpretation
507≝
508λp,p',prog,aai.
509 mk_abstract_interpretation
510  (asm_galois_conn … aai) (option (AssemblerInstr p)) (ass_fetch p p' prog)
511   (ass_instr_map … prog … (instr_map … aai)) (instr_map_ok … aai).
512
513coercion abstract_interpretation_of_asm_abstract_interpretation.
514
515definition non_empty_list : ∀A.list A → bool ≝
516λA,l.match l with [ nil ⇒ false | _ ⇒  true ].
517
518let rec block_cost (p : assembler_params)
519 (prog: AssemblerProgram p) (abs_t : monoid)
520 (instr_m : AssemblerInstr p → abs_t)
521 (prog_c: option ℕ)
522    (program_size: ℕ)
523        on program_size: option abs_t ≝
524match prog_c with
525[ None ⇒ return e … abs_t
526| Some program_counter ⇒
527  match program_size with
528    [ O ⇒ None ?
529    | S program_size' ⇒
530       if eqb program_counter (endmain … prog) then
531        return e … abs_t
532      else
533      ! instr ← fetch … prog program_counter;
534      ! n ← (match emits_labels … instr with
535            [ Some f ⇒ block_cost … prog abs_t instr_m (Some ? (f program_counter)) program_size'
536            | None ⇒ return e …
537            ]);
538      return (op … abs_t (instr_m … instr) n)
539    ]
540].
541
542
543record cost_map_T (dom : DeqSet) (codom : Type[0]) : Type[1] ≝
544{ map_type :> Type[0]
545; empty_map : map_type
546; get_map : map_type → dom → option codom
547; set_map : map_type → dom → codom → map_type
548; get_set_hit : ∀k,v,m.get_map (set_map m k v) k = return v
549; get_set_miss : ∀k1,k2,v,m.(k1 == k2) = false → get_map (set_map m k1 v) k2 = get_map m k2
550}.
551
552
553
554lemma labels_pc_ok : ∀p,prog,l1,l2,i_act,i,lbl,pc,m.
555nth_opt ? pc prog = return i →
556mem ? lbl (labels_pc_of_instr … i (m+pc)) →
557mem ? lbl (labels_pc p prog l1 l2 i_act m).
558#p #instrs #l1 #l2 #iact #i #lbl #pc
559whd in match fetch; normalize nodelta lapply pc -pc
560elim instrs
561[ #pc #m whd in ⊢ (??%% → ?); #EQ destruct]
562#x #xs #IH * [|#pc'] #m  whd in ⊢ (??%% → ?);
563[ #EQ destruct #lbl_addr whd in match (labels_pc ???);
564  /2 by mem_append_l1/
565| #EQ #H2 whd in match (labels_pc ???); @mem_append_l2 @(IH … EQ) //
566]
567qed.
568
569lemma labels_pf_in_act: ∀p,prog,pc.
570  mem CostLabel (in_act p prog)
571  (map (CostLabel×ℕ) CostLabel \fst
572   (labels_pc p (instructions p prog) (call_label_fun p prog)
573    (return_label_fun p prog) (in_act p prog) pc)).
574 #p #prog elim (instructions p prog) normalize /2/
575qed.
576
577lemma labels_pc_return: ∀p,prog,l1,l2,iact,x1,x2.
578 label_of_pc ReturnPostCostLabel l2 x1=return x2 →
579 ∀m.
580   mem … 〈(a_return_post x2),x1〉 (labels_pc p prog l1 l2 iact m).
581 #p #l #l1 #l2 #iact whd in match (labels_pc ???); #x1 #x2 #H elim l
582[ #m @mem_append_l2 @mem_append_l2 whd in H:(??%?);
583  elim l2 in H; [ whd in ⊢ (??%% → ?); #EQ destruct]
584  * #x #y #tl #IH whd in ⊢ (??%? → %); normalize nodelta @eqb_elim
585  normalize nodelta
586  [ #EQ whd in ⊢ (??%% → ?); #EQ2 destruct /2/
587  | #NEQ #H2 %2 @IH // ]
588| #hd #tl #IH #m @mem_append_l2 @IH ]
589qed.
590
591lemma labels_pc_call: ∀p,prog,l1,l2,iact,x1,x2.
592 label_of_pc CallCostLabel l1 x1=return x2 →
593 ∀m.
594   mem … 〈(a_call x2),x1〉 (labels_pc p prog l1 l2 iact m).
595 #p #l #l1 #l2 #iact whd in match (labels_pc ???); #x1 #x2 #H elim l
596[ #m @mem_append_l2 @mem_append_l1 whd in H:(??%?);
597  elim l1 in H; [ whd in ⊢ (??%% → ?); #EQ destruct]
598  * #x #y #tl #IH whd in ⊢ (??%? → %); normalize nodelta @eqb_elim
599  normalize nodelta
600  [ #EQ whd in ⊢ (??%% → ?); #EQ2 destruct /2/
601  | #NEQ #H2 %2 @IH // ]
602| #hd #tl #IH #m @mem_append_l2 @IH ]
603qed.
604
605(*
606lemma labels_pc_bounded : ∀p.∀prog : AssemblerProgram p.∀lbl,pc.∀m.
607mem ? 〈lbl,pc〉 (labels_pc p (instructions … prog) m) →
608(m + pc) < (|(instructions … prog)|).
609#p * #instr #endmain #_  #H1 #H2 elim instr
610[ #H3 @⊥ /2/ ] #x #xs #IH #_ #lbl #pc #m whd in match (labels_pc ???);
611#H cases(mem_append ???? H) -H
612[ whd in match labels_pc_of_instr; normalize nodelta
613  cases x normalize nodelta
614  [ #seq * [|#lab]
615  | #newpc
616  | #cond #newpc #ltrue #lfalse
617  | #lin #io #lout
618  | #f
619  |
620  ]
621  normalize [1,3,6,7: *] * [2,4,6: * [2,4:*] ]
622  #EQ destruct
623*) 
624
625let rec m_foldr (M : Monad) (X,Y : Type[0]) (f : X→Y→M Y) (l : list X) (y : Y) on l : M Y ≝
626match l with
627[ nil ⇒ return y
628| cons x xs ⇒ ! z ← m_foldr M X Y f xs y; f x z
629].
630
631definition static_analisys : ∀p : assembler_params.∀abs_t : monoid.(AssemblerInstr p → abs_t) →
632∀mT : cost_map_T DEQCostLabel abs_t.AssemblerProgram p → option mT ≝
633λp,abs_t,instr_m,mT,prog.
634let prog_size ≝ S (|instructions … prog|) in
635m_foldr Option ?? (λx,m.let 〈z,w〉≝ x in ! k ← block_cost p prog abs_t instr_m (Some ? w) prog_size; 
636                               return set_map … m z k) (labels_pc … (instructions … prog)
637                                         (call_label_fun … prog) (return_label_fun … prog) (in_act … prog) O)
638      (empty_map ?? mT).
639     
640
641(*falso: necessita di no_duplicates*)
642
643definition eq_deq_prod : ∀D1,D2 : DeqSet.D1 × D2 → D1 × D2 → bool ≝
644λD1,D2,x,y.\fst x == \fst y ∧ \snd x == \snd y.
645
646definition DeqProd ≝ λD1 : DeqSet.λD2 : DeqSet.
647mk_DeqSet (D1 × D2) (eq_deq_prod D1 D2) ?.
648@hide_prf
649* #x1 #x2 * #y1 #y2 whd in match eq_deq_prod; normalize nodelta
650% [2: #EQ destruct @andb_Prop >(\b (refl …)) %]
651inversion (? ==?) #EQ1 whd in match (andb ??); #EQ2 destruct
652>(\P EQ1) >(\P EQ2) %
653qed.
654(*
655unification hint  0 ≔ D1,D2 ;
656    X ≟ DeqProd D1 D2
657(* ---------------------------------------- *) ⊢
658    D1 × D2 ≡ carr X.
659
660
661unification hint  0 ≔ D1,D2,p1,p2;
662    X ≟ DeqProd D1 D2
663(* ---------------------------------------- *) ⊢
664    eq_deq_prod D1 D2 p1 p2 ≡ eqb X p1 p2.
665   
666definition deq_prod_to_prod : ∀D1,D2 : DeqSet.DeqProd D1 D2 → D1 × D2 ≝
667λD1,D2,x.x.
668
669coercion deq_prod_to_prod.
670*)
671
672lemma map_mem : ∀A,B,f,l,a.mem A a l → ∃b : B.mem B b (map A B f l)
673∧ b = f a.
674#A #B #f #l elim l [ #a *] #x #xs #IH #a *
675[ #EQ destruct %{(f x)} % // % // | #H cases(IH … H)
676  #b * #H1 #EQ destruct %{(f a)} % // %2 //
677]
678qed.
679
680lemma static_analisys_ok : ∀p,abs_t,instr_m,mT,prog,lbl,pc,map.
681static_analisys p abs_t instr_m mT prog = return map →
682mem … 〈lbl,pc〉 (labels_pc … (instructions … prog) (call_label_fun … prog)
683                   (return_label_fun … prog) (in_act … prog) O) →
684get_map … map lbl =
685block_cost … prog abs_t instr_m (Some ? pc) (S (|(instructions … prog)|)) ∧
686block_cost … prog abs_t instr_m (Some ? pc) (S (|(instructions … prog)|)) ≠ None ?.
687#p #abs_t #instr_m #mT * #prog whd in match static_analisys; normalize nodelta
688#endmain #Hendmain #entry_fun #call_label_fun #return_label_fun #inact
689#nodup generalize in match nodup in ⊢ (∀_.∀_.∀_. (??(????%??)?) → %); #Hnodup lapply nodup -nodup
690lapply (labels_pc ??????) #l elim l [ #x #y #z #w #h * ]
691* #hd1 #hd2 #tl #IH * #H1 #H2 #lbl #pc #map #H
692cases(bind_inversion ????? H) -H #map1 * #EQmap1 normalize nodelta #H
693cases(bind_inversion ????? H) -H #elem * #EQelem whd in ⊢ (??%% → ?); #EQ
694destruct *
695[ #EQ destruct % [ >get_set_hit >EQelem % | >EQelem % whd in ⊢ (??%% → ?); #EQ destruct]
696| #H %
697  [ >get_set_miss [ @(proj1 … (IH …)) //] inversion (? == ?) [2: #_ %]
698    #ABS cases H1 -H1 #H1 @⊥ @H1 >(\P ABS) >mem_to_memb //
699    cases(map_mem … \fst … H) #z1 * #Hz1 #EQ destruct @Hz1
700  | @(proj2 … (IH …)) //
701  ]
702]
703qed.
704
705include "Simulation.ma".
706
707definition terminated_trace : ∀S : abstract_status.∀s1,s3 : S.raw_trace … s1 s3 → Prop ≝
708λS,s1,s3,t.∃s2: S.∃t1 : raw_trace … s1 s2.∃ l,prf.t = t1 @ (t_ind ? s2 s3 s3 l prf … (t_base … s3))
709∧ is_costlabelled_act l.
710
711definition big_op: ∀M: monoid. list M → M ≝
712 λM. foldl … (op … M) (e … M).
713
714lemma big_op_associative:
715 ∀M:monoid. ∀l1,l2.
716  big_op M (l1@l2) = op M (big_op … l1) (big_op … l2).
717 #M #l1 whd in match big_op; normalize nodelta
718 generalize in match (e M) in ⊢ (? → (??%(??%?))); elim l1
719 [ #c #l2 whd in match (append ???); normalize lapply(neutral_r … c)
720   generalize in match c in ⊢ (??%? → ???%); lapply c -c lapply(e M)
721   elim l2 normalize
722   [ #c #c1 #c2 #EQ @sym_eq //
723   | #x #xs #IH #c1 #c2 #c3 #EQ <EQ <IH [% | <is_associative % |]
724   ]
725 | #x #xs #IH #c #l2 @IH
726 ]
727qed.
728
729lemma act_big_op : ∀M,B. ∀act : monoid_action M B.
730 ∀l1,l2,x.
731   act (big_op M (l1@l2)) x = act (big_op … l2) (act (big_op … l1) x).
732 #M #B #act #l1 elim l1
733 [ #l2 #x >act_neutral //
734 | #hd #tl #IH #l2 #x change with ([?]@(tl@l2)) in match ([?]@(tl@l2));
735   >big_op_associative >act_op >IH change with ([hd]@tl) in match ([hd]@tl);
736   >big_op_associative >act_op in ⊢ (???%); %
737 ]
738qed.
739
740lemma monotonicity_of_block_cost : ∀p,prog,abs_t,instr_m,pc,size,k.
741block_cost p prog abs_t instr_m (Some ? pc) size = return k →
742∀size'.size ≤ size' →
743block_cost p prog abs_t instr_m (Some ? pc) size' = return k.
744#p #prog #abs_t #instr_m #pc #size lapply pc elim size
745[ #pc #k whd in ⊢ (??%% → ?); #EQ destruct]
746#n #IH #pc #k whd in ⊢ (??%% → ?); @eqb_elim
747[ #EQ destruct normalize nodelta whd in ⊢ (??%% → ?); #EQ destruct
748  #size' * [2: #m #_] whd in ⊢ (??%%); @eqb_elim try( #_ %) * #H @⊥ @H %
749| #Hpc normalize nodelta #H cases(bind_inversion ????? H) -H #i
750  * #EQi #H cases(bind_inversion ????? H) -H #elem * #EQelem whd in ⊢ (??%% → ?);
751  #EQ destruct #size' *
752  [ whd in ⊢ (??%?); @eqb_elim
753    [ #EQ @⊥ @(absurd ?? Hpc) assumption ]
754    #_ normalize nodelta >EQi >m_return_bind >EQelem %
755  | #m #Hm whd in ⊢ (??%?); @eqb_elim
756    [ #EQ @⊥ @(absurd ?? Hpc) assumption ]
757    #_ normalize nodelta >EQi >m_return_bind
758    cases (emits_labels ??) in EQelem; normalize nodelta
759    [ whd in ⊢ (??%%→ ??%%); #EQ destruct %]
760    #f #EQelem >(IH … EQelem) [2: /2/ ] %
761  ]
762]
763qed.
764
765lemma step_emit : ∀p,p',prog,st1,st2,l,i.
766fetch p prog (pc … st1) = return i →
767eval_vmstate p p' … prog st1 = return 〈l,st2〉 → 
768emits_labels … i = None ? → ∃x.
769match l in ActionLabel return λ_:ActionLabel.(list CostLabel) with 
770[call_act f c ⇒ [a_call c]
771|ret_act x ⇒
772   match x with [None⇒[]|Some c⇒[a_return_post c]]
773|cost_act x ⇒
774   match x with [None⇒[]|Some c⇒[a_non_functional_label c]]
775] = [x] ∧
776 (mem … 〈x,pc … st2〉 (labels_pc p (instructions … prog) (call_label_fun … prog) (return_label_fun … prog) (in_act … prog) O)).
777#p #p' #prog #st1 #st2 #l #i #EQi whd in match eval_vmstate; normalize nodelta
778>EQi >m_return_bind normalize nodelta cases i in EQi; -i normalize nodelta
779[ #seq * [|#lab]
780| #newpc
781| #cond #newpc #ltrue #lfalse
782| #lin #io #lout
783| #f
784|
785]
786#EQi cases(asm_is_io ???) normalize nodelta
787[1,3,5,7,11,13: whd in ⊢ (??%% → ?); #EQ destruct
788|2,4,8,9,12,14: #H cases(bind_inversion ????? H) -H #x1 * #EQx1
789  [3: cases x1 in EQx1; -x1 #EQx1 normalize nodelta
790  |6: #H cases(bind_inversion ????? H) -H #x2 * #EQx2
791  ]
792]
793whd in ⊢ (??%% → ?); #EQ destruct whd in match emits_labels;
794normalize nodelta #EQ destruct % [2,4,6,8,10,12,14: % try % |*:]
795[1,2,4,5,7: @(labels_pc_ok … EQi) normalize /3 by or_introl,or_intror/ ]
796/2 by labels_pc_return, labels_pc_call/
797qed.
798
799lemma step_non_emit : ∀p,p',prog,st1,st2,l,i,f.
800fetch p prog (pc … st1) = return i →
801eval_vmstate p p' … prog st1 = return 〈l,st2〉 → 
802emits_labels … i = Some ? f →
803match l in ActionLabel return λ_:ActionLabel.(list CostLabel) with 
804[call_act f c ⇒ [a_call c]
805|ret_act x ⇒
806   match x with [None⇒[]|Some c⇒[a_return_post c]]
807|cost_act x ⇒
808   match x with [None⇒[]|Some c⇒[a_non_functional_label c]]
809] = [ ] ∧ pc … st2 = f (pc … st1).
810#p #p' #prog #st1 #st2 #l #i #f #EQi whd in match eval_vmstate; normalize nodelta
811>EQi >m_return_bind normalize nodelta cases i in EQi; -i normalize nodelta
812[ #seq * [|#lab]
813| #newpc
814| #cond #newpc #ltrue #lfalse
815| #lin #io #lout
816| #f
817|
818]
819#EQi cases(asm_is_io ???) normalize nodelta
820[1,3,5,7,11,13: whd in ⊢ (??%% → ?); #EQ destruct
821|2,4,8,9,12,14: #H cases(bind_inversion ????? H) -H #x1 * #EQx1
822  [3: cases x1 in EQx1; -x1 #EQx1 normalize nodelta
823  |6: #H cases(bind_inversion ????? H) -H #x2 * #EQx2
824  ]
825]
826whd in ⊢ (??%% → ?); #EQ destruct whd in match emits_labels;
827normalize nodelta #EQ destruct /2 by refl, conj/
828qed.
829
830lemma labels_of_trace_are_in_code :
831∀p,p',prog.∀st1,st2 : vm_ass_state p p'.∀t : raw_trace (asm_operational_semantics p p' prog) … st1 st2.
832∀lbl.
833mem … lbl (get_costlabels_of_trace … t) →
834mem … lbl (map … \fst … (labels_pc … (instructions p prog) (call_label_fun … prog) (return_label_fun … prog) (in_act … prog) O)).
835#p #p' #prog #st1 #st2 #t elim t
836[ #st #lbl *
837| #st1 #st2 #st3 #l whd in ⊢ (% → ?);
838  cases st1 -st1 normalize nodelta [2: * |3: #st1]
839  cases st2 -st2 normalize nodelta [1,4,5: * |2: * #HN1 #HN2 >HN2 -HN2 |6: #st2 * #HN1 #HN2 >HN2 -HN2 |3: #st2 ]
840  [3: * #H1 #H2 #tl #IH #lbl whd in match (get_costlabels_of_trace ????);
841      #H cases(mem_append ???? H) -H [2: #H @IH //]
842      lapply(vm_step_to_eval … H2) whd in match eval_vmstate;
843      normalize nodelta #H cases(bind_inversion ????? H) -H #i * #EQi #_
844      inversion(emits_labels … i)
845      [ #EQemit cases(step_emit … (vm_step_to_eval … H2)) // #x * #EQ1 #EQ2 >EQ1 *
846        [2: *] #EQ destruct cases(map_mem … \fst … EQ2) #y * #H3 #EQ destruct //
847      | #f #EQemit >(proj1 … (step_non_emit … EQi (vm_step_to_eval … H2) … EQemit))
848        *
849      ]
850  |*: #tl #IH #lbl whd in match (get_costlabels_of_trace ????); * // @IH
851]
852qed.
853
854let rec dependent_map (A,B : Type[0]) (l : list A) (f : ∀a : A.mem … a l → B) on l : list B ≝
855(match l return λx.l=x → ? with
856[ nil ⇒ λ_.nil ?
857| cons x xs ⇒ λprf.(f x ?) :: dependent_map A B xs (λx,prf1.f x ?)
858])(refl …).
859[ >prf %% | >prf %2 assumption]
860qed.
861
862lemma dependent_map_append : ∀A,B,l1,l2,f.
863dependent_map A B (l1 @ l2) (λa,prf.f a prf) =
864(dependent_map A B l1 (λa,prf.f a ?)) @ (dependent_map A B l2 (λa,prf.f a ?)).
865[2: @hide_prf /2/ | 3: @hide_prf /2/]
866#A #B #l1 elim l1 normalize /2/
867qed.
868
869lemma rewrite_in_dependent_map : ∀A,B,l1,l2,f.
870        ∀EQ:l1 = l2.
871         dependent_map A B l1 (λa,prf.f a prf) =
872         dependent_map A B l2 (λa,prf.f a ?).
873[2: >EQ // | #A #B #l1 #l2 #f #EQ >EQ in f; #f % ]
874qed.
875
876definition get_pc : ∀p,p'.vm_ass_state p p' → ℕ → option ℕ ≝
877λp,p',st,endmain.match st with
878[ STATE s ⇒ Some ? (pc … s)
879| INITIAL ⇒ None ?
880| FINAL ⇒ Some ? endmain
881].
882
883
884lemma tbase_tind_append : ∀S : abstract_status.∀st1,st2,st3.∀t : raw_trace … st1 st2.
885∀l,prf.∀t'.
886t_base … st1 = t @ t_ind S st2 st3 st1 l prf t' → False.
887#S #st1 #st2 #st3 *
888[ #st #l #prf  #t' normalize #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct ]
889#s1 #s2 #s3 #l1 #prf1 #t2 #l2 #prf2 #t3 normalize #EQ lapply(eq_to_jmeq ??? EQ) -EQ #EQ destruct
890qed.
891
892let rec chop (A : Type[0]) (l : list A) on l : list A ≝
893match l with
894[ nil ⇒ nil ?
895| cons x xs ⇒ match xs with [ nil ⇒ nil ? | cons _ _ ⇒ x :: chop … xs]
896].
897
898lemma chop_append_singleton : ∀A : Type[0].∀x : A.∀l : list A.chop ? (l@ [x]) = l.
899#A #x #l elim l normalize // #y * normalize //
900qed.
901
902lemma chop_mem : ∀A : Type[0].∀x : A.∀l : list A. mem … x (chop ? l) → mem … x l.
903#A #x #l elim l [*] #y * [ normalize /2/] #z #zs #IH * [/2/] /3/
904qed.
905
906definition actionlabel_to_costlabel : ActionLabel → list CostLabel ≝
907λa.match a with
908[ call_act f l ⇒ [a_call l]
909| ret_act opt_l ⇒ match opt_l with [None ⇒ [ ] | Some l ⇒ [a_return_post l]]
910| cost_act opt_l ⇒ match opt_l with [None ⇒ [ ] | Some l ⇒ [a_non_functional_label l]]
911].
912
913lemma get_cost_label_of_trace_tind : ∀S : abstract_status.
914∀st1,st2,st3 : S.∀l,prf,t.
915get_costlabels_of_trace … (t_ind … st1 st2 st3 l prf t) =
916actionlabel_to_costlabel l @ get_costlabels_of_trace … t.
917#S #st1 #st2 #st3 * // qed.
918
919lemma actionlabel_ok :
920 ∀l : ActionLabel.
921  is_costlabelled_act l → ∃c.actionlabel_to_costlabel l = [c].
922* /2/ * /2/ *
923qed.
924
925lemma i_act_in_map :  ∀p,prog,iact,l1,l2.
926mem ? 〈a_non_functional_label iact,O〉 (labels_pc p prog l1 l2 iact O).
927#p #instr #iact #l1 #l2 generalize in match O in ⊢ (???%); elim instr
928[ normalize /2/] #i #xs #IH #m whd in match (labels_pc ???);
929@mem_append_l2 @IH
930qed.
931
932coercion big_op : ∀M:monoid. ∀l: list M. M ≝ big_op on _l: list ? to ?.
933
934lemma static_dynamic_inv :
935(* Given an assembly program *)
936∀p,p',prog.
937
938(* Given an abstraction interpretation framework for the program *)
939∀R: asm_abstract_interpretation p p' prog.
940
941(* If the static analysis does not fail *)
942∀mT,map1. ∀EQmap : static_analisys … (instr_map … R) mT prog = return map1.
943
944(* For every pre_measurable, terminated trace *)
945∀st1,st2. ∀t: raw_trace (asm_operational_semantics … prog) … st1 st2.
946 terminated_trace … t → pre_measurable_trace … t →
947
948(* Let labels be the costlabels observed in the trace (last one excluded) *)
949let labels ≝ chop … (get_costlabels_of_trace …  t) in
950
951(* Let k be the statically computed abstract action of the prefix of the trace
952   up to the first label *)
953∀k.block_cost p prog … (instr_map … R) (get_pc … st1 (endmain … prog)) (S (|(instructions … prog)|)) = return k →
954
955(* Let abs_actions be the list of statically computed abstract actions
956   associated to each label in labels. *)
957∀abs_actions.
958 abs_actions =
959  dependent_map … labels (λlbl,prf.(opt_safe … (get_map … map1 lbl) …)) →
960
961(* Given an abstract state in relation with the first state of the trace *)
962∀a1.R st1 a1 →
963
964(* The final state of the trace is in relation with the one obtained by
965   first applying k to a1, and then applying every semantics in abs_trace. *)
966R st2 (〚abs_actions〛 (〚k〛 a1)).
967
968[2: @hide_prf
969    cases(mem_map ????? (labels_of_trace_are_in_code … (chop_mem … prf))) *
970    #lbl' #pc * #Hmem #EQ destruct   
971    >(proj1 … (static_analisys_ok … EQmap … Hmem))
972    @(proj2 … (static_analisys_ok … EQmap … Hmem))
973]
974#p #p' #prog #R #mT #map1 #EQmap #st1 #st2 #t * #st3 * #t1 *
975#l1 * #prf1 * #EQ destruct #Hlabelled
976>(rewrite_in_dependent_map ??? (get_costlabels_of_trace … t1))
977[2: >get_cost_label_append  >get_cost_label_of_trace_tind >append_nil cases(actionlabel_ok … Hlabelled)
978   #c #EQc >EQc // ]
979lapply Hlabelled lapply prf1 -prf1 lapply l1 -l1 elim t1 -st3
980[ * [3: #st] #l #prf #H1 #_ #k whd in ⊢ (??%? → ?);
981  [3: cases prf
982  |2: whd in ⊢ (??%% → ?); #EQ destruct #labels whd in ⊢ (??%% → ?); #EQlabels
983      #a1 #rel_fin
984      lapply(instr_map_ok … R … prf … (refl …) rel_fin) [ %|] cases st2 in prf; -st2 [3: #st2] *
985      #EQpc #EQ destruct #H >act_neutral >act_neutral normalize in H;
986      <(act_neutral … (act_abs …) a1) @H
987  | @eqb_elim normalize nodelta
988    [ #EQpc whd in ⊢(??%% → ?); #EQ destruct #labels whd in ⊢ (??%% → ?);
989      #EQ destruct #a1 #good_st_a1>act_neutral >act_neutral whd in prf; cases st2 in prf; -st2 [3: #st2]
990      normalize nodelta * >EQpc @eqb_elim [2,4: * #ABS @⊥ @ABS %] #_ #EQ destruct
991      #EQ destruct whd in EQc : (??%%); destruct
992      lapply(instr_map_ok … R … (refl …) good_st_a1)
993      [5: @(FINAL …)
994      |2: whd % [2: % | // ]
995      | whd whd in ⊢ (??%%); @eqb_elim [2: * #ABS @⊥ @ABS assumption | #_ % ]
996      |3,4: skip]
997      whd in ⊢ (% → ?); >act_neutral #H @H
998  | #Hpc lapply prf whd in ⊢ (% → ?); cases st2 in prf; -st2 [3: #st2] #prf
999    normalize nodelta [2:* |3: * #ABS @⊥ lapply ABS -ABS @eqb_elim
1000    [#EQ #_ @(absurd ? EQ Hpc) | #_ #EQ destruct ] ] * #INUTILE #H4
1001    #H cases(bind_inversion ????? H) -H *
1002    [ #seq * [|#lbl1]
1003    | #newpc
1004    | #cond #newpc #ltrue #lfalse
1005    | #lin #io #lout
1006    | #f
1007    |
1008    ]
1009    * #EQfetch  lapply(vm_step_to_eval … H4) whd in match eval_vmstate in ⊢ (% → ?);
1010    normalize nodelta >EQfetch >m_return_bind normalize nodelta
1011    cases(asm_is_io ??) normalize nodelta
1012    [1,3,5,7,11,13: whd in ⊢ (??%% → ?); #EQ destruct
1013    |2,4,8,9,12,14: #H cases(bind_inversion ????? H) -H #x * #_
1014      [3: cases x normalize nodelta
1015      |6: #H cases(bind_inversion ????? H) -H #y * #_
1016      ]
1017    ]
1018    whd in ⊢ (??%% → ?); #EQ destruct [4,8: cases H1 ]
1019    >m_return_bind whd in ⊢ (??%% → ?); #EQ destruct
1020    whd in match (dependent_map ????); #costs #EQ destruct #a1 #good_st_a1
1021    >neutral_r >act_neutral
1022    lapply(instr_map_ok R … (refl …) good_st_a1)
1023    [1,7,13,19,25,31,37: whd in ⊢ (??%%); @eqb_elim normalize nodelta
1024      [1,3,5,7,9,11,13: #EQ cases(absurd ? EQ Hpc) ] #_ whd in match fetch_state;
1025        normalize nodelta
1026        [ >EQfetch in ⊢ (??%?); | >EQfetch in ⊢ (??%?); | >EQfetch in ⊢ (??%?);
1027        | >EQfetch in ⊢ (??%?); | >EQfetch in ⊢ (??%?); | >EQfetch in ⊢ (??%?);
1028        | >EQfetch in ⊢ (??%?); ] %
1029      |3,9,15,21,27,33,39: skip |*: try assumption // ]]]
1030| -st1 * [3: #st1] #st3 #st4 #l [3: *] cases st3 -st3
1031  [1,2,4,5: * #H1 #H2 #tl #_ #l1 #exe @⊥ lapply tl -tl lapply(refl ? (FINAL p p'))
1032    generalize in match (FINAL ??) in ⊢ (??%? → %); #st5 #EQst5 #tl lapply EQst5
1033    lapply exe lapply st2 -st2 -EQst5 elim tl
1034    [ #st #st5 #ABS #EQ destruct cases ABS
1035    | #s1 #s2 #s3 #l2 #H3 #tl1 #IH #s4 #_ #EQ destruct cases H3
1036    ]
1037  ]
1038  #st3 #exe_st1_st3 #tl #IH #l1 #exe_st4_st2 #l1_lab #pre_meas #k whd in ⊢ (??%? → ?);
1039  >rewrite_in_dependent_map [2,5: @get_cost_label_of_trace_tind |3,6: ]
1040  >dependent_map_append
1041  [ @eqb_elim [ #ABS @⊥ cases exe_st1_st3 >ABS @eqb_elim [ #_ #EQ destruct | * #ABS1 @⊥ @ABS1 %] ]
1042    #Hpc normalize nodelta #H cases(bind_inversion ????? H) -H #i * #EQi
1043    inversion(emits_labels ??)
1044    [ #EQemits whd in ⊢ (??%% → ?); #EQ destruct #labels
1045      cases(step_emit … EQi … EQemits)
1046      [4: cases exe_st1_st3  #EQ #H @(vm_step_to_eval … H) |2,3:] #c * #EQc #Hc
1047      whd in match actionlabel_to_costlabel; normalize nodelta
1048      >rewrite_in_dependent_map [2: @EQc |3:] whd in match (dependent_map ????);
1049      @opt_safe_elim #k_c #EQk_c whd in match (dependent_map ????); letin ih_labels ≝ (dependent_map ????)
1050      #EQ destruct #a1 #good_a1 >big_op_associative >act_op @IH
1051    | #f #EQemits normalize nodelta #H cases(bind_inversion ????? H) -H #k' * #EQk' whd in ⊢ (??%% → ?);
1052      #EQ destruct(EQ) #labels cases(step_non_emit … EQi… EQemits)
1053      [4: cases exe_st1_st3  #EQ #H @(vm_step_to_eval … H) |2,3:] #EQl #EQpc
1054      >(rewrite_in_dependent_map ??? []) [2: assumption] whd in match (dependent_map ????);
1055      #EQlabels #a1 #good_a1 >act_op @IH
1056  ]
1057  try //
1058  [2: cases(static_analisys_ok … c … (pc … st3) … EQmap) // #EQ #_ <EQ whd in match (big_op ??);
1059      >neutral_l assumption
1060   |3,6: [ >neutral_r] lapply(instr_map_ok R … (refl …) good_a1)
1061       [1,7: whd in ⊢ (??%?); @eqb_elim
1062         [1,3: #ABS cases(absurd ? ABS Hpc) ] #_ normalize nodelta whd in match fetch_state;
1063         normalize nodelta [ >EQi in ⊢ (??%?); | >EQi in ⊢ (??%?); ] %
1064       |2,8: assumption
1065       |*:
1066       ]
1067       normalize in ⊢ (% → ?); #H @H
1068  |5: whd in match get_pc; normalize nodelta >EQpc >(monotonicity_of_block_cost … EQk') //
1069  |*:  inversion pre_meas in ⊢ ?;
1070    [1,6: #st #c #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in EQ3 : (??%??); destruct
1071    |2,7: #s1 #s2 #s3 #lbl #exe #tl1 #s1_noio * #opt_l #EQ destruct #pre_tl1 #_
1072      #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in EQ3 : (??%??); destruct
1073    |3,8: #s1 #s2 #s3 #lbl #s1_noio #exe #tl1 * #lbl1 #EQ destruct #pre_tl1 #_
1074      #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in EQ3 : (??%??); destruct
1075    |4,9: #s1 #s2 #s3 #lbl #exe #tl1 #s1_noio * #f * #lbl1 #EQ destruct
1076      * #pre_tl1 #_ #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in EQ3 : (??%??); destruct
1077    |5,10: #s1 #s2 #s3 #s4 #s5 #l1 #l2 #exe1 #t1 #t2 #exe2 #noio1 #noio2 #H *
1078    ] //
1079  ]
1080 | whd in ⊢ (??%% → ?); #EQ destruct cases exe_st1_st3 #EQpc_st3 #EQ destruct
1081   #labels whd in match actionlabel_to_costlabel; normalize nodelta
1082   whd in match (dependent_map ????); @opt_safe_elim #k_c #EQk_c letin ih_labels ≝ (dependent_map ????)
1083   change with ([?]@?) in match ([?]@?); #EQ #a1 #good_a1 destruct
1084   >big_op_associative >act_op @IH try //
1085   [  inversion pre_meas in ⊢ ?;
1086     [1,6: #st #c #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in EQ3 : (??%??); destruct
1087     |2,7: #s1 #s2 #s3 #lbl #exe #tl1 #s1_noio * #opt_l #EQ destruct #pre_tl1 #_
1088      #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in EQ3 : (??%??); destruct
1089     |3,8: #s1 #s2 #s3 #lbl #s1_noio #exe #tl1 * #lbl1 #EQ destruct #pre_tl1 #_
1090      #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in EQ3 : (??%??); destruct
1091     |4,9: #s1 #s2 #s3 #lbl #exe #tl1 #s1_noio * #f * #lbl1 #EQ destruct
1092      * #pre_tl1 #_ #EQ1 #EQ2 #EQ3 #EQ4 destruct whd in EQ3 : (??%??); destruct
1093     |5,10: #s1 #s2 #s3 #s4 #s5 #l1 #l2 #exe1 #t1 #t2 #exe2 #noio1 #noio2 #H *
1094     ] //
1095   |  cases(static_analisys_ok … (in_act … prog) … (pc … st3) … EQmap)
1096      [2: lapply EQpc_st3 @eqb_elim [2: #_ #EQ destruct] #EQ #_ >EQ @i_act_in_map ]
1097      #EQ #_ <EQ whd in match (big_op ??); >neutral_l assumption
1098   |  lapply(instr_map_ok R … (refl …) good_a1)
1099     [1: % | assumption |*:] normalize in ⊢ (% → ?); #H @H
1100   ]
1101]
1102qed.
1103
1104definition measurable :
1105 ∀S: abstract_status. ∀si,s1,s3,sn : S. raw_trace … si sn → Prop ≝
1106λS,si,s1,s3,sn,t.
1107 pre_measurable_trace … t ∧
1108 ∃s0,s2:S. ∃ti0 : raw_trace … si s0.∃t12 : raw_trace … s1 s2.∃t3n : raw_trace … s3 sn.
1109 ∃l1,l2,prf1,prf2.
1110  t = ti0 @  t_ind ? s0 s1 sn l1 prf1 … (t12 @ (t_ind ? s2 s3 sn l2 prf2 … t3n))
1111 ∧ is_costlabelled_act l1 ∧ is_costlabelled_act l2 ∧ silent_trace … ti0 ∧ silent_trace … t3n.
1112 
1113lemma chop_cons : ∀A : Type[0].∀x : A.∀xs: list A. xs ≠ [ ] → chop … (x :: xs) = x :: (chop … xs).
1114#A #x * [ #H cases(absurd ?? H) % ] // qed.
1115 
1116(*
1117lemma measurable_terminated:
1118 ∀S,s1,s2,s4,t. measurable S s1 s2 s4 t → terminated_trace … t.
1119 #S #s1 #s2 #s4 #t * #_ * #s3 * #t' * #l1 * #l2 * #prf1 * #prf2 ** #EQ destruct
1120 /6 width=6 by ex_intro, conj/
1121qed. *)
1122
1123lemma execute_mem_label_pc :∀p,p',prog.∀st0,st1 :vm_state p p'.∀l1,c1.
1124actionlabel_to_costlabel l1 = [c1] →
1125vmstep p p' prog l1 st0 st1 →
1126 mem … 〈c1,pc p p' st1〉
1127  (labels_pc … (instructions … prog) (call_label_fun … prog)
1128   (return_label_fun … prog) (in_act … prog) O).
1129#p #p' #prog #st0 #st1 #l1 #c1 #EQc1 #H lapply(vm_step_to_eval … H) -H
1130#H cases(bind_inversion ????? H); #i * #EQi #Hi cases(step_emit … EQi H)
1131[ #c2 whd in match actionlabel_to_costlabel in EQc1; normalize nodelta in EQc1;
1132  >EQc1 * #EQ destruct // ]
1133cases i in Hi;       
1134[ #seq * [|#lbl1]
1135| #newpc
1136| #cond #newpc #ltrue #lfalse
1137| #lin #io #lout
1138| #f
1139|
1140]
1141normalize nodelta cases(asm_is_io ??) normalize nodelta
1142[1,3,5,7,11,13: whd in ⊢ (??%% → ?); #EQ destruct
1143|2,4,8,9,12,14: #H cases(bind_inversion ????? H) -H #x * #_
1144  [3: cases x normalize nodelta
1145  |6: #H cases(bind_inversion ????? H) -H #y *  #_
1146  ]
1147]
1148whd in ⊢ (??%% → ?); #EQ destruct try % normalize in EQc1; destruct
1149qed.
1150
1151theorem static_dynamic :
1152(* Given an assembly program *)
1153∀p,p',prog.
1154
1155(* Given an abstraction interpretation framework for the program *)
1156∀R: asm_abstract_interpretation p p' prog.
1157
1158(* If the static analysis does not fail *)
1159∀mT,map1. ∀EQmap : static_analisys … (instr_map … R) mT prog = return map1.
1160
1161(* For every measurable trace whose second state is st1 or, equivalently,
1162   whose first state after the initial labelled transition is st1 *)
1163∀si,s1,s2,sn. ∀t: raw_trace (asm_operational_semantics … prog) … si sn.
1164 measurable … s1 s2 … t →
1165
1166(* Let labels be the costlabels observed in the trace (last one excluded) *)
1167let labels ≝ chop … (get_costlabels_of_trace …  t) in
1168
1169(* Let abs_actions be the list of statically computed abstract actions
1170   associated to each label in labels. *)
1171∀abs_actions.
1172 abs_actions =
1173  dependent_map … labels (λlbl,prf.(opt_safe … (get_map … map1 lbl) …)) →
1174
1175(* Given an abstract state in relation with the second state of the trace *)
1176∀a1.R s1 a1 →
1177
1178(* The final state of the trace is in relation with the one obtained by
1179   applying every semantics in abs_trace. *)
1180R s2 (〚abs_actions〛 a1).
1181[2: @hide_prf
1182    cases(mem_map ????? (labels_of_trace_are_in_code … (chop_mem … prf))) *
1183    #lbl' #pc * #Hmem #EQ destruct   
1184    >(proj1 … (static_analisys_ok … EQmap … Hmem))
1185    @(proj2 … (static_analisys_ok … EQmap … Hmem))
1186]
1187#p #p' #prog #R #mT #map1 #EQmap #si #s1 #s2 #sn #t #measurable
1188cases measurable #premeas * #s0 * #s3 * #ti0 * #t12 * #t3n *  #l1 * #l2 * #prf1 * #prf2
1189**** #EQ destruct #Hl1 #Hl2 #silent_ti0 #silent_t3n #acts cases(actionlabel_ok … Hl1)
1190#c1 #EQc1 >rewrite_in_dependent_map
1191[2: >get_cost_label_append in ⊢ (??%?); >(get_cost_label_silent_is_empty … silent_ti0) in ⊢ (??%?);
1192     >get_cost_label_of_trace_tind in ⊢ (??%?); >get_cost_label_append in ⊢ (??%?);
1193      >get_cost_label_of_trace_tind in ⊢ (??%?);
1194     >(get_cost_label_silent_is_empty … silent_t3n) in ⊢ (??%?);
1195     >append_nil in ⊢ (??%?); >EQc1 in ⊢ (??%?); whd in ⊢ (??(??%)?);
1196     whd in ⊢ (??(??(???%))?); >chop_cons in ⊢ (??%?); [2: cases daemon] % |3:
1197 ]
1198 whd in ⊢ (???% → ?); @opt_safe_elim #act #EQact #EQacts2 >EQacts2
1199 >(big_op_associative ? [?]) #a1 >act_op #HR
1200letin actsl ≝ (dependent_map ????);
1201@(static_dynamic_inv … EQmap … (t12 @ t_ind … prf2 (t_base …)) … actsl … HR)
1202  [ /6 width=6 by conj, ex_intro/
1203  | cases daemon (* true *)
1204  | cases s1 in prf1 t12; [3: #st1] cases s0 [3,6,9: #st0] *
1205    [2: #H #EQ lapply(refl ? (FINAL p p')) generalize in match (FINAL p p') in ⊢ (??%? → %);
1206        #st' #EQst' #tr lapply prf2 lapply EQst' -EQst' cases tr
1207        [ #st'' #EQ2 >EQ2 *
1208        | #st'' #st''' #st'''' #l3 #ABS #_ #EQ2 >EQ2 in ABS; *
1209        ]
1210    | #H1 #H2 #_ whd in match get_pc; normalize nodelta
1211      <(proj1 ?? (static_analisys_ok … EQmap …))
1212      [ normalize in ⊢ (??%%); >neutral_l @EQact
1213      |
1214      | @execute_mem_label_pc //
1215      ]
1216    | #H #EQ destruct #_ whd in match get_pc; normalize nodelta
1217      <(proj1 ?? (static_analisys_ok … EQmap …))
1218      [ normalize in ⊢ (??%%); >neutral_l @EQact
1219      |
1220      |  whd in EQc1 : (??%%); destruct lapply H @eqb_elim [2: #_ #EQ destruct] #EQ >EQ #_ @i_act_in_map
1221      ]
1222    ]
1223  | >rewrite_in_dependent_map
1224       [2: >get_cost_label_append in ⊢ (??%?); >get_cost_label_of_trace_tind in ⊢ (??%?);
1225       >append_nil in ⊢ (??%?); % |3:]
1226       %
1227   ]
1228qed.
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