source: src/ASM/CostsProof.ma @ 1620

Last change on this file since 1620 was 1620, checked in by sacerdot, 8 years ago

One of the mutual cases of the open proof is practically finished.

File size: 30.6 KB
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1include "ASM/ASMCosts.ma".
2include "ASM/WellLabeled.ma".
3include "ASM/Status.ma".
4include "common/StructuredTraces.ma".
5include "arithmetics/bigops.ma".
6include alias "arithmetics/nat.ma".
7include alias "basics/logic.ma".
8
9let rec compute_max_trace_label_label_cost
10  (cost_labels: BitVectorTrie costlabel 16)
11   (trace_ends_flag: trace_ends_with_ret)
12    (start_status: Status) (final_status: Status)
13      (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag
14        start_status final_status) on the_trace: nat ≝
15  match the_trace with
16  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
17      compute_max_trace_any_label_cost … given_trace
18  ]
19and compute_max_trace_any_label_cost
20  (cost_labels: BitVectorTrie costlabel 16)
21  (trace_ends_flag: trace_ends_with_ret)
22   (start_status: Status) (final_status: Status)
23     (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status)
24       on the_trace: nat ≝
25  match the_trace with
26  [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status
27  | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status
28  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
29     _ _ _ call_trace final_trace ⇒
30      let current_instruction_cost ≝ current_instruction_cost pre_fun_call in
31      let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in
32      let final_trace_cost ≝ compute_max_trace_any_label_cost cost_labels end_flag … final_trace in
33        call_trace_cost + current_instruction_cost + final_trace_cost
34  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
35      let current_instruction_cost ≝ current_instruction_cost status_pre in
36      let tail_trace_cost ≝
37       compute_max_trace_any_label_cost cost_labels end_flag
38        status_init status_end tail_trace
39      in
40        current_instruction_cost + tail_trace_cost
41  ]
42and compute_max_trace_label_return_cost
43  (cost_labels: BitVectorTrie costlabel 16)
44  (start_status: Status) (final_status: Status)
45    (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status)
46      on the_trace: nat ≝
47  match the_trace with
48  [ tlr_base before after trace_to_lift ⇒ compute_max_trace_label_label_cost … trace_to_lift
49  | tlr_step initial labelled final labelled_trace ret_trace ⇒
50      let labelled_cost ≝ compute_max_trace_label_label_cost … labelled_trace in
51      let return_cost ≝ compute_max_trace_label_return_cost … ret_trace in
52        labelled_cost + return_cost
53  ].
54
55(*Useless now?
56(* To be moved *)
57lemma pred_minus_1:
58  ∀m, n: nat.
59  ∀proof: n < m.
60    pred (m - n) = m - n - 1.
61  #m #n
62  cases m
63  [ #proof
64    cases(lt_to_not_zero … proof)
65  | #m' #proof
66    normalize in ⊢ (???%);
67    cases n
68    [ normalize //
69    | #n' normalize
70      cases(m' - n')
71      [ %
72      | #Sm_n'
73        normalize //
74      ]
75    ]
76  ]
77qed.
78
79lemma succ_m_plus_one:
80  ∀m: nat.
81    S m = m + 1.
82 //
83qed.*)
84
85include alias "arithmetics/nat.ma".
86
87(*
88lemma minus_m_n_minus_minus_plus_1:
89  ∀m, n: nat.
90  ∀proof: n < m.
91    m - n = (m - n - 1) + 1.
92 /3 by lt_plus_to_minus_r, plus_minus/
93qed.
94
95lemma plus_minus_rearrangement_1:
96  ∀m, n, o: nat.
97  ∀proof_n_m: n ≤ m.
98  ∀proof_m_o: m ≤ o.
99    (m - n) + (o - m) = o - n.
100  #m #n #o #H1 #H2
101  lapply (minus_to_plus … H1 (refl …)) #K1 >K1
102  lapply (minus_to_plus … H2 (refl …)) #K2 >K2
103  /2 by plus_minus/
104qed.
105
106lemma plus_minus_rearrangement_2:
107  ∀m, n, o: nat. n ≤ m → o ≤ n →
108    (m - n) + (n - o) = m - o.
109 /2 by plus_minus_rearrangement_1/
110qed.
111
112lemma m_le_plus_n_m:
113  ∀m, n: nat.
114    m ≤ n + m.
115  #m #n //
116qed.
117
118lemma n_plus_m_le_o_to_m_le_o:
119  ∀m, n, o: nat.
120    n + m ≤ o → m ≤ o.
121  #m #n #o #assm /2 by le_plus_b/
122qed.
123
124lemma m_minus_n_plus_o_m_minus_n_minus_o:
125  ∀m, n, o: nat.
126    m - (n + o) = m - n - o.
127  #m #n #o /2 by /
128qed.
129*)
130
131let rec compute_max_trace_label_label_cost_is_ok
132  (cost_labels: BitVectorTrie costlabel 16)
133   (trace_ends_flag: trace_ends_with_ret)
134    (start_status: Status) (final_status: Status)
135      (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag
136        start_status final_status) on the_trace:
137          clock … final_status = (compute_max_trace_label_label_cost cost_labels trace_ends_flag start_status final_status the_trace) + (clock … start_status) ≝ ?
138and compute_max_trace_any_label_cost_is_ok
139  (cost_labels: BitVectorTrie costlabel 16)
140  (trace_ends_flag: trace_ends_with_ret)
141   (start_status: Status) (final_status: Status)
142     (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status)
143       on the_trace:
144         clock … final_status = (compute_max_trace_any_label_cost cost_labels trace_ends_flag start_status final_status the_trace) + (clock … start_status) ≝ ?
145and compute_max_trace_label_return_cost_is_ok
146  (cost_labels: BitVectorTrie costlabel 16)
147  (start_status: Status) (final_status: Status)
148    (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status)
149      on the_trace:
150        clock … final_status = (compute_max_trace_label_return_cost cost_labels start_status final_status the_trace) + clock … start_status ≝ ?.
151  [1:
152    cases the_trace
153    #ends_flag #start_status #end_status #any_label_trace #is_costed
154    normalize @compute_max_trace_any_label_cost_is_ok
155  |2:
156    cases the_trace
157    [1,2:
158      #start_status #final_status #is_next #is_not_return try (#is_costed)
159      change with (current_instruction_cost start_status) in ⊢ (???(?%?));
160      cases(is_next) @execute_1_ok
161    |3:
162      #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
163      #status_final #is_next #is_call #is_after_return #call_trace #final_trace
164      change with (
165      let current_instruction_cost ≝ current_instruction_cost status_pre_fun_call in
166      let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in
167      let final_trace_cost ≝ compute_max_trace_any_label_cost cost_labels end_flag … final_trace in
168        call_trace_cost + current_instruction_cost + final_trace_cost) in ⊢ (???(?%?));
169      normalize nodelta;
170      >(compute_max_trace_any_label_cost_is_ok cost_labels end_flag status_after_fun_call
171          status_final final_trace)
172      >(compute_max_trace_label_return_cost_is_ok cost_labels status_start_fun_call
173        status_after_fun_call call_trace)
174      cases(is_next) in match (clock … status_start_fun_call);
175      >(execute_1_ok status_pre_fun_call)
176      <associative_plus in ⊢ (??%?);
177      <commutative_plus in match (
178        compute_max_trace_any_label_cost cost_labels end_flag status_after_fun_call status_final final_trace
179          + compute_max_trace_label_return_cost cost_labels status_start_fun_call status_after_fun_call call_trace);
180      >associative_plus in ⊢ (??%?);
181      <commutative_plus in match (
182        compute_max_trace_any_label_cost cost_labels end_flag status_after_fun_call status_final final_trace
183          + (current_instruction_cost status_pre_fun_call
184              + clock (BitVectorTrie Byte 16) status_pre_fun_call));
185      >associative_plus in ⊢ (??%?);
186      <commutative_plus in match (
187         clock (BitVectorTrie Byte 16) status_pre_fun_call
188           + compute_max_trace_any_label_cost cost_labels end_flag status_after_fun_call status_final final_trace);
189      <(associative_plus (
190         (compute_max_trace_label_return_cost cost_labels status_start_fun_call status_after_fun_call call_trace)))
191      <associative_plus in ⊢ (??%?); %
192    |4:
193      #end_flag #status_pre #status_init #status_end #is_next
194      #trace_any_label #is_other #is_not_costed
195      change with (
196      let current_instruction_cost ≝ current_instruction_cost status_pre in
197      let tail_trace_cost ≝
198       compute_max_trace_any_label_cost cost_labels end_flag
199        status_init status_end trace_any_label
200      in
201        current_instruction_cost + tail_trace_cost) in ⊢ (???(?%?));
202      normalize nodelta;
203      >(compute_max_trace_any_label_cost_is_ok cost_labels end_flag
204         status_init status_end trace_any_label)
205      cases(is_next) in match (clock … status_init);
206      >(execute_1_ok status_pre)
207      >commutative_plus >associative_plus >associative_plus @eq_f
208      @commutative_plus
209    ]
210  |3:
211    cases the_trace
212    [1:
213      #status_before #status_after #trace_to_lift
214      normalize @compute_max_trace_label_label_cost_is_ok
215    |2:
216      #status_initial #status_labelled #status_final #labelled_trace #ret_trace
217      normalize
218      >(compute_max_trace_label_return_cost_is_ok cost_labels status_labelled status_final ret_trace);
219      >(compute_max_trace_label_label_cost_is_ok cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace);
220      <associative_plus in ⊢ (??%?);
221      >commutative_plus in match (
222        compute_max_trace_label_return_cost cost_labels status_labelled status_final ret_trace
223          + compute_max_trace_label_label_cost cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace);
224      %
225    ]
226  ].
227qed.
228
229(* XXX: work below here: *)
230
231let rec compute_paid_trace_any_label
232  (cost_labels: BitVectorTrie costlabel 16)
233  (trace_ends_flag: trace_ends_with_ret)
234   (start_status: Status) (final_status: Status)
235     (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status)
236       on the_trace: nat ≝
237  match the_trace with
238  [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status
239  | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status
240  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
241     _ _ _ call_trace final_trace ⇒
242      let current_instruction_cost ≝ current_instruction_cost pre_fun_call in
243      let final_trace_cost ≝ compute_paid_trace_any_label cost_labels end_flag … final_trace in
244        current_instruction_cost + final_trace_cost
245  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
246      let current_instruction_cost ≝ current_instruction_cost status_pre in
247      let tail_trace_cost ≝
248       compute_paid_trace_any_label cost_labels end_flag
249        status_init status_end tail_trace
250      in
251        current_instruction_cost + tail_trace_cost
252  ].
253
254definition compute_paid_trace_label_label ≝
255 λcost_labels: BitVectorTrie costlabel 16.
256  λtrace_ends_flag: trace_ends_with_ret.
257   λstart_status: Status.
258    λfinal_status: Status.
259     λthe_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag
260      start_status final_status.
261  match the_trace with
262  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
263      compute_paid_trace_any_label … given_trace
264  ].
265
266let rec compute_trace_label_label_cost_using_paid
267  (cost_labels: BitVectorTrie costlabel 16)
268   (trace_ends_flag: trace_ends_with_ret)
269    (start_status: Status) (final_status: Status)
270      (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag
271        start_status final_status) on the_trace: nat ≝
272  match the_trace with
273  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
274      compute_paid_trace_label_label cost_labels … the_trace +
275      compute_trace_any_label_cost_using_paid … given_trace
276  ]
277and compute_trace_any_label_cost_using_paid
278  (cost_labels: BitVectorTrie costlabel 16)
279  (trace_ends_flag: trace_ends_with_ret)
280   (start_status: Status) (final_status: Status)
281     (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status)
282       on the_trace: nat ≝
283  match the_trace with
284  [ tal_base_not_return the_status _ _ _ _ ⇒ 0
285  | tal_base_return the_status _ _ _ ⇒ 0
286  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
287     _ _ _ call_trace final_trace ⇒
288      let call_trace_cost ≝ compute_trace_label_return_cost_using_paid … call_trace in
289      let final_trace_cost ≝ compute_trace_any_label_cost_using_paid cost_labels end_flag … final_trace in
290        call_trace_cost + final_trace_cost
291  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
292     compute_trace_any_label_cost_using_paid cost_labels end_flag
293      status_init status_end tail_trace
294  ]
295and compute_trace_label_return_cost_using_paid
296  (cost_labels: BitVectorTrie costlabel 16)
297  (start_status: Status) (final_status: Status)
298    (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status)
299      on the_trace: nat ≝
300  match the_trace with
301  [ tlr_base before after trace_to_lift ⇒ compute_trace_label_label_cost_using_paid … trace_to_lift
302  | tlr_step initial labelled final labelled_trace ret_trace ⇒
303      let labelled_cost ≝ compute_trace_label_label_cost_using_paid … labelled_trace in
304      let return_cost ≝ compute_trace_label_return_cost_using_paid … ret_trace in
305        labelled_cost + return_cost
306  ].
307
308let rec compute_trace_label_label_cost_using_paid_ok
309  (cost_labels: BitVectorTrie costlabel 16)
310   (trace_ends_flag: trace_ends_with_ret)
311    (start_status: Status) (final_status: Status)
312      (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag
313        start_status final_status) on the_trace:
314 compute_trace_label_label_cost_using_paid cost_labels … the_trace =
315 compute_max_trace_label_label_cost … the_trace ≝ ?
316and compute_trace_any_label_cost_using_paid_ok
317  (cost_labels: BitVectorTrie costlabel 16)
318  (trace_ends_flag: trace_ends_with_ret)
319   (start_status: Status) (final_status: Status)
320     (the_trace: trace_any_label (ASM_abstract_status cost_labels)
321      trace_ends_flag start_status final_status) on the_trace:     
322  compute_paid_trace_any_label cost_labels trace_ends_flag … the_trace
323  +compute_trace_any_label_cost_using_paid cost_labels trace_ends_flag … the_trace
324  =compute_max_trace_any_label_cost cost_labels trace_ends_flag … the_trace ≝ ?
325and compute_trace_label_return_cost_using_paid_ok
326  (cost_labels: BitVectorTrie costlabel 16)
327  (start_status: Status) (final_status: Status)
328    (the_trace: trace_label_return (ASM_abstract_status cost_labels)
329     start_status final_status) on the_trace:
330 compute_trace_label_return_cost_using_paid cost_labels … the_trace =
331 compute_max_trace_label_return_cost cost_labels … the_trace ≝ ?.
332[ cases the_trace #endsf #ss #es #tr #H normalize
333  @compute_trace_any_label_cost_using_paid_ok
334| cases the_trace
335  [ #ss #fs #H1 #H2 #H3 whd in ⊢ (??(?%%)%); <plus_n_O %
336  | #ss #fs #H1 #H2 whd in ⊢ (??(?%%)%); <plus_n_O %
337  | #ef #spfc #ssfc #safc #sf #H1 #H2 #H3 #tr1 #tr2 whd in ⊢ (??(?%%)%);
338    <compute_trace_any_label_cost_using_paid_ok
339    <compute_trace_label_return_cost_using_paid_ok
340    -compute_trace_label_label_cost_using_paid_ok
341    -compute_trace_label_return_cost_using_paid_ok
342    -compute_trace_any_label_cost_using_paid_ok
343    >commutative_plus in ⊢ (???(?%?));
344    >commutative_plus in ⊢ (??(??%)?);
345    >associative_plus >associative_plus in ⊢ (???%); @eq_f2 try %
346    <associative_plus <commutative_plus %
347  | #ef #sp #si #se #H1 #tr #H2 #H3 whd in ⊢ (??(?%%)%); >associative_plus @eq_f2
348    [ % | @compute_trace_any_label_cost_using_paid_ok ]
349  ]
350| cases the_trace
351  [ #sb #sa #tr normalize @compute_trace_label_label_cost_using_paid_ok
352  | #si #sl #sf #tr1 #tr2 normalize @eq_f2
353    [ @compute_trace_label_label_cost_using_paid_ok
354    | @compute_trace_label_return_cost_using_paid_ok ]]]
355qed.
356
357(*
358let rec compute_paid_trace_label_return
359  (cost_labels: BitVectorTrie costlabel 16)
360  (start_status: Status) (final_status: Status)
361    (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status)
362      on the_trace: nat ≝
363 match the_trace with
364  [ tlr_base before after trace_to_lift ⇒ compute_paid_trace_label_label … trace_to_lift
365  | tlr_step initial labelled final labelled_trace ret_trace ⇒
366      let labelled_cost ≝ compute_paid_trace_label_label … labelled_trace in
367      let return_cost ≝ compute_paid_trace_label_return … ret_trace in
368        labelled_cost + return_cost
369  ].
370*)
371
372let rec compute_cost_trace_label_label
373  (cost_labels: BitVectorTrie costlabel 16)
374   (trace_ends_flag: trace_ends_with_ret)
375    (start_status: Status) (final_status: Status)
376      (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag
377        start_status final_status) on the_trace:
378         list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝
379  match the_trace with
380  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
381     let pc ≝ program_counter … initial in
382     let label ≝
383      match lookup_opt … pc cost_labels return λx. match x with [None ⇒ False | Some _ ⇒ True] → costlabel with
384      [ None ⇒ λabs. ⊥
385      | Some l ⇒ λ_. l ] labelled_proof in
386     (mk_Sig … label ?)::compute_cost_trace_any_label … given_trace
387  ]
388and compute_cost_trace_any_label
389  (cost_labels: BitVectorTrie costlabel 16)
390  (trace_ends_flag: trace_ends_with_ret)
391   (start_status: Status) (final_status: Status)
392     (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status)
393       on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝
394  match the_trace with
395  [ tal_base_not_return the_status _ _ _ _ ⇒ []
396  | tal_base_return the_status _ _ _ ⇒ []
397  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
398     _ _ _ call_trace final_trace ⇒
399      let call_cost_trace ≝ compute_cost_trace_label_return … call_trace in
400      let final_cost_trace ≝ compute_cost_trace_any_label cost_labels end_flag … final_trace in
401        call_cost_trace @ final_cost_trace
402  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
403       compute_cost_trace_any_label cost_labels end_flag
404        status_init status_end tail_trace
405  ]
406and compute_cost_trace_label_return
407  (cost_labels: BitVectorTrie costlabel 16)
408  (start_status: Status) (final_status: Status)
409    (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status)
410      on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝
411  match the_trace with
412  [ tlr_base before after trace_to_lift ⇒ compute_cost_trace_label_label … trace_to_lift
413  | tlr_step initial labelled final labelled_trace ret_trace ⇒
414      let labelled_cost ≝ compute_cost_trace_label_label … labelled_trace in
415      let return_cost ≝ compute_cost_trace_label_return … ret_trace in
416        labelled_cost @ return_cost
417  ].
418 [ %{pc} whd in match label;  generalize in match labelled_proof; whd in ⊢ (% → ?);
419  cases (lookup_opt costlabel … pc cost_labels) normalize
420  [ #abs cases abs | // ]
421 | // ]
422qed.
423
424(* ??????????????????????? *)
425axiom block_cost_static_dynamic_ok:
426 ∀cost_labels: BitVectorTrie costlabel 16.
427 ∀trace_ends_flag.
428 ∀start_status: Status.
429 ∀final_status: Status.
430 ∀the_trace:
431  trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag
432   start_status final_status.
433 let mem ≝ code_memory … start_status in
434 let pc ≝ program_counter … start_status in
435 let program_size ≝ 2^16 in
436  block_cost mem cost_labels pc program_size =
437  compute_paid_trace_label_label cost_labels trace_ends_flag
438   start_status final_status the_trace.
439
440(*
441(* To be moved elsewhere*)
442lemma le_S_n_m_to_le_n_m: ∀n,m. S n ≤ m → n ≤ m.
443 #n #m #H change with (pred (S n) ≤ m) @(transitive_le ? (S n)) //
444qed.
445*)
446
447(* This shoudl go in bigops! *)
448theorem bigop_sum_rev: ∀k1,k2,p,B,nil.∀op:Aop B nil.∀f:nat→B.
449 \big[op,nil]_{i<k1+k2|p i} (f i) =
450 op \big[op,nil]_{i<k2|p (i+k1)} (f (i+k1)) \big[op,nil]_{i<k1|p i} (f i).
451 #k1 #k2 #p #B #nil #op #f >bigop_sum >commutative_plus @same_bigop #i @leb_elim normalize
452 [2,4: //
453 | #H1 #H2 <plus_minus_m_m //
454 | #H1 #H2 #H3 <plus_minus_m_m //]
455qed.
456
457(* This is taken by sigma_pi.ma that does not compile now *)
458definition natAop ≝ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n))
459   (λa,b,c.sym_eq ??? (associative_plus a b c)).
460
461definition natACop ≝ mk_ACop nat 0 natAop commutative_plus.
462
463definition natDop ≝ mk_Dop nat 0 natACop times (λn.(sym_eq ??? (times_n_O n)))
464   distributive_times_plus.
465
466unification hint  0 ≔ ;
467   S ≟ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n))
468   (λa,b,c.sym_eq ??? (associative_plus a b c))
469(* ---------------------------------------- *) ⊢
470   plus ≡ op ? ? S.
471
472unification hint  0 ≔ ;
473   S ≟ mk_ACop nat 0 (mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n))
474   (λa,b,c.sym_eq ??? (associative_plus a b c))) commutative_plus
475(* ---------------------------------------- *) ⊢
476   plus ≡ op ? ? S.
477   
478unification hint  0 ≔ ;
479   S ≟ natDop
480(* ---------------------------------------- *) ⊢
481   plus ≡ sum ? ? S.
482
483unification hint  0 ≔ ;
484   S ≟ natDop
485(* ---------------------------------------- *) ⊢
486   times ≡ prod ? ? S.
487
488notation > "Σ_{ ident i < n } f"
489  with precedence 20
490for @{'bigop $n plus 0 (λ${ident i}.true) (λ${ident i}. $f)}.
491
492notation < "Σ_{ ident i < n } f"
493  with precedence 20
494for @{'bigop $n plus 0 (λ${ident i}:$X.true) (λ${ident i}:$Y. $f)}.
495
496axiom code_memory_ro_label_label:
497 ∀cost_labels. ∀ends_flag.
498 ∀initial,final. trace_label_label (ASM_abstract_status cost_labels) ends_flag initial final →
499  code_memory … initial = code_memory … final.
500
501(*
502axiom code_memory_ro_label_return:
503 ∀cost_labels.
504 ∀initial,final. trace_label_return (ASM_abstract_status cost_labels) initial final →
505  code_memory … initial = code_memory … final.
506*)
507
508definition tech_cost_of_label0:
509 ∀cost_labels: BitVectorTrie costlabel 16.
510 ∀cost_map: identifier_map CostTag nat.
511 ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k).
512 ∀ctrace:list (Σk:costlabel.∃pc. lookup_opt costlabel 16 pc cost_labels = Some ? k).
513 ∀i,p. present … cost_map (nth_safe ? i ctrace p).
514 #cost_labels #cost_map #codom_dom #ctrace #i #p
515 cases (nth_safe … i ctrace ?) normalize #id * #id_pc #K
516 lapply (codom_dom … K) #k_pres >(lookup_lookup_present … k_pres)
517 % #abs destruct (abs)
518qed.
519
520include alias "arithmetics/nat.ma".
521include alias "basics/logic.ma".
522
523lemma ltb_rect:
524 ∀P:Type[0].∀n,m. (n < m → P) → (¬ n < m → P) → P.
525 #P #n #m lapply (refl … (ltb n m)) cases (ltb n m) in ⊢ (???% → %); #E #H1 #H2
526 [ @H1 @leb_true_to_le @E | @H2 @leb_false_to_not_le @E ]
527qed.
528
529lemma same_ltb_rect:
530 ∀P,n,m,H1,H2,n',m',H1',H2'.
531  ltb n m = ltb n' m' → (∀x,y. H1 x = H1' y) → (∀x,y. H2 x = H2' y) →
532   ltb_rect P n m H1 H2 =
533   ltb_rect P n' m' H1' H2'.
534 #P #n #m #H1 #H2 #n' #m' #H1' #H2' #E #K1 #K2 whd in ⊢ (??%?);
535 cut (∀xxx,yyy,xxx',yyy'.
536   match ltb n m
537   return λx:bool.
538           eq bool (ltb n m) x
539            → (lt n m → P) → (Not (lt n m) → P) → P
540    with
541    [ true ⇒
542        λE0:eq bool (ltb n m) true.
543         λH10:lt n m → P.
544          λH20:Not (lt n m) → P. H10 (xxx E0)
545    | false ⇒
546        λE0:eq bool (ltb n m) false.
547         λH10:lt n m → P.
548          λH20:Not (lt n m) → P. H20 (yyy E0)]
549    (refl … (ltb n m)) H1 H2 =
550   match ltb n' m'
551   return λx:bool.
552           eq bool (ltb n' m') x
553            → (lt n' m' → P) → (Not (lt n' m') → P) → P
554    with
555    [ true ⇒
556        λE0:eq bool (ltb n' m') true.
557         λH10:lt n' m' → P.
558          λH20:Not (lt n' m') → P. H10 (xxx' E0)
559    | false ⇒
560        λE0:eq bool (ltb n' m') false.
561         λH10:lt n' m' → P.
562          λH20:Not (lt n' m') → P. H20 (yyy' E0)]
563    (refl … (ltb n' m')) H1' H2'
564  ) [2: #X @X]
565 >E cases (ltb n' m') #xxx #yyy #xxx' #yyy' normalize
566 [ @K1 | @K2 ]
567qed.
568
569
570definition tech_cost_of_label:
571 ∀cost_labels: BitVectorTrie costlabel 16.
572 ∀cost_map: identifier_map CostTag nat.
573 ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k).
574 list (Σk:costlabel.∃pc. lookup_opt costlabel 16 pc cost_labels = Some ? k) →
575 nat → nat
576≝ λcost_labels,cost_map,codom_dom,ctrace,i.
577 ltb_rect ? i (|ctrace|)
578 (λH. lookup_present ?? cost_map (nth_safe ? i ctrace H) ?)
579 (λ_.0).
580 @tech_cost_of_label0 @codom_dom
581qed.
582
583lemma shift_nth_safe:
584 ∀T,i,l2,l1,K1,K2.
585  nth_safe T i l1 K1 = nth_safe T (i+|l2|) (l2@l1) K2.
586 #T #i #l2 elim l2 normalize
587  [ #l1 #K1 <plus_n_O #K2 %
588  | #hd #tl #IH #l1 #K1 <plus_n_Sm #K2 change with (? < ?) in K1; change with (? < ?) in K2;
589    whd in ⊢ (???%); @IH ]
590qed.
591
592lemma tech_cost_of_label_shift:
593 ∀cost_labels,cost_map,codom_dom,l1,l2,i.
594  i < |l2| →
595   tech_cost_of_label cost_labels cost_map codom_dom l2 i =
596   tech_cost_of_label cost_labels cost_map codom_dom (l1@l2) (i+|l1|).
597 #cost_labels #cost_Map #codom_dom #l1 #l2 #i #H
598 whd in match tech_cost_of_label; normalize nodelta @same_ltb_rect
599 [ @(ltb_rect ? i (|l2|)) @(ltb_rect ? (i+|l1|) (|l1@l2|)) #K1 #K2
600   whd in match ltb; normalize nodelta
601   [1: >le_to_leb_true try assumption applyS le_to_leb_true //
602   |4: >not_le_to_leb_false try assumption applyS not_le_to_leb_false
603       change with (¬ ? ≤ ?) in K1; applyS K1
604   |2: @⊥ @(absurd (i+|l1| < |l1@l2|)) // >length_append
605       applyS (monotonic_lt_plus_r … (|l1|)) //
606   |3: @⊥ @(absurd ?? K2) >length_append in K1; #K1 /2 by lt_plus_to_lt_l/ ]
607 | #H1 #H2
608   generalize in match (tech_cost_of_label0 ??? (l1@l2) ??);
609   <(shift_nth_safe … H1) #p %
610 | // ]
611qed.
612
613let rec compute_max_trace_label_return_cost_ok_with_trace
614 (cost_labels: BitVectorTrie costlabel 16)
615 (cost_map: identifier_map CostTag nat)
616 (initial: Status) (final: Status)
617 (trace: trace_label_return (ASM_abstract_status cost_labels) initial final)
618 (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k))
619 on trace:
620 ∀dom_codom:(∀k. ∀k_pres:present … cost_map k. ∃pc. lookup_opt … pc cost_labels = Some … k →
621   block_cost (code_memory … initial) cost_labels pc 2^16 = lookup_present … k_pres).
622  let ctrace ≝ compute_cost_trace_label_return … trace in
623  compute_max_trace_label_return_cost … trace =
624   (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i))
625    ≝ ?
626and compute_max_trace_label_label_cost_ok_with_trace
627 (cost_labels: BitVectorTrie costlabel 16)
628 (trace_ends_flag: trace_ends_with_ret)
629 (cost_map: identifier_map CostTag nat)
630 (initial: Status) (final: Status)
631 (trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag initial final)
632 (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k))
633 (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∃pc. lookup_opt … pc cost_labels = Some … k →
634   block_cost (code_memory … initial) cost_labels pc 2^16 = lookup_present … k_pres))
635 on trace:
636  let ctrace ≝ compute_cost_trace_label_label … trace in
637  compute_max_trace_label_label_cost … trace =
638   (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i))
639    ≝ ?.
640cases trace
641[ #sb #sa #tr #dom_codom
642  @compute_max_trace_label_label_cost_ok_with_trace @dom_codom
643| #si #sl #sf #tr1 #tr2 #dom_codom whd in ⊢ (let ctrace ≝ % in ??%?);
644  change with (let ctrace ≝ ? in ? = bigop (|? @ ?|) ?????)
645  >append_length in ⊢ (let ctrace ≝ ? in ???(?%?????));
646  change with (?=?) >bigop_sum_rev >commutative_plus @eq_f2
647  [ >(compute_max_trace_label_return_cost_ok_with_trace … cost_map … codom_dom)
648    -compute_max_trace_label_return_cost_ok_with_trace     
649    [2:lapply (code_memory_ro_label_label … tr1) #E2 <E2 @dom_codom]
650    @same_bigop [//] #i #H #_ -dom_codom @tech_cost_of_label_shift //
651  | >(compute_max_trace_label_label_cost_ok_with_trace … cost_map … codom_dom … dom_codom)
652    @same_bigop [//] #i #H #_
653]
654(*
655lemma
656 compute_max_trace_any_label_cost cost_label trace_ends_flag start_status
657  final_status the_trace =
658 
659
660let rec compute_paid_trace_label_label_cost
661  (cost_labels: BitVectorTrie Byte 16)
662   (trace_ends_flag: trace_ends_with_ret)
663    (start_status: Status) (final_status: Status)
664      (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag
665        start_status final_status) on the_trace: nat ≝
666  match the_trace with
667  [ tll_base _ _ _ given_trace _ ⇒
668      compute_paid_trace_any_label_cost … given_trace
669  ]
670and compute_paid_trace_any_label_cost
671  (cost_labels: BitVectorTrie Byte 16)
672  (trace_ends_flag: trace_ends_with_ret)
673   (start_status: Status) (final_status: Status)
674     (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status)
675       on the_trace: nat ≝
676  match the_trace with
677  [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status
678  | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status
679  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
680     _ _ _ call_trace final_trace ⇒
681      let current_instruction_cost ≝ current_instruction_cost pre_fun_call in
682      let final_trace_cost ≝ compute_paid_trace_any_label_cost cost_labels end_flag … final_trace in
683        current_instruction_cost + final_trace_cost
684  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
685      let current_instruction_cost ≝ current_instruction_cost status_pre in
686      let tail_trace_cost ≝
687       compute_paid_trace_any_label_cost cost_labels end_flag
688        status_init status_end tail_trace
689      in
690        current_instruction_cost + tail_trace_cost
691  ]
692and compute_paid_trace_label_return_cost
693  (cost_labels: BitVectorTrie Byte 16)
694  (start_status: Status) (final_status: Status)
695    (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status)
696      on the_trace: nat ≝
697  match the_trace with
698  [ tlr_base before after trace_to_lift ⇒ compute_paid_trace_label_label_cost … trace_to_lift
699  | tlr_step initial labelled final labelled_trace ret_trace ⇒
700      let labelled_cost ≝ compute_paid_trace_label_label_cost … labelled_trace in
701      let return_cost ≝ compute_paid_trace_label_return_cost … ret_trace in
702        labelled_cost + return_cost
703  ].
704
705let rec trace_lab_rec_cost' (p: trace_lab_ret) : nat.
706 | (call b bf tr af tl) as self ⇒
707    trace_lab_lab_cost_nocall self + trace_lab_ret_cost' tr +
708    trace_lab_rec_cost' tl
709
710theorem main_lemma:
711 ∀p. trace_lab_rec_cost p = trace_lab_rec_cost' p.
712
713axiom lemma1:
714 ∀p: simple_path.
715  traverse_cost_internal (pc (hd p)) … = trace_lab_lab_cost_nocall p.
716
717axiom lemma2:
718 ∀s,l,cost_map.
719  is_labelled l s →
720   traverse_cost_internal s = cost_map l.
721
722axiom main_statement:
723 ∀s.
724 ∀cost_map.
725 let p ≝ compute_simple_path0 s in
726 ∑ (cost_trace p) cost_map = trace_lab_rec_cost' p.
727
728axiom main_statement:
729 ∀s.
730 ∀cost_map.
731 let p ≝ compute_simple_path0 s in
732  execute' (path_length p) s = 〈s',τ〉 →
733   Timer s' - Timer s = ∑ τ cost_map. *)
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