source: src/ASM/CostsProof.ma @ 2516

Last change on this file since 2516 was 2516, checked in by mckinna, 7 years ago

removed typedefs; restored older versions; moved typedefs to compiler.ma
AssemblyProofSplit? *still* doesn't typecheck: disambiguation errors!

File size: 32.4 KB
Line 
1include "ASM/ASMCostsSplit.ma".
2include "ASM/WellLabeled.ma".
3include "ASM/Status.ma".
4include "common/StructuredTraces.ma".
5include "arithmetics/bigops.ma".
6
7include alias "arithmetics/nat.ma".
8include alias "basics/logic.ma".
9
10let rec compute_max_trace_label_label_cost
11  (cm: ?)
12  (cost_labels: BitVectorTrie costlabel 16)
13   (trace_ends_flag: trace_ends_with_ret)
14    (start_status: Status cm) (final_status: Status cm)
15      (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag
16        start_status final_status) on the_trace: nat ≝
17  match the_trace with
18  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
19      compute_max_trace_any_label_cost … given_trace
20  ]
21and compute_max_trace_any_label_cost
22  (cm: ?)
23  (cost_labels: BitVectorTrie costlabel 16)
24  (trace_ends_flag: trace_ends_with_ret)
25   (start_status: Status cm) (final_status: Status cm)
26     (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status)
27       on the_trace: nat ≝
28  match the_trace with
29  [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost cm the_status
30  | tal_base_return the_status _ _ _ ⇒ current_instruction_cost cm the_status
31  | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒
32      let current_instruction_cost ≝ current_instruction_cost cm pre_fun_call in
33      let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in
34        call_trace_cost + current_instruction_cost
35  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
36     _ _ _ call_trace _ final_trace ⇒
37      let current_instruction_cost ≝ current_instruction_cost cm pre_fun_call in
38      let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in
39      let final_trace_cost ≝ compute_max_trace_any_label_cost cm cost_labels end_flag … final_trace in
40        call_trace_cost + current_instruction_cost + final_trace_cost
41  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
42      let current_instruction_cost ≝ current_instruction_cost cm status_pre in
43      let tail_trace_cost ≝
44       compute_max_trace_any_label_cost cm cost_labels end_flag
45        status_init status_end tail_trace
46      in
47        current_instruction_cost + tail_trace_cost
48  ]
49and compute_max_trace_label_return_cost
50  (cm: ?)
51  (cost_labels: BitVectorTrie costlabel 16)
52  (start_status: Status cm) (final_status: Status cm)
53    (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status)
54      on the_trace: nat ≝
55  match the_trace with
56  [ tlr_base before after trace_to_lift ⇒ compute_max_trace_label_label_cost … trace_to_lift
57  | tlr_step initial labelled final labelled_trace ret_trace ⇒
58      let labelled_cost ≝ compute_max_trace_label_label_cost … labelled_trace in
59      let return_cost ≝ compute_max_trace_label_return_cost … ret_trace in
60        labelled_cost + return_cost
61  ].
62
63include alias "arithmetics/nat.ma".
64
65let rec compute_max_trace_label_label_cost_is_ok
66  (cm: ?)
67  (cost_labels: BitVectorTrie costlabel 16)
68   (trace_ends_flag: trace_ends_with_ret)
69    (start_status: Status cm) (final_status: Status cm)
70      (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag
71        start_status final_status) on the_trace:
72          clock … cm … final_status = (compute_max_trace_label_label_cost cm cost_labels trace_ends_flag start_status final_status the_trace) + (clock … cm … start_status) ≝ ?
73and compute_max_trace_any_label_cost_is_ok
74  (cm: ?)
75  (cost_labels: BitVectorTrie costlabel 16)
76  (trace_ends_flag: trace_ends_with_ret)
77   (start_status: Status cm) (final_status: Status cm)
78     (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status)
79       on the_trace:
80         clock … cm … final_status = (compute_max_trace_any_label_cost cm cost_labels trace_ends_flag start_status final_status the_trace) + (clock … cm … start_status) ≝ ?
81and compute_max_trace_label_return_cost_is_ok
82  (cm: ?)
83  (cost_labels: BitVectorTrie costlabel 16)
84  (start_status: Status cm) (final_status: Status cm)
85    (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status)
86      on the_trace:
87        clock … cm … final_status = (compute_max_trace_label_return_cost cm cost_labels start_status final_status the_trace) + clock … cm … start_status ≝ ?.
88  [1:
89    cases the_trace
90    #ends_flag #start_status #end_status #any_label_trace #is_costed
91    normalize @compute_max_trace_any_label_cost_is_ok
92  |2:
93    cases the_trace
94    [1,2:
95      #start_status #final_status #is_next #is_not_return try (#is_costed)
96      change with (current_instruction_cost cm start_status) in ⊢ (???(?%?));
97      cases(is_next) @execute_1_ok_clock
98    |3:
99      #status_pre_fun_call #status_start_fun_call #status_final #is_next
100      #classifier_assm #after_return_assm #call_trace #costed_assm
101      whd in match (compute_max_trace_any_label_cost … (tal_base_call …));
102      >(compute_max_trace_label_return_cost_is_ok … call_trace)
103      >associative_plus @eq_f cases(is_next)
104      @execute_1_ok_clock
105    |4:
106      #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
107      #status_final #is_next #is_call #is_after_return #call_trace #not_costed #final_trace
108      change with (
109      let current_instruction_cost ≝ current_instruction_cost cm status_pre_fun_call in
110      let call_trace_cost ≝ compute_max_trace_label_return_cost cm … call_trace in
111      let final_trace_cost ≝ compute_max_trace_any_label_cost cm cost_labels end_flag … final_trace in
112        call_trace_cost + current_instruction_cost + final_trace_cost) in ⊢ (???(?%?));
113      normalize nodelta;
114      >(compute_max_trace_any_label_cost_is_ok … cost_labels end_flag status_after_fun_call
115          status_final final_trace)
116      >(compute_max_trace_label_return_cost_is_ok … cost_labels status_start_fun_call
117        status_after_fun_call call_trace)
118      cases(is_next) in match (clock … cm status_start_fun_call);
119      >(execute_1_ok_clock cm status_pre_fun_call)
120      <associative_plus in ⊢ (??%?);
121      <commutative_plus in match (
122        compute_max_trace_any_label_cost cm cost_labels end_flag status_after_fun_call status_final final_trace
123          + compute_max_trace_label_return_cost cm cost_labels status_start_fun_call status_after_fun_call call_trace);
124      >associative_plus in ⊢ (??%?); >associative_plus in ⊢ (???%); >associative_plus in ⊢ (???%);
125      @eq_f >commutative_plus in ⊢ (??%?); >associative_plus in ⊢ (??%?);
126      @eq_f @commutative_plus
127    |5:
128      #end_flag #status_pre #status_init #status_end #is_next
129      #trace_any_label #is_other #is_not_costed
130      change with (
131      let current_instruction_cost ≝ current_instruction_cost cm status_pre in
132      let tail_trace_cost ≝
133       compute_max_trace_any_label_cost cm cost_labels end_flag
134        status_init status_end trace_any_label
135      in
136        current_instruction_cost + tail_trace_cost) in ⊢ (???(?%?));
137      normalize nodelta;
138      >(compute_max_trace_any_label_cost_is_ok cm cost_labels end_flag
139         status_init status_end trace_any_label)
140      cases(is_next) in match (clock … cm status_init);
141      >(execute_1_ok_clock … status_pre)
142      >commutative_plus >associative_plus >associative_plus @eq_f
143      @commutative_plus
144    ]
145  |3:
146    cases the_trace
147    [1:
148      #status_before #status_after #trace_to_lift
149      normalize @compute_max_trace_label_label_cost_is_ok
150    |2:
151      #status_initial #status_labelled #status_final #labelled_trace #ret_trace
152      normalize
153      >(compute_max_trace_label_return_cost_is_ok cm cost_labels status_labelled status_final ret_trace);
154      >(compute_max_trace_label_label_cost_is_ok cm cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace);
155      <associative_plus in ⊢ (??%?);
156      >commutative_plus in match (
157        compute_max_trace_label_return_cost cm cost_labels status_labelled status_final ret_trace
158          + compute_max_trace_label_label_cost cm cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace);
159      %
160    ]
161  ]
162qed.
163
164let rec compute_trace_label_label_cost_using_paid
165  (cm: ?)
166  (cost_labels: BitVectorTrie costlabel 16)
167   (trace_ends_flag: trace_ends_with_ret)
168    (start_status: Status cm) (final_status: Status cm)
169      (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag
170        start_status final_status) on the_trace: nat ≝
171  match the_trace with
172  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
173      compute_paid_trace_label_label cm cost_labels … the_trace +
174      compute_trace_any_label_cost_using_paid … given_trace
175  ]
176and compute_trace_any_label_cost_using_paid
177  (cm: ?)
178  (cost_labels: BitVectorTrie costlabel 16)
179  (trace_ends_flag: trace_ends_with_ret)
180   (start_status: Status cm) (final_status: Status cm)
181     (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status)
182       on the_trace: nat ≝
183  match the_trace with
184  [ tal_base_not_return the_status _ _ _ _ ⇒ 0
185  | tal_base_return the_status _ _ _ ⇒ 0
186  | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒
187      compute_trace_label_return_cost_using_paid … call_trace
188  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
189     _ _ _ call_trace _ final_trace ⇒
190      let call_trace_cost ≝ compute_trace_label_return_cost_using_paid … call_trace in
191      let final_trace_cost ≝ compute_trace_any_label_cost_using_paid cm cost_labels end_flag … final_trace in
192        call_trace_cost + final_trace_cost
193  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
194     compute_trace_any_label_cost_using_paid cm cost_labels end_flag
195      status_init status_end tail_trace
196  ]
197and compute_trace_label_return_cost_using_paid
198  (cm: ?)
199  (cost_labels: BitVectorTrie costlabel 16)
200  (start_status: Status cm) (final_status: Status cm)
201    (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status)
202      on the_trace: nat ≝
203  match the_trace with
204  [ tlr_base before after trace_to_lift ⇒ compute_trace_label_label_cost_using_paid … trace_to_lift
205  | tlr_step initial labelled final labelled_trace ret_trace ⇒
206      let labelled_cost ≝ compute_trace_label_label_cost_using_paid … labelled_trace in
207      let return_cost ≝ compute_trace_label_return_cost_using_paid … ret_trace in
208        labelled_cost + return_cost
209  ].
210
211let rec compute_trace_label_label_cost_using_paid_ok
212  (cm: ?)
213  (cost_labels: BitVectorTrie costlabel 16)
214   (trace_ends_flag: trace_ends_with_ret)
215    (start_status: Status cm) (final_status: Status cm)
216      (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag
217        start_status final_status) on the_trace:
218 compute_trace_label_label_cost_using_paid cm cost_labels … the_trace =
219 compute_max_trace_label_label_cost … the_trace ≝ ?
220and compute_trace_any_label_cost_using_paid_ok
221  (cm: ?)
222  (cost_labels: BitVectorTrie costlabel 16)
223  (trace_ends_flag: trace_ends_with_ret)
224   (start_status: Status cm) (final_status: Status cm)
225     (the_trace: trace_any_label (ASM_abstract_status cm cost_labels)
226      trace_ends_flag start_status final_status) on the_trace:     
227  compute_paid_trace_any_label cm cost_labels trace_ends_flag … the_trace
228  +compute_trace_any_label_cost_using_paid cm cost_labels trace_ends_flag … the_trace
229  =compute_max_trace_any_label_cost cm cost_labels trace_ends_flag … the_trace ≝ ?
230and compute_trace_label_return_cost_using_paid_ok
231  (cm: ?)
232  (cost_labels: BitVectorTrie costlabel 16)
233  (start_status: Status cm) (final_status: Status cm)
234    (the_trace: trace_label_return (ASM_abstract_status cm cost_labels)
235     start_status final_status) on the_trace:
236 compute_trace_label_return_cost_using_paid cm cost_labels … the_trace =
237 compute_max_trace_label_return_cost cm cost_labels … the_trace ≝ ?.
238[ cases the_trace #endsf #ss #es #tr #H normalize
239  @compute_trace_any_label_cost_using_paid_ok
240| cases the_trace
241  [ #ss #fs #H1 #H2 #H3 whd in ⊢ (??(?%%)%); <plus_n_O %
242  | #ss #fs #H1 #H2 whd in ⊢ (??(?%%)%); <plus_n_O %
243  |
244    #sp #ss #sf #H1 #H2 #tr1 #tr2 #H3
245    whd in ⊢ (???%); whd in ⊢ (??(??%)?); whd in ⊢ (??(?%?)?);
246    >compute_trace_label_return_cost_using_paid_ok in ⊢ (??%?);
247    >commutative_plus in ⊢ (??%?); @eq_f %
248  | #ef #spfc #ssfc #safc #sf #H1 #H2 #H3 #tr1 #H4 #tr2 whd in ⊢ (??(?%%)%);
249    <compute_trace_any_label_cost_using_paid_ok
250    <compute_trace_label_return_cost_using_paid_ok
251    -compute_trace_label_label_cost_using_paid_ok
252    -compute_trace_label_return_cost_using_paid_ok
253    -compute_trace_any_label_cost_using_paid_ok
254    >commutative_plus in ⊢ (???(?%?));
255    >commutative_plus in ⊢ (??(??%)?);
256    >associative_plus >associative_plus in ⊢ (???%); @eq_f2 try %
257    <associative_plus <commutative_plus %
258  | #ef #sp #si #se #H1 #tr #H2 #H3 whd in ⊢ (??(?%%)%); >associative_plus @eq_f2
259    [ % | @compute_trace_any_label_cost_using_paid_ok ]
260  ]
261| cases the_trace
262  [ #sb #sa #tr normalize @compute_trace_label_label_cost_using_paid_ok
263  | #si #sl #sf #tr1 #tr2 normalize @eq_f2
264    [ @compute_trace_label_label_cost_using_paid_ok
265    | @compute_trace_label_return_cost_using_paid_ok ]]]
266qed.
267
268include alias "ASM/BitVectorTrie.ma".
269
270(*
271let rec compute_cost_trace_label_label
272  (cm: BitVectorTrie Byte 16)
273  (cost_labels: BitVectorTrie costlabel 16)
274   (trace_ends_flag: trace_ends_with_ret)
275    (start_status: Status cm) (final_status: Status cm)
276      (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag
277        start_status final_status) on the_trace:
278         list (Σk:costlabel. ∃pc: Word. lookup_opt … pc cost_labels = Some … k) ≝
279  match the_trace with
280  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
281     let pc ≝ program_counter ? cm initial in
282     let label ≝
283      match lookup_opt … pc cost_labels return λx: option ?. x ≠ None … → costlabel with
284      [ None ⇒ λabs. ⊥
285      | Some l ⇒ λ_. l ] labelled_proof in
286      (mk_Sig ?? label ?)::compute_cost_trace_any_label cm cost_labels ends_flag initial final … given_trace
287  ]
288and compute_cost_trace_any_label
289  (cm: BitVectorTrie Byte 16)
290  (cost_labels: BitVectorTrie costlabel 16)
291  (trace_ends_flag: trace_ends_with_ret)
292   (start_status: Status cm) (final_status: Status cm)
293     (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status)
294       on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝
295  match the_trace with
296  [ tal_base_not_return the_status _ _ _ _ ⇒ [ ]
297  | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒
298      compute_cost_trace_label_return … call_trace
299  | tal_base_return the_status _ _ _ ⇒ [ ]
300  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
301     _ _ _ call_trace _ final_trace ⇒
302      let call_cost_trace ≝ compute_cost_trace_label_return … call_trace in
303      let final_cost_trace ≝ compute_cost_trace_any_label cm cost_labels end_flag … final_trace in
304        call_cost_trace @ final_cost_trace
305  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
306       compute_cost_trace_any_label cm cost_labels end_flag
307        status_init status_end tail_trace
308  ]
309and compute_cost_trace_label_return
310  (cm: BitVectorTrie Byte 16)
311  (cost_labels: BitVectorTrie costlabel 16)
312  (start_status: Status cm) (final_status: Status cm)
313    (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status)
314      on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝
315  match the_trace with
316  [ tlr_base before after trace_to_lift ⇒ compute_cost_trace_label_label cm … trace_to_lift
317  | tlr_step initial labelled final labelled_trace ret_trace ⇒
318      let labelled_cost ≝ compute_cost_trace_label_label cm … labelled_trace in
319      let return_cost ≝ compute_cost_trace_label_return cm … ret_trace in
320        labelled_cost @ return_cost
321  ].
322 [1:
323  %{pc} whd in match label;  generalize in match labelled_proof; whd in ⊢ (% → ?);
324   whd in match (as_costed ??); whd in match (as_label ??); normalize nodelta
325   cases (lookup_opt costlabel … (program_counter … initial) cost_labels)
326   normalize
327  [ #abs cases abs #absurd @⊥ @absurd % | // ]
328 | cases abs #absurd @absurd % ]
329qed.
330*)
331
332include alias "arithmetics/nat.ma".
333include alias "basics/logic.ma".
334
335include alias "arithmetics/bigops.ma".
336
337(* This shoudl go in bigops! *)
338theorem bigop_sum_rev: ∀k1,k2,p,B,nil.∀op:Aop B nil.∀f:nat→B.
339 \big[op,nil]_{i<k1+k2|p i} (f i) =
340 op \big[op,nil]_{i<k2|p (i+k1)} (f (i+k1)) \big[op,nil]_{i<k1|p i} (f i).
341 #k1 #k2 #p #B #nil #op #f >bigop_sum
342 >commutative_plus @same_bigop #i @leb_elim normalize
343 [2,4: //
344 | #H1 #H2 <plus_minus_m_m //
345 | #H1 #H2 #H3 <plus_minus_m_m //]
346qed.
347
348(* This is taken by sigma_pi.ma that does not compile now *)
349definition natAop ≝ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n))
350   (λa,b,c.sym_eq ??? (associative_plus a b c)).
351
352definition natACop ≝ mk_ACop nat 0 natAop commutative_plus.
353
354definition natDop ≝ mk_Dop nat 0 natACop times (λn.(sym_eq ??? (times_n_O n)))
355   distributive_times_plus.
356
357unification hint  0 ≔ ;
358   S ≟ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n))
359   (λa,b,c.sym_eq ??? (associative_plus a b c))
360(* ---------------------------------------- *) ⊢
361   plus ≡ op ? ? S.
362
363unification hint  0 ≔ ;
364   S ≟ mk_ACop nat 0 (mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n))
365   (λa,b,c.sym_eq ??? (associative_plus a b c))) commutative_plus
366(* ---------------------------------------- *) ⊢
367   plus ≡ op ? ? S.
368   
369unification hint  0 ≔ ;
370   S ≟ natDop
371(* ---------------------------------------- *) ⊢
372   plus ≡ sum ? ? S.
373
374unification hint  0 ≔ ;
375   S ≟ natDop
376(* ---------------------------------------- *) ⊢
377   times ≡ prod ? ? S.
378
379notation > "Σ_{ ident i < n } f"
380  with precedence 20
381for @{'bigop $n plus 0 (λ${ident i}.true) (λ${ident i}. $f)}.
382
383notation < "Σ_{ ident i < n } f"
384  with precedence 20
385for @{'bigop $n plus 0 (λ${ident i}:$X.true) (λ${ident i}:$Y. $f)}.
386
387definition tech_cost_of_label0:
388 ∀cost_labels: BitVectorTrie costlabel 16.
389 ∀cost_map: identifier_map CostTag nat.
390 ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k).
391 ∀ctrace:list (Σk:costlabel.∃b.lookup_opt … b cost_labels = Some ? k).
392 ∀i,p. present … cost_map (nth_safe ? i ctrace p).
393 #cost_labels #cost_map #codom_dom #ctrace #i #p
394 cases (nth_safe … i ctrace ?) normalize #id * #id_pc #K
395 lapply (codom_dom … K) #k_pres >(lookup_lookup_present … k_pres)
396 % #abs destruct (abs)
397qed.
398
399include alias "arithmetics/nat.ma".
400include alias "basics/logic.ma".
401
402lemma ltb_rect:
403 ∀P:Type[0].∀n,m. (n < m → P) → (¬ n < m → P) → P.
404 #P #n #m lapply (refl … (ltb n m)) cases (ltb n m) in ⊢ (???% → %); #E #H1 #H2
405 [ @H1 @leb_true_to_le @E | @H2 @leb_false_to_not_le @E ]
406qed.
407
408lemma same_ltb_rect:
409 ∀P,n,m,H1,H2,n',m',H1',H2'.
410  ltb n m = ltb n' m' → (∀x,y. H1 x = H1' y) → (∀x,y. H2 x = H2' y) →
411   ltb_rect P n m H1 H2 =
412   ltb_rect P n' m' H1' H2'.
413 #P #n #m #H1 #H2 #n' #m' #H1' #H2' #E #K1 #K2 whd in ⊢ (??%?);
414 cut (∀xxx,yyy,xxx',yyy'.
415   match ltb n m
416   return λx:bool.
417           eq bool (ltb n m) x
418            → (lt n m → P) → (Not (lt n m) → P) → P
419    with
420    [ true ⇒
421        λE0:eq bool (ltb n m) true.
422         λH10:lt n m → P.
423          λH20:Not (lt n m) → P. H10 (xxx E0)
424    | false ⇒
425        λE0:eq bool (ltb n m) false.
426         λH10:lt n m → P.
427          λH20:Not (lt n m) → P. H20 (yyy E0)]
428    (refl … (ltb n m)) H1 H2 =
429   match ltb n' m'
430   return λx:bool.
431           eq bool (ltb n' m') x
432            → (lt n' m' → P) → (Not (lt n' m') → P) → P
433    with
434    [ true ⇒
435        λE0:eq bool (ltb n' m') true.
436         λH10:lt n' m' → P.
437          λH20:Not (lt n' m') → P. H10 (xxx' E0)
438    | false ⇒
439        λE0:eq bool (ltb n' m') false.
440         λH10:lt n' m' → P.
441          λH20:Not (lt n' m') → P. H20 (yyy' E0)]
442    (refl … (ltb n' m')) H1' H2'
443  ) [2: #X @X]
444 >E cases (ltb n' m') #xxx #yyy #xxx' #yyy' normalize
445 [ @K1 | @K2 ]
446qed.
447
448definition tech_cost_of_label:
449 ∀cost_labels: BitVectorTrie costlabel 16.
450 ∀cost_map: identifier_map CostTag nat.
451 ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k).
452 list (Σk:costlabel.∃b.lookup_opt … b cost_labels = Some ? k) →
453 nat → nat
454≝ λcost_labels,cost_map,codom_dom,ctrace,i.
455 ltb_rect ? i (|ctrace|)
456 (λH. lookup_present ?? cost_map (nth_safe ? i ctrace H) ?)
457 (λ_.0).
458 @tech_cost_of_label0 @codom_dom
459qed.
460
461lemma tech_cost_of_label_shift:
462 ∀cost_labels,cost_map,codom_dom,l1,l2,i.
463  i < |l2| →
464   tech_cost_of_label cost_labels cost_map codom_dom l2 i =
465   tech_cost_of_label cost_labels cost_map codom_dom (l1@l2) (i+|l1|).
466 #cost_labels #cost_Map #codom_dom #l1 #l2 #i #H
467 whd in match tech_cost_of_label; normalize nodelta @same_ltb_rect
468 [ @(ltb_rect ? i (|l2|)) @(ltb_rect ? (i+|l1|) (|l1@l2|)) #K1 #K2
469   whd in match ltb; normalize nodelta
470   [1: >le_to_leb_true try assumption applyS le_to_leb_true / by /
471   |4: >not_le_to_leb_false try assumption applyS not_le_to_leb_false
472       change with (¬ ? ≤ ?) in K1; applyS K1
473   |2: @⊥ @(absurd (i+|l1| < |l1@l2|)) / by / >length_append
474       applyS (monotonic_lt_plus_r … (|l1|)) / by /
475   |3: @⊥ @(absurd ?? K2) >length_append in K1; #K1 /2 by lt_plus_to_lt_l/ ]
476 | #H1 #H2
477   generalize in match (tech_cost_of_label0 ??? (l1@l2) ??);
478   <(shift_nth_safe … H1) #p %
479 | / by / ]
480qed.
481
482lemma tech_cost_of_label_prefix:
483 ∀cost_labels,cost_map,codom_dom,l1,l2,i.
484  i < |l1| →
485   tech_cost_of_label cost_labels cost_map codom_dom l1 i =
486   tech_cost_of_label cost_labels cost_map codom_dom (l1@l2) i.
487 #cost_labels #cost_map #codom_dom #l1 #l2 #i #H
488 whd in match tech_cost_of_label; normalize nodelta @same_ltb_rect
489 [1:
490   whd in match ltb; normalize nodelta
491   >(le_to_leb_true … H) applyS le_to_leb_true
492   >length_append whd in H; >commutative_plus @le_plus_a assumption
493 |2:
494   #K1 #K2
495   generalize in match (tech_cost_of_label0 ??? (l1@l2) ??);
496   <(shift_nth_prefix … l1 i l2 K1 K2) //
497 |3:
498   #_ #_ %
499 ]
500qed.
501   
502let rec compute_trace_label_return_using_paid_ok_with_trace
503 (cm: ?) (cost_labels: BitVectorTrie costlabel 16)
504 (cost_map: identifier_map CostTag nat)
505 (initial: Status cm) (final: Status cm)
506 (trace: trace_label_return (ASM_abstract_status cm cost_labels) initial final)
507 (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k))
508 on trace:
509 ∀unrepeating_witness: tlr_unrepeating … trace.
510 ∀dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k →
511   pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres).
512  let ctrace ≝ flatten_trace_label_return (ASM_abstract_status cm cost_labels) … trace in
513  compute_trace_label_return_cost_using_paid cm … trace =
514   (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i))
515    ≝ ?
516and compute_trace_any_label_using_paid_ok_with_trace
517 (cm: ?) (cost_labels: BitVectorTrie costlabel 16)
518 (trace_ends_flag: trace_ends_with_ret)
519 (cost_map: identifier_map CostTag nat)
520 (initial: Status cm) (final: Status cm)
521 (trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag initial final)
522 (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k))
523 (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k →
524   pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres))
525 on trace:
526  ∀unrepeating_witness: tal_unrepeating … trace.
527  let ctrace ≝ flatten_trace_any_label (ASM_abstract_status cm cost_labels) … trace_ends_flag … trace in
528  compute_trace_any_label_cost_using_paid … trace =
529   (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i))
530    ≝ ?
531and compute_trace_label_label_using_paid_ok_with_trace
532 (cm: ?) (cost_labels: BitVectorTrie costlabel 16)
533 (trace_ends_flag: trace_ends_with_ret)
534 (cost_map: identifier_map CostTag nat)
535 (initial: Status cm) (final: Status cm)
536 (trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag initial final)
537 (unrepeating_witness: tll_unrepeating … trace)
538 (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k))
539 (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k →
540   pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres))
541 on trace:
542 ∀unrepeating_witness: tll_unrepeating … trace.
543  let ctrace ≝ flatten_trace_label_label (ASM_abstract_status cm cost_labels) … trace in
544  compute_trace_label_label_cost_using_paid … trace =
545   (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i))
546    ≝ ?.
547  cases trace normalize nodelta
548  [ #sb #sa #tr #unrepeating_witness #dom_codom whd in ⊢ (??%?);
549    whd in match (flatten_trace_label_return ????);
550    @compute_trace_label_label_using_paid_ok_with_trace
551    assumption
552  | #si #sl #sf #tr1 #tr2 #unrepeating_witness #dom_codom
553    whd in ⊢ (??%?);
554    whd in match (flatten_trace_label_return ????);
555    >append_length >bigop_sum_rev >commutative_plus @eq_f2
556    [ >(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom)
557      -compute_trace_label_return_using_paid_ok_with_trace
558      [1:
559        @same_bigop [//] #i #H #_ -dom_codom @tech_cost_of_label_shift //
560      |2:
561        inversion unrepeating_witness
562        #tll_unrepeating #tlr_unrepeating #_ assumption
563      ]
564    | >(compute_trace_label_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom)
565      -compute_trace_label_label_using_paid_ok_with_trace
566      [1:
567        @same_bigop [//] #i #H #_ @(tech_cost_of_label_prefix … H)
568      |2,3:
569        inversion unrepeating_witness
570        #tll_unrepeating #tlr_unrepeating #_ assumption
571      ]
572    ]
573  |8:
574    #end_flag #start_status #end_status #trace_any_label #costed_assm
575    #unrepeating_witness'
576    whd in ⊢ (??%?); whd in ⊢ (??(?%?)?);
577    >(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom)
578    [1:
579      whd in match (flatten_trace_label_label ?????);
580      >bigop_0 in ⊢ (???%); >commutative_plus @eq_f2
581      [1:
582        @same_bigop [//] #i #H #_ -dom_codom >(plus_n_O i) >plus_n_Sm
583        <(tech_cost_of_label_shift ??? [?] ? i) try assumption <(plus_n_O i) %
584      |2:
585        change with (? = lookup_present ? ? ? ? ?)
586        generalize in match (tech_cost_of_label0 ? ? ? ? ? ?);
587        normalize in match (nth_safe ? ? ? ?);
588        whd in costed_assm; lapply costed_assm whd in match (as_label ??);
589        inversion (lookup_opt ? ? (program_counter … cm start_status) cost_labels)
590        [1:
591          #_ #absurd @⊥ cases absurd #absurd @absurd %
592        |2:
593          normalize nodelta #cost_label #Some_assm #_ #p
594          cases (dom_codom ? p ? Some_assm)
595          cases (block_cost ???)
596          #cost #block_cost_assm
597          cases (block_cost_assm ??? trace_any_label ??)
598          try @refl assumption
599        ]
600      ]
601    |2:
602      assumption
603    ]
604  |3:
605    #start_status #final_status #execute_assm #classifier_assm #costed_assm
606    #unrepeating_witness
607    %
608  |4:
609    #start_status #final_status #execute_assm #classifier_assm #unrepeating_witness
610    %
611  |5:
612    #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
613    #classifier_assm #after_return_assm #trace_label_return #costed_assm #unrepeating_witness
614    whd in ⊢ (??%?);
615    @(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom)
616    assumption
617  |6:
618    #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
619    #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
620    #costed_assm #trace_any_label #unrepeating_witness
621    whd in ⊢ (??%?);
622    >(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom)
623    [1:
624      >(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom)
625      [1:
626        >length_append >bigop_sum_rev >commutative_plus @eq_f2
627        [ @same_bigop [2: #i #H #_ -dom_codom @tech_cost_of_label_shift assumption
628                      |1: #i #H % ]
629        | @same_bigop [#i #H %] #i #H #_ @(tech_cost_of_label_prefix … H)
630        ]
631      |2:
632        inversion unrepeating_witness
633        * #memb_1 #tal_unrepeating #tlr_unrepeating #_ assumption
634      ]
635    |2:
636      inversion unrepeating_witness
637      * #memb_1 #tal_unrepeating #tlr_unrepeating #_ assumption
638    ]
639  |7:
640    #end_flag #status_pre_fun_call #status_start_fun_call #status_final
641    #execute_assm #trace_any_label #classifier_assm #costed_assm #unrepeating_witness
642    whd in ⊢ (??%?);
643    @(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom)
644    inversion unrepeating_witness
645    #memb_1 #tal_unrepeating #_ assumption
646  ]
647qed.
648 
649lemma compute_max_trace_label_return_cost_ok_with_trace_aux:
650  ∀code_memory: BitVectorTrie Byte 16.
651  ∀cost_labels: BitVectorTrie costlabel 16.
652  ∀cost_map: identifier_map CostTag nat.
653  ∀initial: Status code_memory.
654  ∀final: Status code_memory.
655  ∀trace: trace_label_return (ASM_abstract_status code_memory cost_labels) initial final.
656  ∀unrepeating_witness: tlr_unrepeating … trace.
657  ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k).
658  ∀dom_codom: (∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k →
659      pi1 … (block_cost code_memory pc cost_labels) = lookup_present … k_pres).
660        let ctrace ≝ flatten_trace_label_return … trace in
661          clock … code_memory … final =
662            clock … code_memory … initial + (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)).
663  #code_memory #cost_labels #cost_map
664  #initial #final #trace #unrepeating_witness #codom_dom #dom_codom normalize nodelta
665  <compute_trace_label_return_using_paid_ok_with_trace try assumption
666  >commutative_plus >compute_trace_label_return_cost_using_paid_ok //
667qed.
668
669theorem compute_max_trace_label_return_cost_ok_with_trace:
670  ∀code_memory: BitVectorTrie Byte 16.
671  ∀cost_labels: BitVectorTrie costlabel 16.
672  ∀cost_labels_injective:
673   (∀pc,pc',l.
674     lookup_opt costlabel 16 pc cost_labels=Some costlabel l
675      →lookup_opt costlabel 16 pc' cost_labels=Some costlabel l→pc=pc').
676  ∀initial: Status code_memory.
677  ∀final: Status code_memory.
678  ∀trace: trace_label_return (ASM_abstract_status code_memory cost_labels) initial final.
679  ∀unrepeating_witness: tlr_unrepeating … trace.
680    let cost_map ≝ traverse_code code_memory cost_labels cost_labels_injective in
681    let ctrace ≝ flatten_trace_label_return … trace in
682      clock … code_memory … final = clock … code_memory … initial + (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map ? ctrace i)).
683  [1:
684    #code_memory #cost_labels #cost_labels_injective #initial #final #trace #unrepeating_witness
685    @compute_max_trace_label_return_cost_ok_with_trace_aux try assumption
686  |2:
687    skip
688  ]
689  normalize nodelta
690  cases (traverse_code ???)
691  #cost_map * #dom_codom #codom_dom try assumption
692  #pc #k #lookup_opt_assm @(dom_codom … lookup_opt_assm)
693  lapply (sym_eq ? ? ? lookup_opt_assm)
694  -lookup_opt_assm #lookup_opt_assm
695  @nat_of_bitvector_lt_bound
696qed.
697 
698include "utilities/permutations.ma".
699
700lemma tech_cost_sum_eq_as_cost :
701  ∀code_memory: BitVectorTrie Byte 16.
702  ∀cost_labels: BitVectorTrie costlabel 16.
703  ∀cost_map: identifier_map CostTag nat.
704  ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k).
705  ∀trace.
706  (Σ_{i < |trace|}(tech_cost_of_label cost_labels cost_map codom_dom trace i)) =
707  (Σ_{l ∈ trace}(ASM_cost_map code_memory … codom_dom l)).
708#cmem #clab #cmap #codom_dom #trace
709@(list_elim_left … trace)
710[ %
711| #tl #hd #IH >append_length >commutative_plus
712  >(fold_permute … (hd@[tl]) (tl::hd)) [2: @perm_swap_append ]
713  whd in ⊢ (??%%); <IH
714  <tech_cost_of_label_shift [2: %]
715  cut (∀x,y,z,w.x = y → z = w → x + z = y + w) [ // ] #APP @APP -APP
716  [ %
717  | @sym_eq @same_bigop /2 by tech_cost_of_label_prefix/
718  ]
719]
720qed.
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