source: src/ASM/ASMCosts.ma @ 2593

Last change on this file since 2593 was 2593, checked in by mckinna, 8 years ago

Finally chased down wicked failure to close case 1.1: of trace_compute_paid_trace_cl_other, arising from change to type of as_classify in common/StructuredTraces.ma.

Pointless case analysis on instruction in 1.1 thereby also avoided.

File size: 36.3 KB
Line 
1include "ASM/ASM.ma".
2include "ASM/Arithmetic.ma".
3include "ASM/Fetch.ma".
4include "ASM/Interpret.ma". (* includes ASM/AbstractStatus.ma *)
5
6(*include "common/StructuredTraces.ma". (* already included by ASM/AbstractStatus.ma *) *)
7
8(* could this definition be moved to ASM/AbstractStatus.ma ???
9   no: requires ASM/Interpret/execute_1 *)
10
11definition ASM_abstract_status: ∀code_memory. BitVectorTrie costlabel 16 → abstract_status ≝
12  λcode_memory.
13  λcost_labels.
14    mk_abstract_status
15      (Status code_memory)
16      (λs,s'. (execute_1 … s) = s')
17      word_deqset
18      (program_counter …)
19      (λs. Some ? (ASM_classify … s))
20      (λpc.lookup_opt … pc cost_labels)
21      (next_instruction_properly_relates_program_counters code_memory)
22      ? (* (λs.False) *)
23      ?
24      ?.
25  cases daemon (* XXX: is_final predicate, (tail)call_ident function *)
26qed.
27
28lemma Some_inv : ∀A.∀a,b. Some A a = Some A b → a = b.
29  #A#a #b #eq destruct //
30  qed.
31
32definition trace_any_label_length ≝
33  λcode_memory: BitVectorTrie Byte 16.
34  λcost_labels: BitVectorTrie costlabel 16.
35  λtrace_ends_flag: trace_ends_with_ret.
36  λstart_status: Status code_memory.
37  λfinal_status: Status code_memory.
38  λthe_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
39    as_trace_any_label_length' … the_trace.
40
41let rec all_program_counter_list
42  (n: nat)
43    on n ≝
44  match n with
45  [ O ⇒ [ ]
46  | S n' ⇒ (bitvector_of_nat 16 n')::(all_program_counter_list n')
47  ].
48
49lemma all_program_counter_list_bound_eq:
50  ∀bound: nat.
51    |all_program_counter_list bound| = bound.
52  #bound elim bound try %
53  #bound' #inductive_hypothesis normalize
54  >inductive_hypothesis %
55qed.
56
57lemma mem_all_program_counter_list_lt_bound:
58  ∀bound: nat.
59  ∀bound_proof: bound ≤ 2^16.
60  ∀e: Word.
61    memb word_deqset e (all_program_counter_list bound) = true →
62      nat_of_bitvector … e < bound.
63  #bound elim bound
64  [1:
65    #bound_proof #e
66    normalize #absurd destruct(absurd)
67  |2:
68    #bound' #inductive_hypothesis
69    #bound_proof #e
70    change with (if eq_bv ??? then ? else ? = ? → ?)
71    @eq_bv_elim
72    [1:
73      #eq_assm
74      normalize nodelta destruct #_
75      >nat_of_bitvector_bitvector_of_nat_inverse try assumption
76      normalize %
77    |2:
78      #neq_assm normalize nodelta
79      #memb_assm %2 @inductive_hypothesis
80      try assumption
81      @(transitive_le bound' (S bound'))
82      try assumption %2 %
83    ]
84  ]
85qed.
86
87lemma lt_bound_mem_all_program_counter_list:
88  ∀bound: nat.
89  ∀bound_proof: bound ≤ 2^16.
90  ∀e: Word.
91    nat_of_bitvector … e < bound →
92      memb word_deqset e (all_program_counter_list bound) = true.
93  #bound elim bound
94  [1:
95    #bound_proof #e normalize
96    #absurd cases (lt_to_not_zero … absurd)
97  |2:
98    #bound' #inductive_hypothesis
99    #bound_proof #e
100    change with (? → if eq_bv ??? then ? else ? = ?)
101    #assm
102    cases (le_to_or_lt_eq … (nat_of_bitvector 16 e) bound' ?)
103    [1:
104      #e_lt_assm
105      @eq_bv_elim
106      [1:
107        #eq_assm normalize nodelta %
108      |2:
109        #neq_assm normalize nodelta
110        @inductive_hypothesis try assumption
111        @(transitive_le bound' (S bound')) try assumption
112        %2 %
113      ]
114    |2:
115      #relevant >eq_eq_bv normalize nodelta try %
116      destruct >bitvector_of_nat_inverse_nat_of_bitvector %
117    |3:
118      normalize in assm;
119      @le_S_S_to_le assumption
120    ]
121  ]
122qed.
123
124include alias "arithmetics/nat.ma".
125
126lemma execute_1_and_program_counter_after_other_in_lockstep00:
127  ∀code_memory: BitVectorTrie Byte 16.
128  ∀start_status: Status code_memory.
129      let 〈instruction, pc, ticks〉 ≝ fetch code_memory (program_counter … start_status) in
130    ASM_classify0 instruction = cl_other →
131        program_counter ? ? (execute_1 … start_status) =
132          program_counter_after_other pc instruction.
133  #code_memory #start_status
134  whd in match execute_1; normalize nodelta 
135  cases (execute_1' code_memory start_status) #the_status; * #_
136  cases (fetch ? ?); * #instruction #pc #ticks
137  #classify_assm #classify_assm0 @classify_assm assumption
138qed.
139
140lemma execute_1_and_program_counter_after_other_in_lockstep0:
141  ∀code_memory: BitVectorTrie Byte 16.
142  ∀start_status: Status code_memory.
143    ASM_classify … start_status = cl_other →
144      let 〈instruction, pc, ticks〉 ≝ fetch code_memory (program_counter … start_status) in
145        program_counter ? ? (execute_1 … start_status) =
146          program_counter_after_other pc instruction.
147  #code_memory #start_status #classify_assm
148  whd in match execute_1; normalize nodelta
149  cases (execute_1' code_memory start_status) #the_status
150  * #_ whd in classify_assm:(??%?); whd in match (current_instruction ??) in classify_assm;
151  cases (fetch ? ?) in classify_assm; * #instruction #pc #ticks
152  #classify_assm #classify_assm' @classify_assm' assumption
153qed.
154
155lemma execute_1_and_program_counter_after_other_in_lockstep:
156  ∀code_memory: BitVectorTrie Byte 16.
157  ∀cost_labels: BitVectorTrie costlabel 16.
158  ∀start_status: Status code_memory.
159    as_classifier (ASM_abstract_status code_memory cost_labels) start_status cl_other →
160      let 〈instruction, pc, ticks〉 ≝ fetch code_memory (program_counter … start_status) in
161        program_counter ? ? (execute_1 … start_status) =
162          program_counter_after_other pc instruction.
163  #code_memory #cost_labels #start_status #classify_assm
164  change with (Some ? (ASM_classify0 ?) = ?) in classify_assm;
165  destruct (classify_assm)
166  @execute_1_and_program_counter_after_other_in_lockstep0 assumption.
167qed. 
168
169lemma nat_of_bitvector_lt_bound:
170  ∀n: nat.
171  ∀b: BitVector n.
172    nat_of_bitvector n b < 2^n.
173  #n #b cases daemon
174qed.
175
176lemma sublist_tal_pc_list_all_program_counter_list:
177  ∀code_memory: BitVectorTrie Byte 16.
178  ∀cost_labels: BitVectorTrie costlabel 16.
179  ∀start, final: Status code_memory.
180  ∀trace_ends_flag.
181  ∀the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start final.
182    sublist ? (tal_pc_list … the_trace) (all_program_counter_list (2^16)).
183  #code_memory #cost_labels #start #final #trace_ends_flag #the_trace
184  whd in match (sublist ???); #pc #memb_assm
185  @lt_bound_mem_all_program_counter_list try %
186  @nat_of_bitvector_lt_bound
187qed.
188
189corollary tal_pc_list_length_leq_total_program_size:
190  ∀code_memory: BitVectorTrie Byte 16.
191  ∀cost_labels: BitVectorTrie costlabel 16.
192  ∀start, final: Status code_memory.
193  ∀trace_ends_flag.
194  ∀the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start final.
195  ∀unrepeating_witness: tal_unrepeating … the_trace.
196    |tal_pc_list … the_trace| ≤ 2^16.
197  #code_memory #cost_labels #start #final #trace_ends_flag #the_trace #unrepeating_witness
198  <(all_program_counter_list_bound_eq (2^16))
199  @sublist_length
200  [1:
201    @tal_unrepeating_uniqueb
202    assumption
203  |2:
204    @sublist_tal_pc_list_all_program_counter_list
205    assumption
206  ]
207qed.
208
209let rec compute_paid_trace_any_label
210  (code_memory: BitVectorTrie Byte 16) (cost_labels: BitVectorTrie costlabel 16)
211    (trace_ends_flag: trace_ends_with_ret) (start_status: Status code_memory)
212      (final_status: Status code_memory)
213        (the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status)
214       on the_trace: nat ≝
215  match the_trace with
216  [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost … the_status
217  | tal_base_return the_status _ _ _ ⇒ current_instruction_cost … the_status
218  | tal_base_call pre_fun_call start_fun_call final _ _ _ _ call_trace ⇒ current_instruction_cost … pre_fun_call
219  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
220     _ _ _ call_trace _ final_trace ⇒
221      let current_instruction_cost ≝ current_instruction_cost … pre_fun_call in
222      let final_trace_cost ≝ compute_paid_trace_any_label … cost_labels end_flag … final_trace in
223        current_instruction_cost + final_trace_cost
224  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
225      let current_instruction_cost ≝ current_instruction_cost … status_pre in
226      let tail_trace_cost ≝
227       compute_paid_trace_any_label … cost_labels end_flag
228        status_init status_end tail_trace
229      in
230        current_instruction_cost + tail_trace_cost
231  | _ ⇒ ? (* XXX Paolo ??? *)
232  ].
233  cases daemon
234  qed.
235
236definition compute_paid_trace_label_label ≝
237  λcode_memory: BitVectorTrie Byte 16.
238  λcost_labels: BitVectorTrie costlabel 16.
239  λtrace_ends_flag: trace_ends_with_ret.
240  λstart_status: Status code_memory.
241  λfinal_status: Status code_memory.
242  λthe_trace: (trace_label_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status).
243  match the_trace with
244  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
245      compute_paid_trace_any_label … given_trace
246  ].
247
248lemma trace_compute_paid_trace_cl_other:
249  ∀code_memory' : (BitVectorTrie Byte 16).
250  ∀program_counter' : Word.
251  ∀cost_labels : BitVectorTrie costlabel 16.
252  ∀program_size' : ℕ.
253  ∀ticks : ℕ.
254  ∀instruction : instruction.
255  ∀program_counter'' : Word.
256  ∀FETCH : (〈instruction,program_counter'',ticks〉=fetch code_memory' program_counter').
257  let program_counter''' ≝ program_counter_after_other program_counter'' instruction in
258  ∀start_status : (Status code_memory').
259  ∀final_status : (Status code_memory').
260  ∀trace_ends_flag : trace_ends_with_ret.
261  ∀the_trace : (trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status).
262  ∀program_counter_refl : (program_counter' = program_counter … start_status).
263  ∀size_invariant : trace_any_label_length … the_trace ≤S program_size'.
264  ∀classify_assm: ASM_classify0 instruction = cl_other.
265  ∀pi1 : ℕ.
266   (if match lookup_opt costlabel 16 program_counter''' cost_labels with 
267         [None ⇒ true
268         |Some _ ⇒ false
269         ] 
270    then
271      ∀start_status0:Status code_memory'.
272      ∀final_status0:Status code_memory'.
273      ∀trace_ends_flag0:trace_ends_with_ret.
274      ∀the_trace0:trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag0 start_status0 final_status0.
275        trace_any_label_length … the_trace0 ≤ program_size' →
276        program_counter''' = program_counter … start_status0 →
277                  pi1
278                    =compute_paid_trace_any_label … the_trace0
279    else (pi1=O) )
280   → ticks+pi1
281     =compute_paid_trace_any_label … the_trace.
282  #code_memory' #program_counter' #cost_labels #program_size' #ticks #instruction
283  #program_counter'' #FETCH #start_status #final_status #trace_ends_flag #the_trace
284  #program_counter_refl #size_invariant #classify_assm #recursive_block_cost
285  #recursive_assm
286  @(trace_any_label_inv_ind … the_trace)
287    [2:
288      #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_assm
289      #start_status_refl #final_status_refl #the_trace_assm destruct @⊥
290    |3:
291      #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
292      #classifier_assm #after_return_assm #trace_label_return #costed_assm
293      #ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
294      destruct @⊥
295    |4: (* XXX *)
296      cases daemon
297    |5:
298      #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
299      #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
300      #costed_assm #trace_any_label #trace_ends_flag_refl #start_status_refl
301      #final_status_refl #the_trace_refl destruct @⊥
302    |6:
303      #end_flag #status_pre #status_init #status_end #execute_assm #trace_any_label
304      #classifier_assm #costed_assm #trace_ends_refl #start_status_refl #final_status_refl
305      #the_trace_refl
306      destruct
307      whd in match (trace_any_label_length … (tal_step_default …));
308      whd in match (compute_paid_trace_any_label … (tal_step_default …));
309      whd in costed_assm:(?%);
310      generalize in match costed_assm;
311      generalize in match (refl … (lookup_opt … (program_counter … (execute_1 … status_pre)) cost_labels));
312      whd in match (as_label ??);
313      generalize in match (lookup_opt … (program_counter … (execute_1 … status_pre)) cost_labels)
314        in ⊢ (??%? → % → ?);
315      #lookup_assm cases lookup_assm
316      [1:
317        #None_lookup_opt_assm normalize nodelta #_
318        generalize in match recursive_assm;
319        (*lapply (execute_1_and_program_counter_after_other_in_lockstep0 code_memory' ? (Some_inv … classifier_assm))*)
320        lapply (execute_1_and_program_counter_after_other_in_lockstep … classifier_assm)
321        <FETCH normalize nodelta #rewrite_assm >rewrite_assm in None_lookup_opt_assm;
322        #None_lookup_opt_assm <None_lookup_opt_assm
323        normalize nodelta #new_recursive_assm
324        cases(new_recursive_assm (execute_1 code_memory' status_pre) status_end
325          end_flag trace_any_label ? ?) try (>rewrite_assm %)
326        whd in match (current_instruction_cost … status_pre);
327        cut(ticks = \snd (fetch code_memory'
328           (program_counter … status_pre)))
329        [1,3:
330          <FETCH %
331        |2:
332          #ticks_refl_assm
333          >ticks_refl_assm %
334        |4:
335          #ticks_refl_assm
336          change with (S ? ≤ S ?) in size_invariant;
337          lapply (le_S_S_to_le … size_invariant) #assm
338          assumption
339        ]
340      |2:
341        #costlabel #Some_lookup_opt_assm <Some_lookup_opt_assm
342        #absurd
343        cases (not_Some_neq_None_to_False ?? absurd)
344      ]
345    |1:
346      #status_start #status_final #execute_assm #classifier_assm #costed_assm
347      #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
348      destruct
349      whd in match (trace_any_label_length … (tal_base_not_return …));
350      whd in match (compute_paid_trace_any_label … (tal_base_not_return …));
351      whd in costed_assm;
352      generalize in match costed_assm;
353      generalize in match (refl … (lookup_opt … (program_counter … (execute_1 … status_start)) cost_labels));
354      whd in match (as_label ??);
355      generalize in match (lookup_opt … (program_counter … (execute_1 … status_start)) cost_labels)
356        in ⊢ (??%? → % → ?);
357      #lookup_assm cases lookup_assm
358      [1:
359        #None_lookup_opt_assm
360        #absurd @⊥ cases absurd -absurd #absurd @absurd %
361      |2:
362        #costlabel #Some_lookup_opt_assm normalize nodelta #ignore
363        generalize in match recursive_assm;
364        cases classifier_assm -classifier_assm #classifier_assm
365        [2:
366          lapply (execute_1_and_program_counter_after_other_in_lockstep … classifier_assm)
367          <FETCH normalize nodelta #rewrite_assm >rewrite_assm in Some_lookup_opt_assm;
368          #Some_lookup_opt_assm <Some_lookup_opt_assm
369          normalize nodelta #new_recursive_assm >new_recursive_assm
370          cut(ticks = \snd (fetch code_memory'
371             (program_counter … status_start)))
372          [1:
373            <FETCH %
374          |2:
375            #ticks_refl_assm >ticks_refl_assm
376            <plus_n_O %
377          ]
378        |1:
379                (* JHM: wicked failure because Some prevent previous unfolding of ASM_classify here *)
380                @⊥
381                (* *** (* previously: unnecessary case analysis on instruction *)
382          change with ((Some ? (ASM_classify0 ?) = ?) → ?);
383          whd in match (current_instruction code_memory' status_start);
384          <FETCH generalize in match classify_assm;
385          cases instruction
386          [8:
387            #preinstruction normalize nodelta
388            whd in match ASM_classify0; normalize nodelta
389            #contradiction >contradiction #absurd destruct(absurd)
390          ]
391          try(#addr1 #addr2 normalize nodelta #ignore #absurd destruct(absurd))
392          try(#addr normalize nodelta #ignore #absurd destruct(absurd))
393          normalize in ignore; destruct(ignore)
394                *** *)
395        ]
396      ]
397    ]
398  change with (Some ? (ASM_classify0 ?) = ?) in classifier_assm;
399  whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
400  whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
401  <FETCH in classifier_assm; >classify_assm #absurd destruct(absurd)
402qed.
403
404lemma trace_compute_paid_trace_cl_jump:
405  ∀code_memory': BitVectorTrie Byte 16.
406  ∀program_counter': Word.
407  ∀cost_labels: BitVectorTrie costlabel 16.
408  ∀first_time_around: bool.
409  ∀program_size': ℕ.
410  ∀ticks: ℕ.
411  ∀instruction: instruction.
412  ∀program_counter'': Word.
413  ∀FETCH: 〈instruction,program_counter'',ticks〉 = fetch code_memory' program_counter'.
414  ∀start_status: (Status code_memory').
415  ∀final_status: (Status code_memory').
416  ∀trace_ends_flag: trace_ends_with_ret.
417  ∀the_trace: (trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status).
418  ∀program_counter_refl: (program_counter' = program_counter … start_status).
419  ∀classify_assm: ASM_classify0 instruction = cl_jump.
420    ticks
421     =compute_paid_trace_any_label … the_trace.
422  #code_memory' #program_counter' #cost_labels #first_time_around
423  #program_size' #ticks #instruction #program_counter'' #FETCH
424  #start_status #final_status
425  #trace_ends_flag #the_trace #program_counter_refl #classify_assm
426  @(trace_any_label_inv_ind … the_trace)
427  [6:
428    #end_flag #status_pre #status_init #status_end #execute_assm #trace_any_label
429    #classifier_assm #costed_assm #trace_ends_refl #start_status_refl #final_status_refl
430    #the_trace_refl destruct @⊥
431  |1:
432    #status_start #status_final #execute_assm #classifier_assm #costed_assm
433    #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
434    destruct
435    whd in match (trace_any_label_length … (tal_base_not_return …));
436    whd in match (compute_paid_trace_any_label … (tal_base_not_return …));
437    <FETCH %
438  |2:
439    #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_assm
440    #start_status_refl #final_status_refl #the_trace_assm destruct @⊥
441  |3:
442    #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
443    #classifier_assm #after_return_assm #trace_label_return #costed_assm
444    #ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
445    destruct @⊥
446  |4: (* XXX *)
447    cases daemon
448  |5:
449    #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
450    #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
451    #costed_assm #trace_any_label #trace_ends_flag_refl #start_status_refl
452    #final_status_refl #the_trace_refl destruct @⊥
453  ]
454  change with (Some ? (ASM_classify0 ?) = ?) in classifier_assm;
455  whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
456  whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
457  <FETCH in classifier_assm; >classify_assm #absurd destruct(absurd)
458qed.
459
460lemma trace_compute_paid_trace_cl_call:
461  ∀code_memory' : (BitVectorTrie Byte 16).
462  ∀program_counter' : Word.
463  ∀cost_labels : BitVectorTrie costlabel 16.
464  ∀program_size' : ℕ.
465  ∀ticks : ℕ.
466  ∀instruction : instruction.
467  ∀program_counter'' : Word.
468  ∀FETCH : (〈instruction,program_counter'',ticks〉=fetch code_memory' program_counter').
469  ∀start_status : (Status code_memory').
470  ∀final_status : (Status code_memory').
471  ∀trace_ends_flag : trace_ends_with_ret.
472  ∀the_trace : trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
473  ∀program_counter_refl : (program_counter' = program_counter … start_status).
474  ∀size_invariant : trace_any_label_length … the_trace ≤S program_size'.
475  ∀classify_assm: ASM_classify0 instruction = cl_call.
476  (∀pi1:ℕ
477  .if match lookup_opt costlabel 16 program_counter'' cost_labels with 
478      [None ⇒ true | Some _ ⇒ false] 
479   then (∀start_status0:Status code_memory'
480             .∀final_status0:Status code_memory'
481              .∀trace_ends_flag0:trace_ends_with_ret
482               .∀the_trace0:trace_any_label
483                                        (ASM_abstract_status … cost_labels)
484                                        trace_ends_flag0 start_status0 final_status0.
485                trace_any_label_length … the_trace0
486                   ≤ program_size' →
487                program_counter''
488                 =program_counter … start_status0
489                 → pi1
490                   =compute_paid_trace_any_label … the_trace0) 
491   else (pi1=O) 
492   → ticks+pi1
493     =compute_paid_trace_any_label … the_trace).
494  #code_memory' #program_counter' #cost_labels #program_size'
495  #ticks #instruction #program_counter'' #FETCH
496  #start_status #final_status #trace_ends_flag
497  #the_trace #program_counter_refl #size_invariant #classify_assm
498  #recursive_block_cost #recursive_assm
499  @(trace_any_label_inv_ind … the_trace)
500  [6:
501    #end_flag #status_pre #status_init #status_end #execute_assm #trace_any_label
502    #classifier_assm #costed_assm #trace_ends_refl #start_status_refl #final_status_refl
503    #the_trace_refl destruct @⊥
504  |1:
505    #status_start #status_final #execute_assm #classifier_assm #costed_assm
506    #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
507    destruct @⊥
508  |2:
509    #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_assm
510    #start_status_refl #final_status_refl #the_trace_assm destruct @⊥
511  |3:
512    #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
513    #classifier_assm #after_return_assm #trace_label_return #costed_assm
514    #ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
515    destruct
516    whd in match (trace_any_label_length … (tal_base_call …));
517    whd in match (compute_paid_trace_any_label … (tal_base_call …));
518    whd in costed_assm;
519    generalize in match costed_assm;
520    generalize in match (refl … (lookup_opt … (program_counter … status_final) cost_labels));
521    whd in match (as_label ??);
522    generalize in match (lookup_opt … (program_counter … status_final) cost_labels)
523      in ⊢ (??%? → % → ?);
524    #lookup_assm cases lookup_assm
525    [1:
526      #None_lookup_opt normalize nodelta #absurd cases absurd
527      -absurd #absurd @⊥ @absurd %
528    |2:
529      #costlabel #Some_lookup_opt normalize nodelta #ignore
530      generalize in match recursive_assm;
531      cut(program_counter'' = (program_counter … status_final))
532      [1:
533        generalize in match after_return_assm;
534        whd in ⊢ (% → ?); <FETCH normalize nodelta #relevant <relevant %
535      |2:
536        #program_counter_assm >program_counter_assm <Some_lookup_opt
537        normalize nodelta #new_recursive_assm >new_recursive_assm
538        cut(ticks = \snd (fetch code_memory' (program_counter … status_pre_fun_call)))
539        [1:
540          <FETCH %
541        |2:
542          #ticks_refl_assm >ticks_refl_assm
543          <plus_n_O %
544        ]
545      ]
546    ]
547  |4: (* XXX *)
548    cases daemon
549  |5:
550    #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
551    #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
552    #costed_assm #trace_any_label #trace_ends_flag_refl #start_status_refl
553    #final_status_refl #the_trace_refl
554    generalize in match execute_assm; destruct #execute_assm
555    whd in match (trace_any_label_length … (tal_step_call …));
556    whd in match (compute_paid_trace_any_label … (tal_step_call …));
557    whd in costed_assm:(?%);
558    generalize in match costed_assm;
559    generalize in match (refl … (lookup_opt … (program_counter … status_after_fun_call) cost_labels));
560    whd in match (as_label ??);
561    generalize in match (lookup_opt … (program_counter … status_after_fun_call) cost_labels)
562      in ⊢ (??%? → % → ?);
563    #lookup_assm cases lookup_assm
564    [1:
565      #None_lookup_opt_assm normalize nodelta #ignore
566      generalize in match recursive_assm;
567      cut(program_counter'' = program_counter … status_after_fun_call)
568      [1:
569        generalize in match after_return_assm;
570        whd in ⊢ (% → ?); <FETCH normalize nodelta #relevant >relevant %
571      |2:
572        #program_counter_refl >program_counter_refl <None_lookup_opt_assm
573        normalize nodelta #new_recursive_assm
574        cases (new_recursive_assm … trace_any_label ? ?)
575        [1:
576          @plus_right_monotone whd in ⊢ (???%); <FETCH %
577        |2:
578          @le_S_S_to_le @size_invariant
579        |3:
580          %
581        ]
582      ]
583    |2:
584      #cost_label #Some_lookup_opt_assm #absurd
585      cases (not_Some_neq_None_to_False ?? absurd)
586    ]
587  ]
588  try (change with (Some ? (ASM_classify0 ?) = ? ∨ Some ? (ASM_classify0 ?) = ?) in classifier_assm;)
589  try (change with (Some ? (ASM_classify0 ?) = ?) in classifier_assm;)
590  whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
591  whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
592  <FETCH in classifier_assm; >classify_assm #absurd destruct(absurd) cases absurd
593  #absurd destruct(absurd)
594qed.
595
596lemma trace_compute_paid_trace_cl_return:
597  ∀code_memory' : (BitVectorTrie Byte 16).
598  ∀program_counter' : Word.
599  ∀cost_labels : BitVectorTrie costlabel 16.
600  ∀program_size' : ℕ.
601  ∀ticks : ℕ.
602  ∀instruction : instruction.
603  ∀program_counter'' : Word.
604  ∀FETCH : (〈instruction,program_counter'',ticks〉=fetch code_memory' program_counter').
605  ∀start_status : (Status code_memory').
606  ∀final_status : (Status code_memory').
607  ∀trace_ends_flag : trace_ends_with_ret.
608  ∀the_trace : trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
609  ∀program_counter_refl : (program_counter' = program_counter … start_status).
610  ∀classify_assm: ASM_classify0 instruction = cl_return.
611    ticks
612     =compute_paid_trace_any_label … the_trace.
613  #code_memory' #program_counter' #cost_labels #program_size'
614  #ticks #instruction #program_counter'' #FETCH
615  #start_status #final_status #trace_ends_flag
616  #the_trace #program_counter_refl #classify_assm
617  @(trace_any_label_inv_ind … the_trace)
618  [1:
619    #start_status' #final_status' #execute_assm #classifier_assm #costed_assm
620    #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
621    destruct @⊥
622  |2:
623    #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_flag_refl
624    #start_status_refl #final_status_refl #the_trace_refl destruct
625    whd in match (trace_any_label_length … (tal_base_return …));
626    whd in match (compute_paid_trace_any_label … (tal_base_return …));
627    <FETCH %
628  |3:
629    #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
630    #classifier_assm #after_return_assm #trace_label_return #costed_assm
631    #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
632    destruct @⊥
633  |4: (* XXX *)
634    cases daemon
635  |5:
636    #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
637    #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
638    #costed_assm #trace_any_label #trace_ends_flag_refl #start_status_refl
639    #final_status_refl #the_trace_refl
640    destruct @⊥
641  |6:
642    #end_flag #status_pre #status_init #status_end #execute_assm #trace_any_label
643    #classifier_assm #costed_assm #trace_ends_flag_refl #start_status_refl
644    #final_status_refl #the_trace_refl destruct @⊥
645  ]
646  try (change with (Some ? (ASM_classify0 ?) = ? ∨ Some ? (ASM_classify0 ?) = ?) in classifier_assm;)
647  try (change with (Some ? (ASM_classify0 ?) = ?) in classifier_assm;)
648  (*
649  try (change with (ASM_classify0 ? = ? ∨ ASM_classify0 ? = ?) in classifier_assm;)
650  try (change with (ASM_classify0 ? = ?) in classifier_assm;)
651  *)
652  whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
653  whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
654  <FETCH in classifier_assm; >classify_assm
655  #absurd try (destruct(absurd))
656  cases absurd
657  #absurd destruct(absurd)
658qed.
659
660lemma trace_any_label_length_leq_0_to_False:
661  ∀code_memory: BitVectorTrie Byte 16.
662  ∀cost_labels: BitVectorTrie costlabel 16.
663  ∀trace_ends_flag: trace_ends_with_ret.
664  ∀start_status: Status code_memory.
665  ∀final_status: Status code_memory.
666  ∀the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
667   trace_any_label_length … the_trace ≤ 0 → False.
668  #code_memory #cost_labels #trace_ends_flag #start_status #final_status
669  #the_trace
670  cases the_trace /2/
671qed.
672     
673let rec block_cost'
674  (code_memory': BitVectorTrie Byte 16) (program_counter': Word)
675    (program_size: nat) (cost_labels: BitVectorTrie costlabel 16)
676      (first_time_around: bool)
677        on program_size:
678          Σcost_of_block: nat.
679          if (match lookup_opt … program_counter' cost_labels with [ None ⇒ true | _ ⇒ first_time_around ]) then
680            ∀start_status: Status code_memory'.
681            ∀final_status: Status code_memory'.
682            ∀trace_ends_flag.
683            ∀the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
684              trace_any_label_length … the_trace ≤ program_size →
685              program_counter' = program_counter … start_status →
686                cost_of_block = compute_paid_trace_any_label … the_trace
687          else
688            (cost_of_block = 0) ≝
689  match program_size return λx. x = program_size → ? with
690  [ O ⇒ λbase_case. 0 (* XXX: dummy to be inserted here *)
691  | S program_size' ⇒ λstep_case.
692    let 〈instruction, program_counter'', ticks〉 as FETCH ≝ fetch code_memory' program_counter' in
693    let to_continue ≝
694      match lookup_opt … program_counter' cost_labels with
695      [ None ⇒ true
696      | Some _ ⇒ first_time_around
697      ]
698    in
699      ((if to_continue then
700       pi1 … (match instruction return λx. x = instruction → ? with
701        [ RealInstruction real_instruction ⇒ λreal_instruction_refl.
702          match real_instruction return λx. x = real_instruction →
703          Σcost_of_block: nat.
704            ∀start_status: Status code_memory'.
705            ∀final_status: Status code_memory'.
706            ∀trace_ends_flag.
707            ∀the_trace: trace_any_label (ASM_abstract_status code_memory' cost_labels) trace_ends_flag start_status final_status.
708              trace_any_label_length … the_trace ≤ S program_size' →
709              program_counter' = program_counter … start_status →
710                cost_of_block = compute_paid_trace_any_label … the_trace with
711          [ RET                    ⇒ λinstr. ticks
712          | RETI                   ⇒ λinstr. ticks
713          | JC   relative          ⇒ λinstr. ticks
714          | JNC  relative          ⇒ λinstr. ticks
715          | JB   bit_addr relative ⇒ λinstr. ticks
716          | JNB  bit_addr relative ⇒ λinstr. ticks
717          | JBC  bit_addr relative ⇒ λinstr. ticks
718          | JZ   relative          ⇒ λinstr. ticks
719          | JNZ  relative          ⇒ λinstr. ticks
720          | CJNE src_trgt relative ⇒ λinstr. ticks
721          | DJNZ src_trgt relative ⇒ λinstr. ticks
722          | _                      ⇒ λinstr.
723              ticks + block_cost' code_memory' program_counter'' program_size' cost_labels false
724          ] (refl …)
725        | ACALL addr     ⇒ λinstr.
726            ticks + block_cost' code_memory' program_counter'' program_size' cost_labels false
727        | AJMP  addr     ⇒ λinstr.
728          let jump_target ≝ compute_target_of_unconditional_jump program_counter'' instruction in
729            ticks + block_cost' code_memory' jump_target program_size' cost_labels false
730        | LCALL addr     ⇒ λinstr.
731            ticks + block_cost' code_memory' program_counter'' program_size' cost_labels false
732        | LJMP  addr     ⇒ λinstr.
733          let jump_target ≝ compute_target_of_unconditional_jump program_counter'' instruction in
734            ticks + block_cost' code_memory' jump_target program_size' cost_labels false
735        | SJMP  addr     ⇒ λinstr.
736          let jump_target ≝ compute_target_of_unconditional_jump program_counter'' instruction in
737            ticks + block_cost' code_memory' jump_target program_size' cost_labels false
738        | JMP   addr     ⇒ λinstr. (* XXX: actually a call due to use with fptrs *)
739            ticks + block_cost' code_memory' program_counter'' program_size' cost_labels false
740        | MOVC  src trgt ⇒ λinstr.
741            ticks + block_cost' code_memory' program_counter'' program_size' cost_labels false
742        ] (refl …))
743      else
744        0)
745      : Σcost_of_block: nat.
746          match (match lookup_opt … program_counter' cost_labels with [ None ⇒ true | _ ⇒ first_time_around ]) with
747          [ true ⇒
748            ∀start_status: Status code_memory'.
749            ∀final_status: Status code_memory'.
750            ∀trace_ends_flag.
751            ∀the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
752              trace_any_label_length … the_trace ≤ S program_size' →
753              program_counter' = program_counter … start_status →
754                cost_of_block = compute_paid_trace_any_label … the_trace
755          | false ⇒
756            (cost_of_block = 0)
757          ])
758  ] (refl …).
759  [2:
760    change with (if to_continue then ? else (? = 0))
761    >p in ⊢ (match % return ? with [ _ ⇒ ? | _ ⇒ ? ]); normalize nodelta
762    @pi2
763  |1:
764    destruct
765    cases (lookup_opt ????) normalize nodelta
766    [1:
767      #start_status #final_status #trace_ends_flag #the_trace
768      #absurd
769      cases (trace_any_label_length_leq_0_to_False … absurd)
770    |2:
771      #cost cases first_time_around normalize nodelta try %
772      #start_status #final_status #trace_ends_flag #the_trace #absurd
773      cases (trace_any_label_length_leq_0_to_False … absurd)
774    ]
775  |3:
776    change with (if to_continue then ? else (0 = 0))
777    >p normalize nodelta %
778  |5,6:
779    #start_status #final_status #trace_ends_flag #the_trace #size_invariant #program_counter_refl
780    @(trace_compute_paid_trace_cl_return … program_size' … FETCH … the_trace program_counter_refl)
781    destruct %
782  |42,46,47:
783    #start_status #final_status #trace_ends_flag #the_trace #size_invariant #program_counter_refl
784    cases(block_cost' ?????) -block_cost'
785    @(trace_compute_paid_trace_cl_call … program_size' … FETCH … the_trace program_counter_refl size_invariant)
786    destruct %
787  |43,44,45: (* XXX: unconditional jumps *)
788    #start_status #final_status #trace_ends_flag #the_trace #size_invariant #program_counter_refl
789    cases (block_cost' ?????) -block_cost'
790    lapply (trace_compute_paid_trace_cl_other … program_size' … FETCH … the_trace program_counter_refl size_invariant)
791    whd in match (program_counter_after_other ??); normalize nodelta destruct
792    whd in match (is_unconditional_jump ?); normalize nodelta #assm @assm %
793  |24,25,26,27,28,29,30,31,32:
794    #start_status #final_status #trace_ends_flag #the_trace #size_invariant #program_counter_refl
795    @(trace_compute_paid_trace_cl_jump … first_time_around program_size' … FETCH … the_trace program_counter_refl)
796    destruct %
797  |*:
798    #start_status #final_status #trace_ends_flag #the_trace #size_invariant #program_counter_refl
799    cases(block_cost' ?????) -block_cost'
800    lapply (trace_compute_paid_trace_cl_other … program_size' … FETCH … the_trace program_counter_refl size_invariant)     
801    destruct #assm @assm %
802  ]
803qed.
804
805definition block_cost:
806    ∀code_memory': BitVectorTrie Byte 16.
807    ∀program_counter': Word.
808    ∀cost_labels: BitVectorTrie costlabel 16.
809      Σcost_of_block: nat.
810        ∀start_status: Status code_memory'.
811        ∀final_status: Status code_memory'.
812        ∀trace_ends_flag.
813        ∀the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
814        ∀unrepeating_witness: tal_unrepeating … the_trace.
815          program_counter' = program_counter … start_status →
816            cost_of_block = compute_paid_trace_any_label … the_trace ≝
817  λcode_memory: BitVectorTrie Byte 16.
818  λprogram_counter: Word.
819  λcost_labels: BitVectorTrie costlabel 16. ?.
820  cases(block_cost' code_memory program_counter (2^16) cost_labels true)
821  #cost_of_block #block_cost_hyp
822  %{cost_of_block}
823  cases(lookup_opt … cost_labels) in block_cost_hyp;
824  [2: #cost_label] normalize nodelta
825  #hyp #start_status #final_status #trace_ends_flag #the_trace #unrepeating_witness
826  @hyp @tal_pc_list_length_leq_total_program_size try assumption
827qed.
828
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