source: src/ASM/ASMCosts.ma @ 1665

Last change on this file since 1665 was 1665, checked in by mulligan, 9 years ago

progress on closing holes in block_cost' proof
File size: 50.5 KB
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1include "ASM/ASM.ma".
2include "ASM/Arithmetic.ma".
3include "ASM/Fetch.ma".
4include "ASM/Interpret.ma".
5include "common/StructuredTraces.ma".
6
7let rec fetch_program_counter_n
8  (n: nat) (code_memory: BitVectorTrie Byte 16) (program_counter: Word)
9    on n: option Word ≝
10  match n with
11  [ O ⇒ Some … program_counter
12  | S n ⇒
13    match fetch_program_counter_n n code_memory program_counter with
14    [ None ⇒ None …
15    | Some tail_pc ⇒
16      let 〈instr, program_counter, ticks〉 ≝ fetch code_memory tail_pc in
17        if ltb (nat_of_bitvector … tail_pc) (nat_of_bitvector … program_counter) then
18          Some … program_counter
19        else
20          None Word (* XXX: overflow! *)
21    ]
22  ].
23   
24definition reachable_program_counter: BitVectorTrie Byte 16 → nat → Word → Prop ≝
25  λcode_memory: BitVectorTrie Byte 16.
26  λprogram_size: nat.
27  λprogram_counter: Word.
28    (∃n: nat. Some … program_counter = fetch_program_counter_n n code_memory (zero 16)) ∧
29        nat_of_bitvector 16 program_counter < program_size.
30   
31definition good_program: ∀code_memory: BitVectorTrie Byte 16. ∀total_program_size: nat. Prop ≝
32  λcode_memory: BitVectorTrie Byte 16.
33  λtotal_program_size: nat.
34  ∀program_counter: Word.
35  ∀good_program_counter_witness: reachable_program_counter code_memory total_program_size program_counter.
36  let 〈instruction, program_counter', ticks〉 ≝ fetch code_memory program_counter in
37    match instruction with
38    [ RealInstruction instr ⇒
39      match instr with
40      [ RET                    ⇒ True
41      | JC   relative          ⇒ True (* XXX: see below *)
42      | JNC  relative          ⇒ True (* XXX: see below *)
43      | JB   bit_addr relative ⇒ True
44      | JNB  bit_addr relative ⇒ True
45      | JBC  bit_addr relative ⇒ True
46      | JZ   relative          ⇒ True
47      | JNZ  relative          ⇒ True
48      | CJNE src_trgt relative ⇒ True
49      | DJNZ src_trgt relative ⇒ True
50      | _                      ⇒
51        nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
52          nat_of_bitvector … program_counter' < total_program_size
53      ]
54    | LCALL addr         ⇒
55      match addr return λx. bool_to_Prop (is_in … [[ addr16 ]] x) → Prop with
56      [ ADDR16 addr ⇒ λaddr16: True.
57          reachable_program_counter code_memory total_program_size addr ∧
58            nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
59              nat_of_bitvector … program_counter' < total_program_size
60      | _ ⇒ λother: False. ⊥
61      ] (subaddressing_modein … addr)
62    | ACALL addr         ⇒
63      match addr return λx. bool_to_Prop (is_in … [[ addr11 ]] x) → Prop with
64      [ ADDR11 addr ⇒ λaddr11: True.
65        let 〈pc_bu, pc_bl〉 ≝ split … 8 8 program_counter' in
66        let 〈thr, eig〉 ≝ split … 3 8 addr in
67        let 〈fiv, thr'〉 ≝ split … 5 3 pc_bu in
68        let new_program_counter ≝ (fiv @@ thr) @@ pc_bl in
69          reachable_program_counter code_memory total_program_size new_program_counter ∧
70            nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
71              nat_of_bitvector … program_counter' < total_program_size
72      | _ ⇒ λother: False. ⊥
73      ] (subaddressing_modein … addr)
74    | AJMP  addr         ⇒
75      match addr return λx. bool_to_Prop (is_in … [[ addr11 ]] x) → Prop with
76      [ ADDR11 addr ⇒ λaddr11: True.
77        let 〈pc_bu, pc_bl〉 ≝ split … 8 8 program_counter' in
78        let 〈nu, nl〉 ≝ split … 4 4 pc_bu in
79        let bit ≝ get_index' … O ? nl in
80        let 〈relevant1, relevant2〉 ≝ split … 3 8 addr in
81        let new_addr ≝ (nu @@ (bit ::: relevant1)) @@ relevant2 in
82        let 〈carry, new_program_counter〉 ≝ half_add 16 program_counter new_addr in
83          reachable_program_counter code_memory total_program_size new_program_counter
84      | _ ⇒ λother: False. ⊥
85      ] (subaddressing_modein … addr)
86    | LJMP  addr         ⇒
87      match addr return λx. bool_to_Prop (is_in … [[ addr16 ]] x) → Prop with
88      [ ADDR16 addr ⇒ λaddr16: True.
89          reachable_program_counter code_memory total_program_size addr
90      | _ ⇒ λother: False. ⊥
91      ] (subaddressing_modein … addr)
92    | SJMP  addr     ⇒
93      match addr return λx. bool_to_Prop (is_in … [[ relative ]] x) → Prop with
94      [ RELATIVE addr ⇒ λrelative: True.
95        let 〈carry, new_program_counter〉 ≝ half_add … program_counter' (sign_extension addr) in
96          reachable_program_counter code_memory total_program_size new_program_counter
97      | _ ⇒ λother: False. ⊥
98      ] (subaddressing_modein … addr)
99    | JMP   addr     ⇒ (* XXX: JMP is used for fptrs and unconstrained *)
100      nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
101        nat_of_bitvector … program_counter' < total_program_size
102    | MOVC  src trgt ⇒
103        nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
104          nat_of_bitvector … program_counter' < total_program_size
105    ].
106  cases other
107qed.
108       
109lemma is_a_decidable:
110  ∀hd.
111  ∀element.
112    is_a hd element = true ∨ is_a hd element = false.
113  #hd #element //
114qed.
115
116lemma mem_decidable:
117  ∀n: nat.
118  ∀v: Vector addressing_mode_tag n.
119  ∀element: addressing_mode_tag.
120    mem … eq_a n v element = true ∨
121      mem … eq_a … v element = false.
122  #n #v #element //
123qed.
124
125lemma eq_a_elim:
126  ∀tag.
127  ∀hd.
128  ∀P: bool → Prop.
129    (tag = hd → P (true)) →
130      (tag ≠ hd → P (false)) →
131        P (eq_a tag hd).
132  #tag #hd #P
133  cases tag
134  cases hd
135  #true_hyp #false_hyp
136  try @false_hyp
137  try @true_hyp
138  try %
139  #absurd destruct(absurd)
140qed.
141 
142lemma is_a_true_to_is_in:
143  ∀n: nat.
144  ∀x: addressing_mode.
145  ∀tag: addressing_mode_tag.
146  ∀supervector: Vector addressing_mode_tag n.
147  mem addressing_mode_tag eq_a n supervector tag →
148    is_a tag x = true →
149      is_in … supervector x.
150  #n #x #tag #supervector
151  elim supervector
152  [1:
153    #absurd cases absurd
154  |2:
155    #n' #hd #tl #inductive_hypothesis
156    whd in match (mem … eq_a (S n') (hd:::tl) tag);
157    @eq_a_elim normalize nodelta
158    [1:
159      #tag_hd_eq #irrelevant
160      whd in match (is_in (S n') (hd:::tl) x);
161      <tag_hd_eq #is_a_hyp >is_a_hyp normalize nodelta
162      @I
163    |2:
164      #tag_hd_neq
165      whd in match (is_in (S n') (hd:::tl) x);
166      change with (
167        mem … eq_a n' tl tag)
168          in match (fold_right … n' ? false tl);
169      #mem_hyp #is_a_hyp
170      cases(is_a hd x)
171      [1:
172        normalize nodelta //
173      |2:
174        normalize nodelta
175        @inductive_hypothesis assumption
176      ]
177    ]
178  ]
179qed.
180
181lemma is_in_subvector_is_in_supervector:
182  ∀m, n: nat.
183  ∀subvector: Vector addressing_mode_tag m.
184  ∀supervector: Vector addressing_mode_tag n.
185  ∀element: addressing_mode.
186    subvector_with … eq_a subvector supervector →
187      is_in m subvector element → is_in n supervector element.
188  #m #n #subvector #supervector #element
189  elim subvector
190  [1:
191    #subvector_with_proof #is_in_proof
192    cases is_in_proof
193  |2:
194    #n' #hd' #tl' #inductive_hypothesis #subvector_with_proof
195    whd in match (is_in … (hd':::tl') element);
196    cases (is_a_decidable hd' element)
197    [1:
198      #is_a_true >is_a_true
199      #irrelevant
200      whd in match (subvector_with … eq_a (hd':::tl') supervector) in subvector_with_proof;
201      @(is_a_true_to_is_in … is_a_true)
202      lapply(subvector_with_proof)
203      cases(mem … eq_a n supervector hd') //
204    |2:
205      #is_a_false >is_a_false normalize nodelta
206      #assm
207      @inductive_hypothesis
208      [1:
209        generalize in match subvector_with_proof;
210        whd in match (subvector_with … eq_a (hd':::tl') supervector);
211        cases(mem_decidable n supervector hd')
212        [1:
213          #mem_true >mem_true normalize nodelta
214          #assm assumption
215        |2:
216          #mem_false >mem_false #absurd
217          cases absurd
218        ]
219      |2:
220        assumption
221      ]
222    ]
223  ]
224qed.
225
226let rec member_addressing_mode_tag
227  (n: nat) (v: Vector addressing_mode_tag n) (a: addressing_mode_tag)
228    on v: Prop ≝
229  match v with
230  [ VEmpty ⇒ False
231  | VCons n' hd tl ⇒
232      bool_to_Prop (eq_a hd a) ∨ member_addressing_mode_tag n' tl a
233  ].
234 
235let rec subaddressing_mode_elim_type
236  (T: Type[2]) (m: nat) (fixed_v: Vector addressing_mode_tag m)
237    (Q: addressing_mode → T → Prop)
238      (p_addr11:            ∀w: Word11.      is_in m fixed_v (ADDR11 w)        → T)
239      (p_addr16:            ∀w: Word.        is_in m fixed_v (ADDR16 w)        → T)
240      (p_direct:            ∀w: Byte.        is_in m fixed_v (DIRECT w)        → T)
241      (p_indirect:          ∀w: Bit.         is_in m fixed_v (INDIRECT w)      → T)
242      (p_ext_indirect:      ∀w: Bit.         is_in m fixed_v (EXT_INDIRECT w)  → T)
243      (p_acc_a:                              is_in m fixed_v ACC_A             → T)
244      (p_register:          ∀w: BitVector 3. is_in m fixed_v (REGISTER w)      → T)
245      (p_acc_b:                              is_in m fixed_v ACC_B             → T)
246      (p_dptr:                               is_in m fixed_v DPTR              → T)
247      (p_data:              ∀w: Byte.        is_in m fixed_v (DATA w)          → T)
248      (p_data16:            ∀w: Word.        is_in m fixed_v (DATA16 w)        → T)
249      (p_acc_dptr:                           is_in m fixed_v ACC_DPTR          → T)
250      (p_acc_pc:                             is_in m fixed_v ACC_PC            → T)
251      (p_ext_indirect_dptr:                  is_in m fixed_v EXT_INDIRECT_DPTR → T)
252      (p_indirect_dptr:                      is_in m fixed_v INDIRECT_DPTR     → T)
253      (p_carry:                              is_in m fixed_v CARRY             → T)
254      (p_bit_addr:          ∀w: Byte.        is_in m fixed_v (BIT_ADDR w)      → T)
255      (p_n_bit_addr:        ∀w: Byte.        is_in m fixed_v (N_BIT_ADDR w)    → T)
256      (p_relative:          ∀w: Byte.        is_in m fixed_v (RELATIVE w)      → T)
257        (n: nat) (v: Vector addressing_mode_tag n) (proof: subvector_with … eq_a v fixed_v)
258      on v: Prop ≝
259  match v return λo: nat. λv': Vector addressing_mode_tag o. o = n → v ≃ v' → ? with
260  [ VEmpty         ⇒ λm_refl. λv_refl.
261      ∀addr: addressing_mode. ∀p: is_in m fixed_v addr.
262        Q addr (
263        match addr return λx: addressing_mode. is_in … fixed_v x → T with 
264        [ ADDR11 x          ⇒ p_addr11 x
265        | ADDR16 x          ⇒ p_addr16 x
266        | DIRECT x          ⇒ p_direct x
267        | INDIRECT x        ⇒ p_indirect x
268        | EXT_INDIRECT x    ⇒ p_ext_indirect x
269        | ACC_A             ⇒ p_acc_a
270        | REGISTER x        ⇒ p_register x
271        | ACC_B             ⇒ p_acc_b
272        | DPTR              ⇒ p_dptr
273        | DATA x            ⇒ p_data x
274        | DATA16 x          ⇒ p_data16 x
275        | ACC_DPTR          ⇒ p_acc_dptr
276        | ACC_PC            ⇒ p_acc_pc
277        | EXT_INDIRECT_DPTR ⇒ p_ext_indirect_dptr
278        | INDIRECT_DPTR     ⇒ p_indirect_dptr
279        | CARRY             ⇒ p_carry
280        | BIT_ADDR x        ⇒ p_bit_addr x
281        | N_BIT_ADDR x      ⇒ p_n_bit_addr x
282        | RELATIVE x        ⇒ p_relative x
283        ] p)
284  | VCons n' hd tl ⇒ λm_refl. λv_refl.
285    let tail_call ≝ subaddressing_mode_elim_type T m fixed_v Q p_addr11
286      p_addr16 p_direct p_indirect p_ext_indirect p_acc_a
287        p_register p_acc_b p_dptr p_data p_data16 p_acc_dptr
288          p_acc_pc p_ext_indirect_dptr p_indirect_dptr p_carry
289            p_bit_addr p_n_bit_addr p_relative n' tl ?
290    in
291    match hd return λa: addressing_mode_tag. a = hd → ? with
292    [ addr11            ⇒ λhd_refl. (∀w. Q (ADDR11 w) (p_addr11 w ?)) → tail_call
293    | addr16            ⇒ λhd_refl. (∀w. Q (ADDR16 w) (p_addr16 w ?)) → tail_call
294    | direct            ⇒ λhd_refl. (∀w. Q (DIRECT w) (p_direct w ?)) → tail_call
295    | indirect          ⇒ λhd_refl. (∀w. Q (INDIRECT w) (p_indirect w ?)) → tail_call
296    | ext_indirect      ⇒ λhd_refl. (∀w. Q (EXT_INDIRECT w) (p_ext_indirect w ?)) → tail_call
297    | acc_a             ⇒ λhd_refl. (Q ACC_A (p_acc_a ?)) → tail_call
298    | registr           ⇒ λhd_refl. (∀w. Q (REGISTER w) (p_register w ?)) → tail_call
299    | acc_b             ⇒ λhd_refl. (Q ACC_A (p_acc_b ?)) → tail_call
300    | dptr              ⇒ λhd_refl. (Q DPTR (p_dptr ?)) → tail_call
301    | data              ⇒ λhd_refl. (∀w. Q (DATA w) (p_data w ?)) → tail_call
302    | data16            ⇒ λhd_refl. (∀w. Q (DATA16 w) (p_data16 w ?)) → tail_call
303    | acc_dptr          ⇒ λhd_refl. (Q ACC_DPTR (p_acc_dptr ?)) → tail_call
304    | acc_pc            ⇒ λhd_refl. (Q ACC_PC (p_acc_pc ?)) → tail_call
305    | ext_indirect_dptr ⇒ λhd_refl. (Q EXT_INDIRECT_DPTR (p_ext_indirect_dptr ?)) → tail_call
306    | indirect_dptr     ⇒ λhd_refl. (Q INDIRECT_DPTR (p_indirect_dptr ?)) → tail_call
307    | carry             ⇒ λhd_refl. (Q CARRY (p_carry ?)) → tail_call
308    | bit_addr          ⇒ λhd_refl. (∀w. Q (BIT_ADDR w) (p_bit_addr w ?)) → tail_call
309    | n_bit_addr        ⇒ λhd_refl. (∀w. Q (N_BIT_ADDR w) (p_n_bit_addr w ?)) → tail_call
310    | relative          ⇒ λhd_refl. (∀w. Q (RELATIVE w) (p_relative w ?)) → tail_call
311    ] (refl … hd)
312  ] (refl … n) (refl_jmeq … v).
313  [20:
314    generalize in match proof; destruct
315    whd in match (subvector_with … eq_a (hd:::tl) fixed_v);
316    cases (mem … eq_a m fixed_v hd) normalize nodelta
317    [1:
318      whd in match (subvector_with … eq_a tl fixed_v);
319      #assm assumption
320    |2:
321      normalize in ⊢ (% → ?);
322      #absurd cases absurd
323    ]
324  ]
325  @(is_in_subvector_is_in_supervector … proof)
326  destruct @I
327qed.
328
329lemma subaddressing_mode_elim':
330  ∀T: Type[2].
331  ∀n: nat.
332  ∀o: nat.
333  ∀Q: addressing_mode → T → Prop.
334  ∀fixed_v: Vector addressing_mode_tag (n + o).
335  ∀P1,P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13,P14,P15,P16,P17,P18,P19.
336  ∀v1: Vector addressing_mode_tag n.
337  ∀v2: Vector addressing_mode_tag o.
338  ∀fixed_v_proof: fixed_v = v1 @@ v2.
339  ∀subaddressing_mode_proof.
340    subaddressing_mode_elim_type T (n + o) fixed_v Q P1 P2 P3 P4 P5 P6 P7
341      P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 (n + o) (v1 @@ v2) subaddressing_mode_proof.
342  #T #n #o #Q #fixed_v #P1 #P2 #P3 #P4 #P5 #P6 #P7 #P8 #P9 #P10
343  #P11 #P12 #P13 #P14 #P15 #P16 #P17 #P18 #P19 #v1 #v2 #fixed_v_proof
344  cases daemon
345qed.
346
347axiom subaddressing_mode_elim:
348  ∀T: Type[2].
349  ∀m: nat.
350  ∀n: nat.
351  ∀Q: addressing_mode → T → Prop.
352  ∀fixed_v: Vector addressing_mode_tag m.
353  ∀P1,P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13,P14,P15,P16,P17,P18,P19.
354  ∀v: Vector addressing_mode_tag n.
355  ∀proof.
356    subaddressing_mode_elim_type T m fixed_v Q P1 P2 P3 P4 P5 P6 P7
357      P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 n v proof.
358
359definition current_instruction0 ≝
360  λcode_memory: BitVectorTrie Byte 16.
361  λprogram_counter: Word.
362    \fst (\fst (fetch … code_memory program_counter)).
363
364definition current_instruction ≝
365  λs: Status.
366    current_instruction0 (code_memory … s) (program_counter … s).
367
368definition ASM_classify0: instruction → status_class ≝
369  λi: instruction.
370  match i with
371   [ RealInstruction pre ⇒
372     match pre with
373      [ RET ⇒ cl_return
374      | JZ _ ⇒ cl_jump
375      | JNZ _ ⇒ cl_jump
376      | JC _ ⇒ cl_jump
377      | JNC _ ⇒ cl_jump
378      | JB _ _ ⇒ cl_jump
379      | JNB _ _ ⇒ cl_jump
380      | JBC _ _ ⇒ cl_jump
381      | CJNE _ _ ⇒ cl_jump
382      | DJNZ _ _ ⇒ cl_jump
383      | _ ⇒ cl_other
384      ]
385   | ACALL _ ⇒ cl_call
386   | LCALL _ ⇒ cl_call
387   | JMP _ ⇒ cl_call
388   | AJMP _ ⇒ cl_jump
389   | LJMP _ ⇒ cl_jump
390   | SJMP _ ⇒ cl_jump
391   | _ ⇒ cl_other
392   ].
393
394definition ASM_classify: Status → status_class ≝
395  λs: Status.
396    ASM_classify0 (current_instruction s).
397
398definition current_instruction_is_labelled ≝
399  λcost_labels: BitVectorTrie costlabel 16.
400  λs: Status.
401  let pc ≝ program_counter … s in
402    match lookup_opt … pc cost_labels with
403    [ None ⇒ False
404    | _    ⇒ True
405    ].
406
407definition next_instruction_properly_relates_program_counters ≝
408  λbefore: Status.
409  λafter : Status.
410  let size ≝ current_instruction_cost before in
411  let pc_before ≝ program_counter … before in
412  let pc_after ≝ program_counter … after in
413  let sum ≝ \snd (half_add … pc_before (bitvector_of_nat … size)) in
414    sum = pc_after.
415
416definition ASM_abstract_status: BitVectorTrie costlabel 16 → abstract_status ≝
417 λcost_labels.
418  mk_abstract_status
419   Status
420   (λs,s'. (execute_1 s) = s')
421   (λs,class. ASM_classify s = class)
422   (current_instruction_is_labelled cost_labels)
423   next_instruction_properly_relates_program_counters.
424
425let rec trace_any_label_length
426  (cost_labels: BitVectorTrie costlabel 16) (trace_ends_flag: trace_ends_with_ret)
427    (start_status: Status) (final_status: Status)
428      (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status)
429        on the_trace: nat ≝
430  match the_trace with
431  [ tal_base_not_return the_status _ _ _ _ ⇒ 0
432  | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒ 0
433  | tal_base_return the_status _ _ _ ⇒ 0
434  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final _ _ _ call_trace _ final_trace ⇒
435      let tail_length ≝ trace_any_label_length … final_trace in
436      let pc_difference ≝ nat_of_bitvector … (program_counter … after_fun_call) - nat_of_bitvector … (program_counter … pre_fun_call) in       
437        pc_difference + tail_length
438  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
439      let tail_length ≝ trace_any_label_length … tail_trace in
440      let pc_difference ≝ nat_of_bitvector … (program_counter … status_init) - nat_of_bitvector … (program_counter … status_pre) in
441        pc_difference + tail_length       
442  ].
443
444let rec compute_paid_trace_any_label
445  (cost_labels: BitVectorTrie costlabel 16)
446  (trace_ends_flag: trace_ends_with_ret)
447   (start_status: Status) (final_status: Status)
448     (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status)
449       on the_trace: nat ≝
450  match the_trace with
451  [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status
452  | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status
453  | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒ current_instruction_cost pre_fun_call
454  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
455     _ _ _ call_trace _ final_trace ⇒
456      let current_instruction_cost ≝ current_instruction_cost pre_fun_call in
457      let final_trace_cost ≝ compute_paid_trace_any_label cost_labels end_flag … final_trace in
458        current_instruction_cost + final_trace_cost
459  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
460      let current_instruction_cost ≝ current_instruction_cost status_pre in
461      let tail_trace_cost ≝
462       compute_paid_trace_any_label cost_labels end_flag
463        status_init status_end tail_trace
464      in
465        current_instruction_cost + tail_trace_cost
466  ].
467
468definition compute_paid_trace_label_label ≝
469 λcost_labels: BitVectorTrie costlabel 16.
470  λtrace_ends_flag: trace_ends_with_ret.
471   λstart_status: Status.
472    λfinal_status: Status.
473     λthe_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag
474      start_status final_status.
475  match the_trace with
476  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
477      compute_paid_trace_any_label … given_trace
478  ].
479
480include alias "arithmetics/nat.ma".
481include alias "basics/logic.ma".
482
483lemma plus_right_monotone:
484  ∀m, n, o: nat.
485    m = n → m + o = n + o.
486  #m #n #o #refl >refl %
487qed.
488
489axiom as_execute_preserves_code_memory:
490  ∀cost_labels : BitVectorTrie costlabel 16.
491  ∀start_status: ASM_abstract_status cost_labels.
492  ∀final_status: ASM_abstract_status cost_labels.
493    as_execute … start_status final_status →
494      code_memory … start_status = code_memory … final_status.
495
496axiom trace_label_return_preserves_code_memory:
497  ∀cost_labels : BitVectorTrie costlabel 16.
498  ∀start_status: ASM_abstract_status cost_labels.
499  ∀final_status: ASM_abstract_status cost_labels.
500  ∀the_trace   : trace_label_return … start_status final_status.
501    code_memory … start_status = code_memory … final_status.
502
503(* XXX: indexing bug re-emerges! *)
504example fetch_half_add:
505  ∀cost_labels: BitVectorTrie costlabel 16.
506  ∀the_status : ASM_abstract_status cost_labels.
507    \snd (\fst (fetch (code_memory … the_status) (program_counter … the_status))) =
508      \snd (half_add 16 (program_counter (BitVectorTrie Byte 16) the_status)
509        (bitvector_of_nat … (\snd (fetch (code_memory … the_status) (program_counter … the_status))))).
510  cases daemon
511qed.
512
513axiom half_add_minus_right:
514  ∀size : nat.
515  ∀left : BitVector size.
516  ∀right: nat.
517  nat_of_bitvector … (\snd (half_add … left (bitvector_of_nat … right))) -
518    nat_of_bitvector … left = right.
519
520lemma minus_plus_cancel:
521  ∀m, n : nat.
522  ∀proof: n ≤ m.
523    (m - n) + n = m.
524  #m #n #proof /2 by plus_minus/
525qed.
526
527let rec block_cost'
528  (code_memory': BitVectorTrie Byte 16) (program_counter': Word)
529    (program_size: nat) (total_program_size: nat) (cost_labels: BitVectorTrie costlabel 16)
530      (reachable_program_counter_witness: reachable_program_counter code_memory' total_program_size program_counter')
531        (good_program_witness: good_program code_memory' total_program_size) (first_time_around: bool)
532          on program_size:
533          total_program_size ≤ program_size + nat_of_bitvector … program_counter' →
534          Σcost_of_block: nat.
535            ∀start_status: Status.
536            ∀final_status: Status.
537            ∀trace_ends_flag.
538            ∀the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status.
539              code_memory' = code_memory … start_status →
540                program_counter' = program_counter … start_status →
541                    (∃n: nat. trace_any_label_length … the_trace + (S n) = total_program_size) ∧
542                      cost_of_block = compute_paid_trace_any_label cost_labels trace_ends_flag start_status final_status the_trace ≝
543  match program_size return λprogram_size: nat. total_program_size ≤ program_size + nat_of_bitvector … program_counter' → ? with
544  [ O ⇒ λbase_case. ⊥
545  | S program_size' ⇒ λrecursive_case.
546    let 〈instruction, program_counter'', ticks〉 as FETCH ≝ fetch code_memory' program_counter' in
547    let to_continue ≝
548      match lookup_opt … program_counter'' cost_labels with
549      [ None ⇒ true
550      | Some _ ⇒ first_time_around
551      ]
552    in
553      if to_continue then
554        match instruction return λx. x = instruction → ? with
555        [ RealInstruction real_instruction ⇒ λreal_instruction_refl.
556          match real_instruction return λx. x = real_instruction →
557          Σcost_of_block: nat.
558            ∀start_status: Status.
559            ∀final_status: Status.
560            ∀trace_ends_flag.
561            ∀the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status.
562              code_memory' = code_memory … start_status →
563                program_counter' = program_counter … start_status →
564                    (∃n: nat. trace_any_label_length … the_trace + (S n) = total_program_size) ∧
565                      cost_of_block = compute_paid_trace_any_label cost_labels trace_ends_flag start_status final_status the_trace with
566          [ RET                    ⇒ λinstr. ticks
567          | JC   relative          ⇒ λinstr. ticks
568          | JNC  relative          ⇒ λinstr. ticks
569          | JB   bit_addr relative ⇒ λinstr. ticks
570          | JNB  bit_addr relative ⇒ λinstr. ticks
571          | JBC  bit_addr relative ⇒ λinstr. ticks
572          | JZ   relative          ⇒ λinstr. ticks
573          | JNZ  relative          ⇒ λinstr. ticks
574          | CJNE src_trgt relative ⇒ λinstr. ticks
575          | DJNZ src_trgt relative ⇒ λinstr. ticks
576          | _                      ⇒ λinstr.
577              ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
578          ] (refl …)
579        | ACALL addr     ⇒ λinstr.
580            ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
581        | AJMP  addr     ⇒ λinstr. ticks
582        | LCALL addr     ⇒ λinstr.
583            ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
584        | LJMP  addr     ⇒ λinstr. ticks
585        | SJMP  addr     ⇒ λinstr. ticks
586        | JMP   addr     ⇒ λinstr. (* XXX: actually a call due to use with fptrs *)
587            ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
588        | MOVC  src trgt ⇒ λinstr.
589            ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
590        ] (refl …)
591      else
592        (0 : (Σn: nat. ?))
593  ].
594  [1:
595    cases reachable_program_counter_witness #_ #hyp
596    @(absurd (total_program_size < total_program_size) … (not_le_Sn_n …))
597    @(le_to_lt_to_lt … base_case hyp)
598  |4:
599    #start_status #final_status #trace_ends_flag #the_trace #code_memory_refl #program_counter_refl
600    cases(block_cost' ?????????) -block_cost'
601    #recursive_block_cost #recursive_assm
602    @(trace_any_label_inv_ind … the_trace)
603    [1:
604      #start_status' #final_status' #execute_assm #classifier_assm #costed_assm
605      #ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl destruct
606      cases classifier_assm
607      whd in match (as_classifier ???);
608      whd in ⊢ (??%? → ?);
609      whd in match current_instruction; normalize nodelta
610      whd in match current_instruction0; normalize nodelta
611      #absurd @⊥ <FETCH in absurd; normalize nodelta
612      #absurd destruct(absurd)
613    |2:
614      #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_assm
615      #start_status_refl #final_status_refl #the_trace_assm destruct
616      whd in classifier_assm;
617      whd in classifier_assm:(??%?);
618      whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
619      whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
620      @⊥ <FETCH in classifier_assm; normalize nodelta
621      #absurd destruct(absurd)
622    |3:
623      #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
624      #classifier_assm #after_return_assm #trace_label_return #costed_assm
625      #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
626      destruct
627      whd in match (compute_paid_trace_any_label … (tal_base_call …));
628      whd in match (trace_any_label_length … (tal_base_call …));
629      cases(recursive_assm status_pre_fun_call status_final doesnt_end_with_ret
630        (tal_base_call (ASM_abstract_status cost_labels) status_pre_fun_call
631          (execute_1 status_pre_fun_call) status_final
632            (refl Status (execute_1 status_pre_fun_call)) classifier_assm
633              after_return_assm trace_label_return costed_assm) ? ?)
634      [1:
635        * #recursive_n #recursive_trace_total #recursive_block_cost_assm %
636        [1:
637          %{(pred total_program_size)}
638          whd in ⊢ (??%?);
639          >S_pred [1: % ]
640          <recursive_trace_total
641          @lt_O_S
642        |2:
643          >recursive_block_cost_assm
644          whd in ⊢ (??(??%)?); <FETCH
645          (* XXX: ??? *)
646        ]
647      |2:
648        %
649      |3:
650        cut(program_counter'' = \snd (\fst (fetch (code_memory … status_pre_fun_call) (program_counter … status_pre_fun_call))))
651        [1:
652          cases FETCH %
653        |2:
654          #program_counter_hyp'' >program_counter_hyp''
655          whd in after_return_assm; <after_return_assm
656          whd in match (current_instruction_cost status_pre_fun_call);
657          @fetch_half_add
658        ]
659      ]
660      [1:
661    |4:
662      #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
663      #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
664      #costed_assm #trace_any_label #trace_ends_flag_refl #start_status_refl
665      #final_status_refl #the_trace_refl
666      generalize in match execute_assm; destruct #execute_assm
667      whd in match (compute_paid_trace_any_label … (tal_step_call …));
668      whd in match (trace_any_label_length … (tal_step_call …));
669      cases(recursive_assm status_after_fun_call status_final end_flag trace_any_label ? ?)
670      [1:
671        * #recursive_n #recursive_trace_total #recursive_block_cost_assm %
672        [1:
673          <recursive_trace_total %
674          [1:
675            @(recursive_n - (current_instruction_cost status_pre_fun_call))
676          |2:
677            whd in after_return_assm; cases after_return_assm
678            >half_add_minus_right in ⊢ (??%?);
679            >commutative_plus in ⊢ (???%);
680            >commutative_plus in ⊢ (??%?);
681            <associative_plus in ⊢ (??%?);
682            @plus_right_monotone
683            cut(current_instruction_cost status_pre_fun_call ≤ recursive_n)
684            [1:
685              (* XXX: ??? *)
686            |2:
687              #current_instruction_cost_assm
688              <minus_Sn_m in ⊢ (??%?); [2: assumption ]
689              >minus_plus_cancel [1: % ]
690              @le_S assumption
691            ]
692          ]
693        |2:
694          >recursive_block_cost_assm
695          @plus_right_monotone
696          whd in match (current_instruction_cost status_pre_fun_call);
697          <FETCH %
698        ]
699      |2:
700        lapply (trace_label_return_preserves_code_memory … (execute_1 status_pre_fun_call) status_after_fun_call)
701        #hyp cases (hyp trace_label_return)
702        @as_execute_preserves_code_memory whd %
703      |3:
704        cut(program_counter'' = \snd (\fst (fetch (code_memory … status_pre_fun_call) (program_counter … status_pre_fun_call))))
705        [1:
706          cases FETCH %
707        |2:
708          #program_counter_hyp'' >program_counter_hyp''
709          whd in after_return_assm; <after_return_assm
710          whd in match (current_instruction_cost status_pre_fun_call);
711          @fetch_half_add
712        ]
713      ]
714    |5:
715      #end_flag #status_pre #status_init #status_end #execute_assm #the_trace'
716      #classifier_assm #costed_assm #trace_ends_flag_refl #start_status_refl
717      #final_status_refl #trace_refl
718      generalize in match the_trace; destruct #the_trace
719      whd in classifier_assm;
720      whd in classifier_assm:(??%?);
721      whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
722      whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
723      @⊥ <FETCH in classifier_assm; normalize nodelta
724      #absurd destruct(absurd)
725    ]
726  |7,8,9,10,13,16,19,22,25,28,31,34,37,40,41,42,43,44,45,46,47,48,49,52,55,58,61,
727    64,67,70,73,76,79,82,85,88,91,94,97,100,101,104,108,107:
728    cases daemon (* XXX: massive terms *)
729  |2:
730    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
731    lapply(good_program_witness program_counter' reachable_program_counter_witness)
732    <FETCH normalize nodelta <instr normalize nodelta
733    @(subaddressing_mode_elim … [[addr11]] … [[addr11]]) [1: // ] #new_addr
734    cases (split … 8 8 program_counter'') #pc_bu #pc_bl normalize nodelta
735    cases (split … 3 8 new_addr) #thr #eig normalize nodelta
736    cases (split … 5 3 pc_bu) #fiv #thr' normalize nodelta * * * * #n'
737    #_ #_ #program_counter_lt' #program_counter_lt_tps'     
738    @(transitive_le
739      total_program_size
740      ((S program_size') + nat_of_bitvector … program_counter')
741      (program_size' + nat_of_bitvector … program_counter'') recursive_case)
742    normalize in match (S program_size' + nat_of_bitvector … program_counter');
743    >plus_n_Sm
744    @monotonic_le_plus_r
745    change with (
746      nat_of_bitvector … program_counter' <
747        nat_of_bitvector … program_counter'')
748    assumption
749  |3:
750    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
751    lapply(good_program_witness program_counter' reachable_program_counter_witness)
752    <FETCH normalize nodelta <instr normalize nodelta
753    @(subaddressing_mode_elim … [[addr11]] … [[addr11]]) [1: // ] #new_addr
754    cases (split … 8 8 program_counter'') #pc_bu #pc_bl normalize nodelta
755    cases (split … 3 8 new_addr) #thr #eig normalize nodelta
756    cases (split … 5 3 pc_bu) #fiv #thr' normalize nodelta * * * * #n'
757    #_ #_ #program_counter_lt' #program_counter_lt_tps'
758    %
759    [1:
760      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
761      <FETCH normalize nodelta whd in match ltb; normalize nodelta
762      >(le_to_leb_true … program_counter_lt') %
763    |2:
764      assumption
765    ]
766  |5:
767    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
768    lapply(good_program_witness program_counter' reachable_program_counter_witness)
769      <FETCH normalize nodelta <instr normalize nodelta
770    @(subaddressing_mode_elim … [[addr16]] … [[addr16]]) [1: // ] #new_addr
771    * * * * #n'
772    #_ #_ #program_counter_lt' #program_counter_lt_tps'     
773    @(transitive_le
774      total_program_size
775      ((S program_size') + nat_of_bitvector … program_counter')
776      (program_size' + nat_of_bitvector … program_counter'') recursive_case)
777    normalize in match (S program_size' + nat_of_bitvector … program_counter');
778    >plus_n_Sm
779    @monotonic_le_plus_r
780    change with (
781      nat_of_bitvector … program_counter' <
782        nat_of_bitvector … program_counter'')
783    assumption
784  |6:
785    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
786    lapply(good_program_witness program_counter' reachable_program_counter_witness)
787      <FETCH normalize nodelta <instr normalize nodelta
788    @(subaddressing_mode_elim … [[addr16]] … [[addr16]]) [1: // ] #new_addr
789    * * * * #n'
790    #_ #_ #program_counter_lt' #program_counter_lt_tps'
791    %
792    [1:
793      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
794      <FETCH normalize nodelta whd in match ltb; normalize nodelta
795      >(le_to_leb_true … program_counter_lt') %
796    |2:
797      assumption
798    ]
799  |11,14: (* JMP and MOVC *)
800    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
801    lapply(good_program_witness program_counter' reachable_program_counter_witness)
802    <FETCH normalize nodelta <instr normalize nodelta
803    try(<real_instruction_refl <instr normalize nodelta) *
804    #pc_pc_lt_hyp' #pc_tps_lt_hyp'
805    @(transitive_le
806      total_program_size
807      ((S program_size') + nat_of_bitvector … program_counter')
808      (program_size' + nat_of_bitvector … program_counter'') recursive_case)
809    normalize in match (S program_size' + nat_of_bitvector … program_counter');
810    >plus_n_Sm
811    @monotonic_le_plus_r
812    change with (
813      nat_of_bitvector … program_counter' <
814        nat_of_bitvector … program_counter'')
815    assumption
816  |12,15:
817    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
818    lapply(good_program_witness program_counter' reachable_program_counter_witness)
819    <FETCH normalize nodelta <instr normalize nodelta *
820    #program_counter_lt' #program_counter_lt_tps' %
821    [1,3:
822      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
823      <FETCH normalize nodelta whd in match ltb; normalize nodelta
824      >(le_to_leb_true … program_counter_lt') %
825    |2,4:
826      assumption
827    ]
828  |17,20,23,26,29,32,35,38,50,53,56,59,62,65,68,71,74,77,80,83,86,89,92,95,98,
829   102,105:
830    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
831    lapply(good_program_witness program_counter' reachable_program_counter_witness)
832    <FETCH normalize nodelta
833    <real_instruction_refl <instr normalize nodelta *
834    #pc_pc_lt_hyp' #pc_tps_lt_hyp'
835    @(transitive_le
836      total_program_size
837      ((S program_size') + nat_of_bitvector … program_counter')
838      (program_size' + nat_of_bitvector … program_counter'') recursive_case)
839    normalize in match (S program_size' + nat_of_bitvector … program_counter');
840    >plus_n_Sm
841    @monotonic_le_plus_r
842    change with (
843      nat_of_bitvector … program_counter' <
844        nat_of_bitvector … program_counter'')
845    assumption
846  |18,21,24,27,30,33,36,39,51,54,57,60,63,66,69,72,75,78,
847    81,84,87,90,93,96,99,103,106:
848    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
849    lapply(good_program_witness program_counter' reachable_program_counter_witness)
850    <FETCH normalize nodelta <real_instruction_refl <instr normalize nodelta *
851    #program_counter_lt' #program_counter_lt_tps' %
852    try assumption
853    [*:
854      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
855      <FETCH normalize nodelta whd in match ltb; normalize nodelta
856      >(le_to_leb_true … program_counter_lt') %
857    ]
858  [1:
859    #start_status #final_status #trace_ends_flag #the_trace #code_memory_refl #program_counter_refl
860    cases(block_cost' ?????? ?? ?) -block_cost'
861    #recursive_block_cost #recursive_assm
862    @(trace_any_label_inv_ind … the_trace)
863    [1:
864      #start_status' #final_status' #execute_assm #classifier_assm #costed_assm
865      #ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl destruct
866      cases classifier_assm
867      whd in match (as_classifier ? ? ?);
868      whd in ⊢ (??%? → ?);
869      whd in match current_instruction; normalize nodelta
870      whd in match current_instruction0; normalize nodelta
871      #absurd @⊥ >p1 in absurd; normalize nodelta
872      #absurd destruct(absurd)
873    |2:
874      #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_assm
875      #start_status_refl #final_status_refl #the_trace_assm destruct
876      whd in classifier_assm;
877      whd in classifier_assm:(??%?);
878      whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
879      whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
880      @⊥
881      >p1 in classifier_assm; normalize nodelta
882      #absurd destruct(absurd)
883    |3:
884      #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
885      #classifier_assm #after_return_assm #trace_label_return #costed_assm
886      #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
887      destruct
888      cases(recursive_assm status_pre_fun_call status_final doesnt_end_with_ret
889        (tal_base_call (ASM_abstract_status cost_labels) status_pre_fun_call
890          (execute_1 status_pre_fun_call) status_final
891          (refl Status (execute_1 status_pre_fun_call)) classifier_assm
892          after_return_assm trace_label_return costed_assm) ? ?)
893      [2: %
894      |3:
895      |1:
896      ]
897    |4:
898      #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
899      #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
900      #costed_assm #trace_any_label #trace_ends_refl #start_status_refl #final_status_refl
901      #the_trace_refl destruct whd in ⊢ (??(???%)); whd in match (trace_any_label_length … (tal_step_call … trace_any_label));
902    |5:
903      #end_flag #status_pre #status_init #status_end #execute_assm #the_trace
904      #classifier_assm #costed_assm #ends_flag_refl #start_status_refl #final_status_refl
905      #the_trace_refl -the_trace_refl destruct
906      whd in classifier_assm;
907      whd in classifier_assm:(??%?);
908      whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
909      whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
910      @⊥ >p1 in classifier_assm; normalize nodelta
911      #absurd destruct(absurd)
912    ]
913  [1:
914    #start_status #final_status #trace_ends_flag #the_trace
915    #code_memory_refl #program_counter_refl
916    cases(block_cost' ?????? ?? ?) -block_cost'
917    #recursive_block_cost #recursive_assm
918    @(trace_any_label_inv_ind … the_trace)
919    [1:
920      #start_status' #final_status' #execute_assm #not_return_assm
921      #costed_assm #trace_ends_flag_assm #start_status_assm #final_status_assm
922      #the_trace_assm
923      cases daemon (* XXX: bug in notion of traces here *)
924    |2:
925
926    |3:
927      cases daemon (* XXX *)
928    |4:
929    ]
930  |108:
931    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
932    lapply(good_program_witness program_counter' reachable_program_counter_witness)
933    <FETCH normalize nodelta <real_instruction_refl <instr normalize nodelta *
934    #program_counter_lt' #program_counter_lt_tps' %
935    try assumption
936    [*:
937      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
938      <FETCH normalize nodelta whd in match ltb; normalize nodelta
939      >(le_to_leb_true … program_counter_lt') %
940    ]
941  [1:
942    cases reachable_program_counter_witness #_ #hyp
943    @(absurd (total_program_size < total_program_size) … (not_le_Sn_n …))
944    @(le_to_lt_to_lt … base_case hyp)
945  |2:
946    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
947    lapply(good_program_witness program_counter reachable_program_counter_witness)
948      <FETCH normalize nodelta <instr normalize nodelta
949    @(subaddressing_mode_elim … [[addr11]] … [[addr11]]) [1: // ] #new_addr
950    cases (split … 8 8 program_counter') #pc_bu #pc_bl normalize nodelta
951    cases (split … 3 8 new_addr) #thr #eig normalize nodelta
952    cases (split … 5 3 pc_bu) #fiv #thr' normalize nodelta * * * * #n'
953    #_ #_ #program_counter_lt' #program_counter_lt_tps'     
954    @(transitive_le
955      total_program_size
956      ((S program_size') + nat_of_bitvector … program_counter)
957      (program_size' + nat_of_bitvector … program_counter') recursive_case)
958    normalize in match (S program_size' + nat_of_bitvector … program_counter);
959    >plus_n_Sm
960    @monotonic_le_plus_r
961    change with (
962      nat_of_bitvector … program_counter <
963        nat_of_bitvector … program_counter')
964    assumption
965  |3:
966    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
967    lapply(good_program_witness program_counter reachable_program_counter_witness)
968      <FETCH normalize nodelta <instr normalize nodelta
969    @(subaddressing_mode_elim … [[addr11]] … [[addr11]]) [1: // ] #new_addr
970    cases (split … 8 8 program_counter') #pc_bu #pc_bl normalize nodelta
971    cases (split … 3 8 new_addr) #thr #eig normalize nodelta
972    cases (split … 5 3 pc_bu) #fiv #thr' normalize nodelta * * * * #n'
973    #_ #_ #program_counter_lt' #program_counter_lt_tps'
974    %
975    [1:
976      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
977      <FETCH normalize nodelta whd in match ltb; normalize nodelta
978      >(le_to_leb_true … program_counter_lt') %
979    |2:
980      assumption
981    ]
982  |4:
983    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
984    lapply(good_program_witness program_counter reachable_program_counter_witness)
985      <FETCH normalize nodelta <instr normalize nodelta
986    @(subaddressing_mode_elim … [[addr16]] … [[addr16]]) [1: // ] #new_addr
987    * * * * #n'
988    #_ #_ #program_counter_lt' #program_counter_lt_tps'     
989    @(transitive_le
990      total_program_size
991      ((S program_size') + nat_of_bitvector … program_counter)
992      (program_size' + nat_of_bitvector … program_counter') recursive_case)
993    normalize in match (S program_size' + nat_of_bitvector … program_counter);
994    >plus_n_Sm
995    @monotonic_le_plus_r
996    change with (
997      nat_of_bitvector … program_counter <
998        nat_of_bitvector … program_counter')
999    assumption
1000  |5:
1001    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1002    lapply(good_program_witness program_counter reachable_program_counter_witness)
1003      <FETCH normalize nodelta <instr normalize nodelta
1004    @(subaddressing_mode_elim … [[addr16]] … [[addr16]]) [1: // ] #new_addr
1005    * * * * #n'
1006    #_ #_ #program_counter_lt' #program_counter_lt_tps'
1007    %
1008    [1:
1009      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
1010      <FETCH normalize nodelta whd in match ltb; normalize nodelta
1011      >(le_to_leb_true … program_counter_lt') %
1012    |2:
1013      assumption
1014    ]
1015  |6,8: (* JMP and MOVC *)
1016    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1017    lapply(good_program_witness program_counter reachable_program_counter_witness)
1018    <FETCH normalize nodelta <instr normalize nodelta
1019    try(<real_instruction_refl <instr normalize nodelta) *
1020    #pc_pc_lt_hyp' #pc_tps_lt_hyp'
1021    @(transitive_le
1022      total_program_size
1023      ((S program_size') + nat_of_bitvector … program_counter)
1024      (program_size' + nat_of_bitvector … program_counter') recursive_case)
1025    normalize in match (S program_size' + nat_of_bitvector … program_counter);
1026    >plus_n_Sm
1027    @monotonic_le_plus_r
1028    change with (
1029      nat_of_bitvector … program_counter <
1030        nat_of_bitvector … program_counter')
1031    assumption
1032  |7,9:
1033    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1034    lapply(good_program_witness program_counter reachable_program_counter_witness)
1035    <FETCH normalize nodelta <instr normalize nodelta *
1036    #program_counter_lt' #program_counter_lt_tps' %
1037    [1,3:
1038      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
1039      <FETCH normalize nodelta whd in match ltb; normalize nodelta
1040      >(le_to_leb_true … program_counter_lt') %
1041    |2,4:
1042      assumption
1043    ]
1044  |11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,
1045    55,57,59,61,63:
1046    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1047    lapply(good_program_witness program_counter reachable_program_counter_witness)
1048    <FETCH normalize nodelta <real_instruction_refl <instr normalize nodelta *
1049    #program_counter_lt' #program_counter_lt_tps' %
1050    try assumption
1051    [*:
1052      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
1053      <FETCH normalize nodelta whd in match ltb; normalize nodelta
1054      >(le_to_leb_true … program_counter_lt') %
1055    ]
1056  |10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,
1057    54,56,58,60,62:
1058    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1059    lapply(good_program_witness program_counter reachable_program_counter_witness)
1060    <FETCH normalize nodelta
1061    <real_instruction_refl <instr normalize nodelta *
1062    #pc_pc_lt_hyp' #pc_tps_lt_hyp'
1063    @(transitive_le
1064      total_program_size
1065      ((S program_size') + nat_of_bitvector … program_counter)
1066      (program_size' + nat_of_bitvector … program_counter') recursive_case)
1067    normalize in match (S program_size' + nat_of_bitvector … program_counter);
1068    >plus_n_Sm
1069    @monotonic_le_plus_r
1070    change with (
1071      nat_of_bitvector … program_counter <
1072        nat_of_bitvector … program_counter')
1073    assumption
1074  ]
1075qed.
1076
1077definition block_cost ≝
1078  λcode_memory: BitVectorTrie Byte 16.
1079  λprogram_counter: Word.
1080  λtotal_program_size: nat.
1081  λcost_labels: BitVectorTrie costlabel 16.
1082  λreachable_program_counter_witness: reachable_program_counter code_memory total_program_size program_counter.
1083  λgood_program_witness: good_program code_memory total_program_size.
1084    block_cost' code_memory program_counter total_program_size total_program_size cost_labels
1085      reachable_program_counter_witness good_program_witness true ?.
1086  @le_plus_n_r
1087qed.
1088
1089lemma fetch_program_counter_n_Sn:
1090  ∀instruction: instruction.
1091  ∀program_counter, program_counter': Word.
1092  ∀ticks, n: nat.
1093  ∀code_memory: BitVectorTrie Byte 16.
1094    Some … program_counter = fetch_program_counter_n n code_memory (zero 16) →
1095      〈instruction,program_counter',ticks〉 = fetch code_memory program_counter →
1096        nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' →
1097          Some … program_counter' = fetch_program_counter_n (S n) code_memory (zero …).
1098  #instruction #program_counter #program_counter' #ticks #n #code_memory
1099  #fetch_program_counter_n_hyp #fetch_hyp #lt_hyp
1100  whd in match (fetch_program_counter_n (S n) code_memory (zero …));
1101  <fetch_program_counter_n_hyp normalize nodelta
1102  <fetch_hyp normalize nodelta
1103  change with (
1104    leb (S (nat_of_bitvector … program_counter)) (nat_of_bitvector … program_counter')
1105  ) in match (ltb (nat_of_bitvector … program_counter) (nat_of_bitvector … program_counter'));
1106  >(le_to_leb_true … lt_hyp) %
1107qed.
1108
1109let rec traverse_code_internal
1110  (code_memory: BitVectorTrie Byte 16) (cost_labels: BitVectorTrie costlabel 16)
1111    (program_counter: Word) (fixed_program_size: nat) (program_size: nat)
1112      (reachable_program_counter_witness:
1113          ∀lbl: costlabel.
1114          ∀program_counter: Word. Some … lbl = lookup_opt … program_counter cost_labels →
1115            reachable_program_counter code_memory fixed_program_size program_counter)
1116        (good_program_witness: good_program code_memory fixed_program_size)
1117        on program_size: identifier_map CostTag nat ≝
1118  match program_size with
1119  [ O ⇒ empty_map …
1120  | S program_size ⇒
1121    let 〈instruction, new_program_counter, ticks〉 as FETCH ≝ fetch … code_memory program_counter in
1122    let cost_mapping ≝ traverse_code_internal code_memory cost_labels new_program_counter fixed_program_size program_size ? good_program_witness in
1123    match lookup_opt … program_counter cost_labels return λx. x = lookup_opt … program_counter cost_labels → ? with
1124    [ None ⇒ λlookup_refl. cost_mapping
1125    | Some lbl ⇒ λlookup_refl.
1126      let cost ≝ block_cost code_memory program_counter fixed_program_size cost_labels ? good_program_witness in
1127        add … cost_mapping lbl cost
1128    ] (refl … (lookup_opt … program_counter cost_labels))
1129  ].
1130  [1:
1131    @(reachable_program_counter_witness lbl)
1132    assumption
1133  |2:
1134    assumption
1135  ]
1136qed.
1137
1138definition traverse_code ≝
1139  λcode_memory: BitVectorTrie Byte 16.
1140  λcost_labels: BitVectorTrie costlabel 16.
1141  λprogram_size: nat.
1142  λreachable_program_counter_witness:
1143          ∀lbl: costlabel.
1144          ∀program_counter: Word. Some … lbl = lookup_opt … program_counter cost_labels →
1145            reachable_program_counter code_memory program_size program_counter.
1146  λgood_program_witness: good_program code_memory program_size.
1147    traverse_code_internal code_memory cost_labels (zero …) program_size program_size reachable_program_counter_witness good_program_witness.
1148 
1149definition compute_costs ≝
1150  λprogram: list Byte.
1151  λcost_labels: BitVectorTrie costlabel 16.
1152  λreachable_program_witness.
1153  λgood_program_witness: good_program (load_code_memory program) (|program| + 1).
1154    let program_size ≝ |program| + 1 in
1155    let code_memory ≝ load_code_memory program in
1156      traverse_code code_memory cost_labels program_size reachable_program_witness ?.
1157  @good_program_witness
1158qed.
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