source: src/ASM/ASMCosts.ma @ 1632

Last change on this file since 1632 was 1625, checked in by mulligan, 8 years ago

before christmas

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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
7definition current_instruction0 ≝
8  λmem,pc. \fst (\fst (fetch … mem pc)).
9
10definition current_instruction ≝
11 λs:Status. current_instruction0 (code_memory … s) (program_counter … s).
12
13definition ASM_classify0: instruction → status_class ≝
14 λi.
15  match i with
16   [ RealInstruction pre ⇒
17     match pre with
18      [ RET ⇒ cl_return
19      | JZ _ ⇒ cl_jump
20      | JNZ _ ⇒ cl_jump
21      | JC _ ⇒ cl_jump
22      | JNC _ ⇒ cl_jump
23      | JB _ _ ⇒ cl_jump
24      | JNB _ _ ⇒ cl_jump
25      | JBC _ _ ⇒ cl_jump
26      | CJNE _ _ ⇒ cl_jump
27      | DJNZ _ _ ⇒ cl_jump
28      | _ ⇒ cl_other
29      ]
30   | ACALL _ ⇒ cl_call
31   | LCALL _ ⇒ cl_call
32   | JMP _ ⇒ cl_call
33   | AJMP _ ⇒ cl_jump
34   | LJMP _ ⇒ cl_jump
35   | SJMP _ ⇒ cl_jump
36   | _ ⇒ cl_other
37   ].
38
39definition ASM_classify: Status → status_class ≝
40 λs.ASM_classify0 (current_instruction s).
41
42definition current_instruction_is_labelled ≝
43  λcost_labels: BitVectorTrie costlabel 16.
44  λs: Status.
45  let pc ≝ program_counter … s in
46    match lookup_opt … pc cost_labels with
47    [ None ⇒ False
48    | _    ⇒ True
49    ].
50
51definition label_of_current_instruction:
52 BitVectorTrie costlabel 16 → Status → list costlabel
53 ≝
54  λcost_labels,s.
55  let pc ≝ program_counter … s in
56    match lookup_opt … pc cost_labels with
57    [ None ⇒ []
58    | Some l ⇒ [l]
59    ].
60
61definition next_instruction_properly_relates_program_counters ≝
62  λbefore: Status.
63  λafter : Status.
64  let size ≝ current_instruction_cost before in
65  let pc_before ≝ program_counter … before in
66  let pc_after ≝ program_counter … after in
67  let sum ≝ \snd (half_add … pc_before (bitvector_of_nat … size)) in
68    sum = pc_after.
69
70definition ASM_abstract_status: BitVectorTrie costlabel 16 → abstract_status ≝
71 λcost_labels.
72  mk_abstract_status
73   Status
74   (λs,s'. (execute_1 s) = s')
75   (λs,class. ASM_classify s = class)
76   (current_instruction_is_labelled cost_labels)
77   next_instruction_properly_relates_program_counters.
78
79(* To be moved in ASM/arithmetic.ma *)
80definition addr16_of_addr11: Word → Word11 → Word ≝
81  λpc: Word.
82  λa: Word11.
83  let 〈pc_upper, ignore〉 ≝ split … 8 8 pc in
84  let 〈n1, n2〉 ≝ split … 4 4 pc_upper in
85  let 〈b123, b〉 ≝ split … 3 8 a in
86  let b1 ≝ get_index_v … b123 0 ? in
87  let b2 ≝ get_index_v … b123 1 ? in
88  let b3 ≝ get_index_v … b123 2 ? in
89  let p5 ≝ get_index_v … n2 0 ? in
90    (n1 @@ [[ p5; b1; b2; b3 ]]) @@ b.
91  //
92qed.
93   
94definition good_program_counter: BitVectorTrie Byte 16 → Word → nat → Prop ≝
95  λcode_memory: BitVectorTrie Byte 16.
96  λprogram_counter: Word.
97  λprogram_size: nat.
98    ∃n: nat.
99      let tail_program_counter ≝ fetch_program_counter_n n code_memory (zero 16) in
100        program_counter = fetch_program_counter_n (S n) code_memory (zero 16) ∧
101          nat_of_bitvector 16 program_counter ≤ program_size ∧
102            nat_of_bitvector 16 tail_program_counter < nat_of_bitvector 16 program_counter.
103
104definition good_program: BitVectorTrie Byte 16 → Word → nat → Prop ≝
105  λcode_memory: BitVectorTrie Byte 16.
106  λprogram_counter: Word.
107  λtotal_program_size: nat.
108  let 〈instruction, program_counter, ticks〉 ≝ fetch code_memory program_counter in
109    match instruction with
110    [ RealInstruction instr ⇒
111      match instr with
112      [ RET                    ⇒ True
113      | JC   relative          ⇒ True (* XXX: see below *)
114      | JNC  relative          ⇒ True (* XXX: see below *)
115      | JB   bit_addr relative ⇒ True
116      | JNB  bit_addr relative ⇒ True
117      | JBC  bit_addr relative ⇒ True
118      | JZ   relative          ⇒ True
119      | JNZ  relative          ⇒ True
120      | CJNE src_trgt relative ⇒ True
121      | DJNZ src_trgt relative ⇒ True
122      | _                      ⇒
123          good_program_counter code_memory program_counter total_program_size
124      ]
125    | LCALL addr         ⇒
126      match addr return λx. bool_to_Prop (is_in … [[ addr16 ]] x) → Prop with
127      [ ADDR16 addr ⇒ λaddr16: True.
128          good_program_counter code_memory addr total_program_size ∧
129            good_program_counter code_memory program_counter total_program_size
130      | _ ⇒ λother: False. ⊥
131      ] (subaddressing_modein … addr)
132    | ACALL addr         ⇒
133      match addr return λx. bool_to_Prop (is_in … [[ addr11 ]] x) → Prop with
134      [ ADDR11 addr ⇒ λaddr11: True.
135        let 〈pc_bu, pc_bl〉 ≝ split … 8 8 program_counter in
136        let 〈thr, eig〉 ≝ split … 3 8 addr in
137        let 〈fiv, thr'〉 ≝ split … 5 3 pc_bu in
138        let new_program_counter ≝ (fiv @@ thr) @@ pc_bl in
139          good_program_counter code_memory new_program_counter total_program_size ∧
140            good_program_counter code_memory program_counter total_program_size
141      | _ ⇒ λother: False. ⊥
142      ] (subaddressing_modein … addr)
143    | AJMP  addr         ⇒
144      match addr return λx. bool_to_Prop (is_in … [[ addr11 ]] x) → Prop with
145      [ ADDR11 addr ⇒ λaddr11: True.
146        let 〈pc_bu, pc_bl〉 ≝ split … 8 8 program_counter in
147        let 〈nu, nl〉 ≝ split … 4 4 pc_bu in
148        let bit ≝ get_index' … O ? nl in
149        let 〈relevant1, relevant2〉 ≝ split … 3 8 addr in
150        let new_addr ≝ (nu @@ (bit ::: relevant1)) @@ relevant2 in
151        let 〈carry, new_program_counter〉 ≝ half_add 16 program_counter new_addr in
152          (good_program_counter code_memory new_program_counter total_program_size) ∧
153            (good_program_counter code_memory program_counter total_program_size)
154      | _ ⇒ λother: False. ⊥
155      ] (subaddressing_modein … addr)
156    | LJMP  addr         ⇒
157      match addr return λx. bool_to_Prop (is_in … [[ addr16 ]] x) → Prop with
158      [ ADDR16 addr ⇒ λaddr16: True.
159          good_program_counter code_memory addr total_program_size ∧
160            good_program_counter code_memory program_counter total_program_size
161      | _ ⇒ λother: False. ⊥
162      ] (subaddressing_modein … addr)
163    | SJMP  addr     ⇒
164      match addr return λx. bool_to_Prop (is_in … [[ relative ]] x) → Prop with
165      [ RELATIVE addr ⇒ λrelative: True.
166        let 〈carry, new_program_counter〉 ≝ half_add … program_counter (sign_extension addr) in
167          good_program_counter code_memory new_program_counter total_program_size ∧
168            good_program_counter code_memory program_counter total_program_size
169      | _ ⇒ λother: False. ⊥
170      ] (subaddressing_modein … addr)
171    | JMP   addr     ⇒ True (* XXX: should recurse here, but dptr in JMP ... *)
172    | MOVC  src trgt ⇒
173        good_program_counter code_memory program_counter total_program_size
174    ].
175  cases other
176qed.
177
178lemma is_in_tail_to_is_in_cons_hd_tl:
179  ∀n: nat.
180  ∀the_vect: Vector addressing_mode_tag n.
181  ∀h: addressing_mode_tag.
182  ∀element: addressing_mode.
183    is_in n the_vect element → is_in (S n) (h:::the_vect) element.
184  #n #the_vect #h #element #assm
185  normalize cases (is_a h element) normalize nodelta
186  //
187qed.
188
189axiom is_in_subvector_is_in_supervector:
190  ∀m, n: nat.
191  ∀subvector: Vector addressing_mode_tag m.
192  ∀supervector: Vector addressing_mode_tag n.
193  ∀element: addressing_mode.
194    subvector_with … eq_a subvector supervector →
195      is_in m subvector element → is_in n supervector element.
196
197let rec member_addressing_mode_tag
198  (n: nat) (v: Vector addressing_mode_tag n) (a: addressing_mode_tag)
199    on v: Prop ≝
200  match v with
201  [ VEmpty ⇒ False
202  | VCons n' hd tl ⇒
203      bool_to_Prop (eq_a hd a) ∨ member_addressing_mode_tag n' tl a
204  ].
205 
206let rec subaddressing_mode_elim_type
207  (T: Type[2]) (m: nat) (fixed_v: Vector addressing_mode_tag m)
208    (Q: addressing_mode → T → Prop)
209      (p_addr11:            ∀w: Word11.      is_in m fixed_v (ADDR11 w)        → T)
210      (p_addr16:            ∀w: Word.        is_in m fixed_v (ADDR16 w)        → T)
211      (p_direct:            ∀w: Byte.        is_in m fixed_v (DIRECT w)        → T)
212      (p_indirect:          ∀w: Bit.         is_in m fixed_v (INDIRECT w)      → T)
213      (p_ext_indirect:      ∀w: Bit.         is_in m fixed_v (EXT_INDIRECT w)  → T)
214      (p_acc_a:                              is_in m fixed_v ACC_A             → T)
215      (p_register:          ∀w: BitVector 3. is_in m fixed_v (REGISTER w)      → T)
216      (p_acc_b:                              is_in m fixed_v ACC_B             → T)
217      (p_dptr:                               is_in m fixed_v DPTR              → T)
218      (p_data:              ∀w: Byte.        is_in m fixed_v (DATA w)          → T)
219      (p_data16:            ∀w: Word.        is_in m fixed_v (DATA16 w)        → T)
220      (p_acc_dptr:                           is_in m fixed_v ACC_DPTR          → T)
221      (p_acc_pc:                             is_in m fixed_v ACC_PC            → T)
222      (p_ext_indirect_dptr:                  is_in m fixed_v EXT_INDIRECT_DPTR → T)
223      (p_indirect_dptr:                      is_in m fixed_v INDIRECT_DPTR     → T)
224      (p_carry:                              is_in m fixed_v CARRY             → T)
225      (p_bit_addr:          ∀w: Byte.        is_in m fixed_v (BIT_ADDR w)      → T)
226      (p_n_bit_addr:        ∀w: Byte.        is_in m fixed_v (N_BIT_ADDR w)    → T)
227      (p_relative:          ∀w: Byte.        is_in m fixed_v (RELATIVE w)      → T)
228        (n: nat) (v: Vector addressing_mode_tag n) (proof: subvector_with … eq_a v fixed_v)
229      on v: Prop ≝
230  match v return λo: nat. λv': Vector addressing_mode_tag o. o = n → v ≃ v' → ? with
231  [ VEmpty         ⇒ λm_refl. λv_refl.
232      ∀addr: addressing_mode. ∀p: is_in m fixed_v addr.
233        Q addr (
234        match addr return λx: addressing_mode. is_in … fixed_v x → T with 
235        [ ADDR11 x          ⇒ p_addr11 x
236        | ADDR16 x          ⇒ p_addr16 x
237        | DIRECT x          ⇒ p_direct x
238        | INDIRECT x        ⇒ p_indirect x
239        | EXT_INDIRECT x    ⇒ p_ext_indirect x
240        | ACC_A             ⇒ p_acc_a
241        | REGISTER x        ⇒ p_register x
242        | ACC_B             ⇒ p_acc_b
243        | DPTR              ⇒ p_dptr
244        | DATA x            ⇒ p_data x
245        | DATA16 x          ⇒ p_data16 x
246        | ACC_DPTR          ⇒ p_acc_dptr
247        | ACC_PC            ⇒ p_acc_pc
248        | EXT_INDIRECT_DPTR ⇒ p_ext_indirect_dptr
249        | INDIRECT_DPTR     ⇒ p_indirect_dptr
250        | CARRY             ⇒ p_carry
251        | BIT_ADDR x        ⇒ p_bit_addr x
252        | N_BIT_ADDR x      ⇒ p_n_bit_addr x
253        | RELATIVE x        ⇒ p_relative x
254        ] p)
255  | VCons n' hd tl ⇒ λm_refl. λv_refl.
256    let tail_call ≝ subaddressing_mode_elim_type T m fixed_v Q p_addr11
257      p_addr16 p_direct p_indirect p_ext_indirect p_acc_a
258        p_register p_acc_b p_dptr p_data p_data16 p_acc_dptr
259          p_acc_pc p_ext_indirect_dptr p_indirect_dptr p_carry
260            p_bit_addr p_n_bit_addr p_relative n' tl ?
261    in
262    match hd return λa: addressing_mode_tag. a = hd → ? with
263    [ addr11            ⇒ λhd_refl. (∀w. Q (ADDR11 w) (p_addr11 w ?)) → tail_call
264    | addr16            ⇒ λhd_refl. (∀w. Q (ADDR16 w) (p_addr16 w ?)) → tail_call
265    | direct            ⇒ λhd_refl. (∀w. Q (DIRECT w) (p_direct w ?)) → tail_call
266    | indirect          ⇒ λhd_refl. (∀w. Q (INDIRECT w) (p_indirect w ?)) → tail_call
267    | ext_indirect      ⇒ λhd_refl. (∀w. Q (EXT_INDIRECT w) (p_ext_indirect w ?)) → tail_call
268    | acc_a             ⇒ λhd_refl. (Q ACC_A (p_acc_a ?)) → tail_call
269    | registr           ⇒ λhd_refl. (∀w. Q (REGISTER w) (p_register w ?)) → tail_call
270    | acc_b             ⇒ λhd_refl. (Q ACC_A (p_acc_b ?)) → tail_call
271    | dptr              ⇒ λhd_refl. (Q DPTR (p_dptr ?)) → tail_call
272    | data              ⇒ λhd_refl. (∀w. Q (DATA w) (p_data w ?)) → tail_call
273    | data16            ⇒ λhd_refl. (∀w. Q (DATA16 w) (p_data16 w ?)) → tail_call
274    | acc_dptr          ⇒ λhd_refl. (Q ACC_DPTR (p_acc_dptr ?)) → tail_call
275    | acc_pc            ⇒ λhd_refl. (Q ACC_PC (p_acc_pc ?)) → tail_call
276    | ext_indirect_dptr ⇒ λhd_refl. (Q EXT_INDIRECT_DPTR (p_ext_indirect_dptr ?)) → tail_call
277    | indirect_dptr     ⇒ λhd_refl. (Q INDIRECT_DPTR (p_indirect_dptr ?)) → tail_call
278    | carry             ⇒ λhd_refl. (Q CARRY (p_carry ?)) → tail_call
279    | bit_addr          ⇒ λhd_refl. (∀w. Q (BIT_ADDR w) (p_bit_addr w ?)) → tail_call
280    | n_bit_addr        ⇒ λhd_refl. (∀w. Q (N_BIT_ADDR w) (p_n_bit_addr w ?)) → tail_call
281    | relative          ⇒ λhd_refl. (∀w. Q (RELATIVE w) (p_relative w ?)) → tail_call
282    ] (refl … hd)
283  ] (refl … n) (refl_jmeq … v).
284  [20:
285    generalize in match proof; destruct
286    whd in match (subvector_with … eq_a (hd:::tl) fixed_v);
287    cases (mem … eq_a m fixed_v hd) normalize nodelta
288    [1:
289      whd in match (subvector_with … eq_a tl fixed_v);
290      #assm assumption
291    |2:
292      normalize in ⊢ (% → ?);
293      #absurd cases absurd
294    ]
295  ]
296  @(is_in_subvector_is_in_supervector n m v fixed_v … proof)
297  destruct @I
298qed.
299
300lemma subaddressing_mode_elim':
301  ∀T: Type[2].
302  ∀m: nat.
303  ∀fixed_v: Vector addressing_mode_tag m.
304  ∀Q: addressing_mode → T → Prop.
305  ∀P1,P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13,P14,P15,P16,P17,P18,P19.
306  ∀n: nat.
307 
308  ∀v: Vector addressing_mode_tag n.
309  ∀proof.
310    subaddressing_mode_elim_type T m fixed_v Q P1 P2 P3 P4 P5 P6 P7
311      P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 n v proof.
312  #T #m #fixed_v #Q #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 #H14 #H15 #H16 #H17 #H18 #H19 #n #v
313  elim v
314  [ #proof normalize
315   
316qed.
317
318(*
319lemma subaddressing_mode_elim:
320  ∀T:Type[2].
321  ∀P1: Word11 → T.
322  ∀P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13,P14,P15,P16,P17,P18,P19: False → T.
323  ∀addr: addressing_mode.
324  ∀p: is_in 1 [[ addr11 ]] addr.
325  ∀Q: addressing_mode → T → Prop.
326    (∀w. Q (ADDR11 w) (P1 w)) →
327      Q addr (
328        match addr return λx:addressing_mode. (is_in 1 [[addr11]] x → T) with 
329        [ ADDR11 (x:Word11) ⇒ λH:True. P1 x
330        | ADDR16 _ ⇒ λH:False. P2 H
331        | DIRECT _ ⇒ λH:False. P3 H
332        | INDIRECT _ ⇒ λH:False. P4 H
333        | EXT_INDIRECT _ ⇒ λH:False. P5 H
334        | ACC_A ⇒ λH:False. P6 H
335        | REGISTER _ ⇒ λH:False. P7 H
336        | ACC_B ⇒ λH:False. P8 H
337        | DPTR ⇒ λH:False. P9 H
338        | DATA _ ⇒ λH:False. P10 H
339        | DATA16 _ ⇒ λH:False. P11 H
340        | ACC_DPTR ⇒ λH:False. P12 H
341        | ACC_PC ⇒ λH:False. P13 H
342        | EXT_INDIRECT_DPTR ⇒ λH:False. P14 H
343        | INDIRECT_DPTR ⇒ λH:False. P15 H
344        | CARRY ⇒ λH:False. P16 H
345        | BIT_ADDR _ ⇒ λH:False. P17 H
346        | N_BIT_ADDR _ ⇒ λH:False. P18 H   
347        | RELATIVE _ ⇒ λH:False. P19 H
348        ] p).
349  #T #P1 #P2 #P3 #P4 #P5 #P6 #P7 #P8 #P9 #P10 #P11 #P12 #P13
350  #P14 #P15 #P16 #P17 #P18 #P19
351  * try #x1 try #x2 try #x3 try #x4
352  try (@⊥ assumption) normalize @x4
353qed. *)
354
355include alias "arithmetics/nat.ma".
356
357lemma lt_n_o_to_plus_m_n_lt_plus_m_o:
358  ∀m, n, o: nat.
359    n < o → m + n < m + o.
360  #m #n #o #assm /2 by monotonic_le_plus_r/
361qed.
362
363axiom fetch_program_counter_n_technical:
364  ∀code_memory: BitVectorTrie Byte 16.
365  ∀program_counter, program_counter': Word.
366  ∀instruction: instruction.
367  ∀ticks, n: nat.
368  program_counter' = \snd (\fst (fetch code_memory program_counter)) →
369    program_counter' = fetch_program_counter_n (S n) code_memory (zero …) →
370      program_counter = fetch_program_counter_n n code_memory (zero …).
371   
372let rec block_cost
373  (code_memory: BitVectorTrie Byte 16) (program_counter: Word)
374    (program_size: nat) (total_program_size: nat) (cost_labels: BitVectorTrie costlabel 16)
375      (good_program_witness: good_program code_memory program_counter total_program_size)
376        on program_size: total_program_size ≤ program_size + nat_of_bitvector … program_counter → nat ≝
377  match program_size return λprogram_size: nat. total_program_size ≤ program_size + nat_of_bitvector … program_counter → nat with
378  [ O ⇒ λbase_case. 0
379  | S program_size' ⇒ λrecursive_case.
380    let 〈instruction, program_counter', ticks〉 as FETCH ≝ fetch code_memory program_counter in
381      match lookup_opt … program_counter' cost_labels return λx: option costlabel. nat with
382      [ None   ⇒
383        match instruction return λx. x = instruction → ? with
384        [ RealInstruction instruction ⇒ λreal_instruction.
385          match instruction return λx. x = instruction → ? with
386          [ RET                    ⇒ λinstr. ticks
387          | JC   relative          ⇒ λinstr. ticks
388          | JNC  relative          ⇒ λinstr. ticks
389          | JB   bit_addr relative ⇒ λinstr. ticks
390          | JNB  bit_addr relative ⇒ λinstr. ticks
391          | JBC  bit_addr relative ⇒ λinstr. ticks
392          | JZ   relative          ⇒ λinstr. ticks
393          | JNZ  relative          ⇒ λinstr. ticks
394          | CJNE src_trgt relative ⇒ λinstr. ticks
395          | DJNZ src_trgt relative ⇒ λinstr. ticks
396          | _                      ⇒ λinstr.
397              ticks + block_cost code_memory program_counter' program_size' total_program_size cost_labels ? ?
398          ] (refl … instruction)
399        | ACALL addr     ⇒ λinstr.
400            ticks + block_cost code_memory program_counter' program_size' total_program_size cost_labels ? ?
401        | AJMP  addr     ⇒ λinstr. ticks
402        | LCALL addr     ⇒ λinstr.
403            ticks + block_cost code_memory program_counter' program_size' total_program_size cost_labels ? ?
404        | LJMP  addr     ⇒ λinstr. ticks
405        | SJMP  addr     ⇒ λinstr. ticks
406        | JMP   addr     ⇒ λinstr. (* XXX: actually a call due to use with fptrs *)
407            ticks + block_cost code_memory program_counter' program_size' total_program_size cost_labels ? ?
408        | MOVC  src trgt ⇒ λinstr.
409            ticks + block_cost code_memory program_counter' program_size' total_program_size cost_labels ? ?
410        ] (refl … instruction)
411      | Some _ ⇒ ticks
412      ]
413  ].
414  [1:
415    generalize in match good_program_witness;
416    whd in match good_program; normalize nodelta
417    cases FETCH normalize nodelta
418    cases instr normalize nodelta
419    @(subaddressing_mode_elim … [[ addr11 ]]) #new_addr
420    cases (split … 8 8 program_counter') #pc_bu #pc_bl normalize nodelta
421    cases (split … 3 8 new_addr) #thr #eig normalize nodelta
422    cases (split … 5 3 pc_bu) #fiv #thr' normalize nodelta
423    #assm cases assm #ignore
424    whd in match good_program_counter; normalize nodelta * #n * *
425    #program_counter_eq' #program_counter_lt_total_program_size
426    #fetch_n_leq_program_counter'
427    @(transitive_le
428      total_program_size
429      ((S program_size') + nat_of_bitvector … program_counter)
430      (program_size' + nat_of_bitvector … program_counter') recursive_case)
431    whd in ⊢ (?%?);
432    change with (
433      program_size' + (nat_of_bitvector … program_counter) <
434        program_size' + (nat_of_bitvector … program_counter'))
435    @lt_n_o_to_plus_m_n_lt_plus_m_o
436    >(fetch_program_counter_n_technical code_memory program_counter
437      program_counter' instruction ticks n)
438    /2 by pair_destruct_2/
439  |7:
440    generalize in match good_program_witness;
441    whd in match good_program; normalize nodelta
442    cases FETCH normalize nodelta
443    cases instr normalize nodelta
444    @(subaddressing_mode_elim … [[ acc_dptr; acc_pc ]]) #new_addr
445    cases (split … 8 8 program_counter') #pc_bu #pc_bl normalize nodelta
446    cases (split … 3 8 new_addr) #thr #eig normalize nodelta
447    cases (split … 5 3 pc_bu) #fiv #thr' normalize nodelta
448    #assm cases assm #ignore
449    whd in match good_program_counter; normalize nodelta * #n * *
450    #program_counter_eq' #program_counter_lt_total_program_size
451    #fetch_n_leq_program_counter'
452    @(transitive_le
453      total_program_size
454      ((S program_size') + nat_of_bitvector … program_counter)
455      (program_size' + nat_of_bitvector … program_counter') recursive_case)
456    whd in ⊢ (?%?);
457    change with (
458      program_size' + (nat_of_bitvector … program_counter) <
459        program_size' + (nat_of_bitvector … program_counter'))
460    @lt_n_o_to_plus_m_n_lt_plus_m_o
461    >(fetch_program_counter_n_technical code_memory program_counter
462      program_counter' instruction ticks n)
463    /2 by pair_destruct_2/
464  |3,5,7,9,11:
465    (* XXX etc. need the subaddressing_mode_elim generalizing *)
466  |2:
467    generalize in match good_program_witness;
468    whd in match (good_program code_memory program_counter total_program_size);
469    cases FETCH normalize nodelta
470    cases instr normalize nodelta
471    @subaddressing_mode_elim #new_addr
472    cases (split … 8 8 program_counter') #pc_bu #pc_bl normalize nodelta
473    cases (split … 3 8 new_addr) #thr #eig normalize nodelta
474    cases (split … 5 3 pc_bu) #fiv #thr' normalize nodelta
475    #assm cases assm #ignore #good_program_counter
476    whd in match (good_program code_memory program_counter' total_program_size);
477    cases(fetch code_memory program_counter') #instruction_program_counter'' #ticks''
478    cases(instruction_program_counter'') #instruction'' #program_counter'' normalize nodelta
479   
480  [2:
481    (* generalize in match good_program_witness; *)
482    whd in match good_program; normalize nodelta
483    cases FETCH normalize nodelta
484    cases (fetch code_memory program_counter') #instruction_program_counter' #ticks' normalize nodelta
485    cases instruction_program_counter' #instruction' #program_counter'' normalize nodelta
486    cases acall normalize nodelta
487    cases addr #subaddressing_mode cases subaddressing_mode
488    try (#assm #absurd normalize in absurd; cases absurd)
489    try (#absurd normalize in absurd; cases absurd)
490    normalize nodelta
491    cases instruction'
492    try (#assm normalize nodelta)
493    [7:
494      #irrelevant
495      whd in match good_program_counter; normalize nodelta
496     
497     
498     
499     
500     
501     
502     
503     
504     
505(* XXX: use memoisation here in the future *)
506let rec block_cost
507  (code_memory: BitVectorTrie Byte 16) (cost_labels: BitVectorTrie costlabel 16)
508    (program_counter: Word) (program_size: nat) (total_program_size: nat)
509      (size_invariant: total_program_size ≤ code_memory_size)
510        (pc_invariant: nat_of_bitvector … program_counter < total_program_size)
511          on program_size: total_program_size - nat_of_bitvector … program_counter ≤ program_size → nat ≝
512  match program_size return λprogram_size: nat. total_program_size - nat_of_bitvector … program_counter ≤ program_size → nat with
513  [ O ⇒ λbase_case. 0 (* XXX: change from ⊥ before *)
514  | S program_size ⇒ λrecursive_case.
515    let 〈instr, newpc, ticks〉 ≝ fetch … code_memory program_counter in
516      match lookup_opt … newpc cost_labels return λx: option costlabel. nat with
517      [ None ⇒
518          let classify ≝ ASM_classify0 instr in
519          match classify return λx. classify = x → ? with
520          [ cl_jump ⇒ λclassify_refl. ticks
521          | cl_call ⇒ λclassify_refl. (* ite here *)
522              ticks + (block_cost code_memory cost_labels newpc program_size total_program_size size_invariant ? final_instr_invariant ?)
523          | cl_return ⇒ λclassify_refl. ticks
524          | cl_other ⇒ λclassify_refl. (* ite here *)
525              ticks + (block_cost code_memory cost_labels newpc program_size total_program_size size_invariant ? final_instr_invariant ?)
526          ] (refl … classify)
527        | Some _ ⇒ ticks
528      ]
529  ].
530
531let rec traverse_code_internal
532  (program: list Byte) (mem: BitVectorTrie Byte 16)
533    (cost_labels: BitVectorTrie costlabel 16) (pc: Word) (program_size: nat)
534      on program: identifier_map CostTag nat ≝
535 let 〈instr,newpc,ticks〉 ≝ fetch … mem pc in
536 match program with
537 [ nil ⇒ empty_map …
538 | cons hd tl ⇒
539   match lookup_opt … pc cost_labels with
540   [ None ⇒ traverse_code_internal tl mem cost_labels newpc program_size
541   | Some lbl ⇒
542     let cost ≝ block_cost mem cost_labels pc program_size in
543     let cost_mapping ≝ traverse_code_internal tl mem cost_labels newpc program_size in
544       add … cost_mapping lbl cost ]].
545
546definition traverse_code ≝
547  λprogram: list Byte.
548  λmem: BitVectorTrie Byte 16.
549  λcost_labels.
550  λprogram_size: nat.
551    traverse_code_internal program mem cost_labels (zero …) program_size.
552
553definition compute_costs ≝
554  λprogram: list Byte.
555  λcost_labels: BitVectorTrie costlabel 16.
556  λhas_main: bool.
557  let program_size ≝ |program| + 1 in
558  let memory ≝ load_code_memory program in
559   traverse_code program memory cost_labels program_size.
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