source: src/ASM/ASMCosts.ma @ 1903

Last change on this file since 1903 was 1903, checked in by mulligan, 8 years ago

Small changes prior to experiment
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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 well_labelling: BitVectorTrie costlabel 16 → Prop ≝
32  λcost_labels.
33    injective … (λx. lookup_opt … x cost_labels).
34   
35definition good_program: ∀code_memory: BitVectorTrie Byte 16. ∀total_program_size: nat. Prop ≝
36  λcode_memory: BitVectorTrie Byte 16.
37  λtotal_program_size: nat.
38  ∀program_counter: Word.
39  ∀good_program_counter_witness: reachable_program_counter code_memory total_program_size program_counter.
40  let 〈instruction, program_counter', ticks〉 ≝ fetch code_memory program_counter in
41    match instruction with
42    [ RealInstruction instr ⇒
43      match instr with
44      [ RET                    ⇒ True
45      | JC   relative          ⇒ True (* XXX: see below *)
46      | JNC  relative          ⇒ True (* XXX: see below *)
47      | JB   bit_addr relative ⇒ True
48      | JNB  bit_addr relative ⇒ True
49      | JBC  bit_addr relative ⇒ True
50      | JZ   relative          ⇒ True
51      | JNZ  relative          ⇒ True
52      | CJNE src_trgt relative ⇒ True
53      | DJNZ src_trgt relative ⇒ True
54      | _                      ⇒
55        nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
56          nat_of_bitvector … program_counter' < total_program_size
57      ]
58    | LCALL addr         ⇒
59      match addr return λx. bool_to_Prop (is_in … [[ addr16 ]] x) → Prop with
60      [ ADDR16 addr ⇒ λaddr16: True.
61          reachable_program_counter code_memory total_program_size addr ∧
62            nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
63              nat_of_bitvector … program_counter' < total_program_size
64      | _ ⇒ λother: False. ⊥
65      ] (subaddressing_modein … addr)
66    | ACALL addr         ⇒
67      match addr return λx. bool_to_Prop (is_in … [[ addr11 ]] x) → Prop with
68      [ ADDR11 addr ⇒ λaddr11: True.
69        let 〈pc_bu, pc_bl〉 ≝ split … 8 8 program_counter' in
70        let 〈thr, eig〉 ≝ split … 3 8 addr in
71        let 〈fiv, thr'〉 ≝ split … 5 3 pc_bu in
72        let new_program_counter ≝ (fiv @@ thr) @@ pc_bl in
73          reachable_program_counter code_memory total_program_size new_program_counter ∧
74            nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
75              nat_of_bitvector … program_counter' < total_program_size
76      | _ ⇒ λother: False. ⊥
77      ] (subaddressing_modein … addr)
78    | AJMP  addr         ⇒
79      match addr return λx. bool_to_Prop (is_in … [[ addr11 ]] x) → Prop with
80      [ ADDR11 addr ⇒ λaddr11: True.
81        let 〈pc_bu, pc_bl〉 ≝ split … 8 8 program_counter' in
82        let 〈nu, nl〉 ≝ split … 4 4 pc_bu in
83        let bit ≝ get_index' … O ? nl in
84        let 〈relevant1, relevant2〉 ≝ split … 3 8 addr in
85        let new_addr ≝ (nu @@ (bit ::: relevant1)) @@ relevant2 in
86        let 〈carry, new_program_counter〉 ≝ half_add 16 program_counter new_addr in
87          reachable_program_counter code_memory total_program_size new_program_counter
88      | _ ⇒ λother: False. ⊥
89      ] (subaddressing_modein … addr)
90    | LJMP  addr         ⇒
91      match addr return λx. bool_to_Prop (is_in … [[ addr16 ]] x) → Prop with
92      [ ADDR16 addr ⇒ λaddr16: True.
93          reachable_program_counter code_memory total_program_size addr
94      | _ ⇒ λother: False. ⊥
95      ] (subaddressing_modein … addr)
96    | SJMP  addr     ⇒
97      match addr return λx. bool_to_Prop (is_in … [[ relative ]] x) → Prop with
98      [ RELATIVE addr ⇒ λrelative: True.
99        let 〈carry, new_program_counter〉 ≝ half_add … program_counter' (sign_extension addr) in
100          reachable_program_counter code_memory total_program_size new_program_counter
101      | _ ⇒ λother: False. ⊥
102      ] (subaddressing_modein … addr)
103    | JMP   addr     ⇒ (* XXX: JMP is used for fptrs and unconstrained *)
104      nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
105        nat_of_bitvector … program_counter' < total_program_size
106    | MOVC  src trgt ⇒
107        nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' ∧
108          nat_of_bitvector … program_counter' < total_program_size
109    ].
110  cases other
111qed.
112       
113lemma is_a_decidable:
114  ∀hd.
115  ∀element.
116    is_a hd element = true ∨ is_a hd element = false.
117  #hd #element //
118qed.
119
120lemma mem_decidable:
121  ∀n: nat.
122  ∀v: Vector addressing_mode_tag n.
123  ∀element: addressing_mode_tag.
124    mem … eq_a n v element = true ∨
125      mem … eq_a … v element = false.
126  #n #v #element //
127qed.
128
129lemma eq_a_elim:
130  ∀tag.
131  ∀hd.
132  ∀P: bool → Prop.
133    (tag = hd → P (true)) →
134      (tag ≠ hd → P (false)) →
135        P (eq_a tag hd).
136  #tag #hd #P
137  cases tag
138  cases hd
139  #true_hyp #false_hyp
140  try @false_hyp
141  try @true_hyp
142  try %
143  #absurd destruct(absurd)
144qed.
145 
146lemma is_a_true_to_is_in:
147  ∀n: nat.
148  ∀x: addressing_mode.
149  ∀tag: addressing_mode_tag.
150  ∀supervector: Vector addressing_mode_tag n.
151  mem addressing_mode_tag eq_a n supervector tag →
152    is_a tag x = true →
153      is_in … supervector x.
154  #n #x #tag #supervector
155  elim supervector
156  [1:
157    #absurd cases absurd
158  |2:
159    #n' #hd #tl #inductive_hypothesis
160    whd in match (mem … eq_a (S n') (hd:::tl) tag);
161    @eq_a_elim normalize nodelta
162    [1:
163      #tag_hd_eq #irrelevant
164      whd in match (is_in (S n') (hd:::tl) x);
165      <tag_hd_eq #is_a_hyp >is_a_hyp normalize nodelta
166      @I
167    |2:
168      #tag_hd_neq
169      whd in match (is_in (S n') (hd:::tl) x);
170      change with (
171        mem … eq_a n' tl tag)
172          in match (fold_right … n' ? false tl);
173      #mem_hyp #is_a_hyp
174      cases(is_a hd x)
175      [1:
176        normalize nodelta //
177      |2:
178        normalize nodelta
179        @inductive_hypothesis assumption
180      ]
181    ]
182  ]
183qed.
184
185lemma is_in_subvector_is_in_supervector:
186  ∀m, n: nat.
187  ∀subvector: Vector addressing_mode_tag m.
188  ∀supervector: Vector addressing_mode_tag n.
189  ∀element: addressing_mode.
190    subvector_with … eq_a subvector supervector →
191      is_in m subvector element → is_in n supervector element.
192  #m #n #subvector #supervector #element
193  elim subvector
194  [1:
195    #subvector_with_proof #is_in_proof
196    cases is_in_proof
197  |2:
198    #n' #hd' #tl' #inductive_hypothesis #subvector_with_proof
199    whd in match (is_in … (hd':::tl') element);
200    cases (is_a_decidable hd' element)
201    [1:
202      #is_a_true >is_a_true
203      #irrelevant
204      whd in match (subvector_with … eq_a (hd':::tl') supervector) in subvector_with_proof;
205      @(is_a_true_to_is_in … is_a_true)
206      lapply(subvector_with_proof)
207      cases(mem … eq_a n supervector hd') //
208    |2:
209      #is_a_false >is_a_false normalize nodelta
210      #assm
211      @inductive_hypothesis
212      [1:
213        generalize in match subvector_with_proof;
214        whd in match (subvector_with … eq_a (hd':::tl') supervector);
215        cases(mem_decidable n supervector hd')
216        [1:
217          #mem_true >mem_true normalize nodelta
218          #assm assumption
219        |2:
220          #mem_false >mem_false #absurd
221          cases absurd
222        ]
223      |2:
224        assumption
225      ]
226    ]
227  ]
228qed.
229
230let rec member_addressing_mode_tag
231  (n: nat) (v: Vector addressing_mode_tag n) (a: addressing_mode_tag)
232    on v: Prop ≝
233  match v with
234  [ VEmpty ⇒ False
235  | VCons n' hd tl ⇒
236      bool_to_Prop (eq_a hd a) ∨ member_addressing_mode_tag n' tl a
237  ].
238 
239let rec subaddressing_mode_elim_type
240  (T: Type[2]) (m: nat) (fixed_v: Vector addressing_mode_tag m)
241    (Q: addressing_mode → T → Prop)
242      (p_addr11:            ∀w: Word11.      is_in m fixed_v (ADDR11 w)        → T)
243      (p_addr16:            ∀w: Word.        is_in m fixed_v (ADDR16 w)        → T)
244      (p_direct:            ∀w: Byte.        is_in m fixed_v (DIRECT w)        → T)
245      (p_indirect:          ∀w: Bit.         is_in m fixed_v (INDIRECT w)      → T)
246      (p_ext_indirect:      ∀w: Bit.         is_in m fixed_v (EXT_INDIRECT w)  → T)
247      (p_acc_a:                              is_in m fixed_v ACC_A             → T)
248      (p_register:          ∀w: BitVector 3. is_in m fixed_v (REGISTER w)      → T)
249      (p_acc_b:                              is_in m fixed_v ACC_B             → T)
250      (p_dptr:                               is_in m fixed_v DPTR              → T)
251      (p_data:              ∀w: Byte.        is_in m fixed_v (DATA w)          → T)
252      (p_data16:            ∀w: Word.        is_in m fixed_v (DATA16 w)        → T)
253      (p_acc_dptr:                           is_in m fixed_v ACC_DPTR          → T)
254      (p_acc_pc:                             is_in m fixed_v ACC_PC            → T)
255      (p_ext_indirect_dptr:                  is_in m fixed_v EXT_INDIRECT_DPTR → T)
256      (p_indirect_dptr:                      is_in m fixed_v INDIRECT_DPTR     → T)
257      (p_carry:                              is_in m fixed_v CARRY             → T)
258      (p_bit_addr:          ∀w: Byte.        is_in m fixed_v (BIT_ADDR w)      → T)
259      (p_n_bit_addr:        ∀w: Byte.        is_in m fixed_v (N_BIT_ADDR w)    → T)
260      (p_relative:          ∀w: Byte.        is_in m fixed_v (RELATIVE w)      → T)
261        (n: nat) (v: Vector addressing_mode_tag n) (proof: subvector_with … eq_a v fixed_v)
262      on v: Prop ≝
263  match v return λo: nat. λv': Vector addressing_mode_tag o. o = n → v ≃ v' → ? with
264  [ VEmpty         ⇒ λm_refl. λv_refl.
265      ∀addr: addressing_mode. ∀p: is_in m fixed_v addr.
266        Q addr (
267        match addr return λx: addressing_mode. is_in … fixed_v x → T with 
268        [ ADDR11 x          ⇒ p_addr11 x
269        | ADDR16 x          ⇒ p_addr16 x
270        | DIRECT x          ⇒ p_direct x
271        | INDIRECT x        ⇒ p_indirect x
272        | EXT_INDIRECT x    ⇒ p_ext_indirect x
273        | ACC_A             ⇒ p_acc_a
274        | REGISTER x        ⇒ p_register x
275        | ACC_B             ⇒ p_acc_b
276        | DPTR              ⇒ p_dptr
277        | DATA x            ⇒ p_data x
278        | DATA16 x          ⇒ p_data16 x
279        | ACC_DPTR          ⇒ p_acc_dptr
280        | ACC_PC            ⇒ p_acc_pc
281        | EXT_INDIRECT_DPTR ⇒ p_ext_indirect_dptr
282        | INDIRECT_DPTR     ⇒ p_indirect_dptr
283        | CARRY             ⇒ p_carry
284        | BIT_ADDR x        ⇒ p_bit_addr x
285        | N_BIT_ADDR x      ⇒ p_n_bit_addr x
286        | RELATIVE x        ⇒ p_relative x
287        ] p)
288  | VCons n' hd tl ⇒ λm_refl. λv_refl.
289    let tail_call ≝ subaddressing_mode_elim_type T m fixed_v Q p_addr11
290      p_addr16 p_direct p_indirect p_ext_indirect p_acc_a
291        p_register p_acc_b p_dptr p_data p_data16 p_acc_dptr
292          p_acc_pc p_ext_indirect_dptr p_indirect_dptr p_carry
293            p_bit_addr p_n_bit_addr p_relative n' tl ?
294    in
295    match hd return λa: addressing_mode_tag. a = hd → ? with
296    [ addr11            ⇒ λhd_refl. (∀w. Q (ADDR11 w) (p_addr11 w ?)) → tail_call
297    | addr16            ⇒ λhd_refl. (∀w. Q (ADDR16 w) (p_addr16 w ?)) → tail_call
298    | direct            ⇒ λhd_refl. (∀w. Q (DIRECT w) (p_direct w ?)) → tail_call
299    | indirect          ⇒ λhd_refl. (∀w. Q (INDIRECT w) (p_indirect w ?)) → tail_call
300    | ext_indirect      ⇒ λhd_refl. (∀w. Q (EXT_INDIRECT w) (p_ext_indirect w ?)) → tail_call
301    | acc_a             ⇒ λhd_refl. (Q ACC_A (p_acc_a ?)) → tail_call
302    | registr           ⇒ λhd_refl. (∀w. Q (REGISTER w) (p_register w ?)) → tail_call
303    | acc_b             ⇒ λhd_refl. (Q ACC_A (p_acc_b ?)) → tail_call
304    | dptr              ⇒ λhd_refl. (Q DPTR (p_dptr ?)) → tail_call
305    | data              ⇒ λhd_refl. (∀w. Q (DATA w) (p_data w ?)) → tail_call
306    | data16            ⇒ λhd_refl. (∀w. Q (DATA16 w) (p_data16 w ?)) → tail_call
307    | acc_dptr          ⇒ λhd_refl. (Q ACC_DPTR (p_acc_dptr ?)) → tail_call
308    | acc_pc            ⇒ λhd_refl. (Q ACC_PC (p_acc_pc ?)) → tail_call
309    | ext_indirect_dptr ⇒ λhd_refl. (Q EXT_INDIRECT_DPTR (p_ext_indirect_dptr ?)) → tail_call
310    | indirect_dptr     ⇒ λhd_refl. (Q INDIRECT_DPTR (p_indirect_dptr ?)) → tail_call
311    | carry             ⇒ λhd_refl. (Q CARRY (p_carry ?)) → tail_call
312    | bit_addr          ⇒ λhd_refl. (∀w. Q (BIT_ADDR w) (p_bit_addr w ?)) → tail_call
313    | n_bit_addr        ⇒ λhd_refl. (∀w. Q (N_BIT_ADDR w) (p_n_bit_addr w ?)) → tail_call
314    | relative          ⇒ λhd_refl. (∀w. Q (RELATIVE w) (p_relative w ?)) → tail_call
315    ] (refl … hd)
316  ] (refl … n) (refl_jmeq … v).
317  [20:
318    generalize in match proof; destruct
319    whd in match (subvector_with … eq_a (hd:::tl) fixed_v);
320    cases (mem … eq_a m fixed_v hd) normalize nodelta
321    [1:
322      whd in match (subvector_with … eq_a tl fixed_v);
323      #assm assumption
324    |2:
325      normalize in ⊢ (% → ?);
326      #absurd cases absurd
327    ]
328  ]
329  @(is_in_subvector_is_in_supervector … proof)
330  destruct @I
331qed.
332
333(* XXX: todo *)
334lemma subaddressing_mode_elim':
335  ∀T: Type[2].
336  ∀n: nat.
337  ∀o: nat.
338  ∀v1: Vector addressing_mode_tag n.
339  ∀v2: Vector addressing_mode_tag o.
340  ∀Q: addressing_mode → T → Prop.
341  ∀fixed_v: Vector addressing_mode_tag (n + o).
342  ∀P1,P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13,P14,P15,P16,P17,P18,P19.
343  ∀fixed_v_proof: fixed_v = v1 @@ v2.
344  ∀subaddressing_mode_proof.
345    subaddressing_mode_elim_type T (n + o) fixed_v Q P1 P2 P3 P4 P5 P6 P7
346      P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 (n + o) (v1 @@ v2) subaddressing_mode_proof.
347  #T #n #o #v1 #v2
348  elim v1 cases v2
349  [1:
350    #Q #fixed_v #P1 #P2 #P3 #P4 #P5 #P6 #P7 #P8 #P9 #P10
351    #P11 #P12 #P13 #P14 #P15 #P16 #P17 #P18 #P19 #fixed_v_proof
352    #subaddressing_mode_proof destruct normalize
353    #addr #absurd cases absurd
354  |2:
355    #n' #hd #tl #Q #fixed_v #P1 #P2 #P3 #P4 #P5 #P6 #P7 #P8 #P9 #P10
356    #P11 #P12 #P13 #P14 #P15 #P16 #P17 #P18 #P19 #fixed_v_proof
357    destruct normalize in match ([[]]@@hd:::tl);
358  ]
359  cases daemon
360qed.
361
362(* XXX: todo *)
363axiom subaddressing_mode_elim:
364  ∀T: Type[2].
365  ∀m: nat.
366  ∀n: nat.
367  ∀Q: addressing_mode → T → Prop.
368  ∀fixed_v: Vector addressing_mode_tag m.
369  ∀P1,P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13,P14,P15,P16,P17,P18,P19.
370  ∀v: Vector addressing_mode_tag n.
371  ∀proof.
372    subaddressing_mode_elim_type T m fixed_v Q P1 P2 P3 P4 P5 P6 P7
373      P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 n v proof.
374
375definition current_instruction_is_labelled ≝
376  λcode_memory.
377  λcost_labels: BitVectorTrie costlabel 16.
378  λs: Status code_memory.
379  let pc ≝ program_counter … code_memory s in
380    match lookup_opt … pc cost_labels with
381    [ None ⇒ False
382    | _    ⇒ True
383    ].
384
385definition next_instruction_properly_relates_program_counters ≝
386  λcode_memory.
387  λbefore: Status code_memory.
388  λafter : Status code_memory.
389    let program_counter_before ≝ program_counter ? code_memory before in
390    let 〈instruction, program_counter_after, ticks〉 ≝ fetch code_memory program_counter_before in
391      program_counter ? code_memory after = program_counter_after.
392(* XXX: why defined like this?
393  let size ≝ current_instruction_cost code_memory before in
394  let pc_before ≝ program_counter … code_memory before in
395  let pc_after ≝ program_counter … code_memory after in
396  let sum ≝ \snd (half_add … pc_before (bitvector_of_nat … size)) in
397    sum = pc_after.
398*)
399
400definition ASM_abstract_status: ∀code_memory. BitVectorTrie costlabel 16 → abstract_status ≝
401  λcode_memory.
402  λcost_labels.
403    mk_abstract_status
404      (Status code_memory)
405      (λs,s'. (execute_1 … s) = s')
406      (λs,class. ASM_classify … s = class)
407      (current_instruction_is_labelled … cost_labels)
408      (next_instruction_properly_relates_program_counters code_memory).
409
410let rec trace_any_label_length
411  (code_memory: BitVectorTrie Byte 16) (cost_labels: BitVectorTrie costlabel 16)
412    (trace_ends_flag: trace_ends_with_ret) (start_status: Status code_memory)
413      (final_status: Status code_memory)
414      (the_trace: trace_any_label (ASM_abstract_status code_memory cost_labels) trace_ends_flag start_status final_status)
415        on the_trace: nat ≝
416  match the_trace with
417  [ tal_base_not_return the_status _ _ _ _ ⇒ 0
418  | tal_base_call pre_fun_call start_fun_call final _ _ _ _ call_trace ⇒ 0
419  | tal_base_return the_status _ _ _ ⇒ 0
420  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final _ _ _ call_trace _ final_trace ⇒
421      let tail_length ≝ trace_any_label_length … final_trace in
422      let pc_difference ≝ nat_of_bitvector … (program_counter … after_fun_call) - nat_of_bitvector … (program_counter … pre_fun_call) in       
423        pc_difference + tail_length
424  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
425      let tail_length ≝ trace_any_label_length … tail_trace in
426      let pc_difference ≝ nat_of_bitvector … (program_counter … status_init) - nat_of_bitvector … (program_counter … status_pre) in
427        pc_difference + tail_length       
428  ].
429
430let rec compute_paid_trace_any_label
431  (code_memory: BitVectorTrie Byte 16) (cost_labels: BitVectorTrie costlabel 16)
432    (trace_ends_flag: trace_ends_with_ret) (start_status: Status code_memory)
433      (final_status: Status code_memory)
434        (the_trace: trace_any_label (ASM_abstract_status code_memory cost_labels) trace_ends_flag start_status final_status)
435       on the_trace: nat ≝
436  match the_trace with
437  [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost … the_status
438  | tal_base_return the_status _ _ _ ⇒ current_instruction_cost … the_status
439  | tal_base_call pre_fun_call start_fun_call final _ _ _ _ call_trace ⇒ current_instruction_cost … pre_fun_call
440  | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final
441     _ _ _ call_trace _ final_trace ⇒
442      let current_instruction_cost ≝ current_instruction_cost … pre_fun_call in
443      let final_trace_cost ≝ compute_paid_trace_any_label … cost_labels end_flag … final_trace in
444        current_instruction_cost + final_trace_cost
445  | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒
446      let current_instruction_cost ≝ current_instruction_cost … status_pre in
447      let tail_trace_cost ≝
448       compute_paid_trace_any_label … cost_labels end_flag
449        status_init status_end tail_trace
450      in
451        current_instruction_cost + tail_trace_cost
452  ].
453
454definition compute_paid_trace_label_label ≝
455  λcode_memory: BitVectorTrie Byte 16.
456  λcost_labels: BitVectorTrie costlabel 16.
457  λtrace_ends_flag: trace_ends_with_ret.
458  λstart_status: Status code_memory.
459  λfinal_status: Status code_memory.
460  λthe_trace: trace_label_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
461  match the_trace with
462  [ tll_base ends_flag initial final given_trace labelled_proof ⇒
463      compute_paid_trace_any_label … given_trace
464  ].
465
466include alias "arithmetics/nat.ma".
467include alias "basics/logic.ma".
468
469lemma plus_right_monotone:
470  ∀m, n, o: nat.
471    m = n → m + o = n + o.
472  #m #n #o #refl >refl %
473qed.
474
475lemma plus_left_monotone:
476  ∀m, n, o: nat.
477    m = n → o + m = o + n.
478  #m #n #o #refl destruct %
479qed.
480
481lemma minus_plus_cancel:
482  ∀m, n : nat.
483  ∀proof: n ≤ m.
484    (m - n) + n = m.
485  #m #n #proof /2 by plus_minus/
486qed.
487
488(* XXX: indexing bug *)
489lemma fetch_twice_fetch_execute_1:
490  ∀code_memory: BitVectorTrie Byte 16.
491  ∀start_status: Status code_memory.
492    ASM_classify code_memory start_status = cl_other →
493    \snd (\fst (fetch code_memory (program_counter … start_status))) =
494      program_counter … (execute_1 … start_status).
495  #code_memory #start_status #classify_assm
496  whd in match execute_1; normalize nodelta
497  cases (execute_1' code_memory start_status) #the_status
498  * #_ #classify_assm' @classify_assm' assumption
499qed-.
500
501lemma reachable_program_counter_to_0_lt_total_program_size:
502  ∀code_memory: BitVectorTrie Byte 16.
503  ∀program_counter: Word.
504  ∀total_program_size: nat.
505    reachable_program_counter code_memory total_program_size program_counter →
506      0 < total_program_size.
507  #code_memory #program_counter #total_program_size
508  whd in match reachable_program_counter; normalize nodelta * * #some_n
509  #_ cases (nat_of_bitvector 16 program_counter)
510  [1:
511    #assm assumption
512  |2:
513    #new_pc @ltn_to_ltO
514  ]
515qed.
516
517lemma trace_compute_paid_trace_cl_other:
518  ∀code_memory' : (BitVectorTrie Byte 16).
519  ∀program_counter' : Word.
520  ∀total_program_size : ℕ.
521  ∀cost_labels : (BitVectorTrie costlabel 16).
522  ∀reachable_program_counter_witness : (reachable_program_counter code_memory' total_program_size program_counter').
523  ∀good_program_witness : (good_program code_memory' total_program_size).
524  ∀program_size' : ℕ.
525  ∀recursive_case : (total_program_size≤S program_size'+nat_of_bitvector 16 program_counter').
526  ∀ticks : ℕ.
527  ∀instruction : instruction.
528  ∀program_counter'' : Word.
529  ∀FETCH : (〈instruction,program_counter'',ticks〉=fetch code_memory' program_counter').
530  ∀start_status : (Status code_memory').
531  ∀final_status : (Status code_memory').
532  ∀trace_ends_flag : trace_ends_with_ret.
533  ∀the_trace : (trace_any_label (ASM_abstract_status code_memory' cost_labels) trace_ends_flag start_status final_status).
534  ∀program_counter_refl : (program_counter' = program_counter (BitVectorTrie Byte 16) code_memory' start_status).
535  ∀classify_assm: ASM_classify0 instruction = cl_other.
536  ∀pi1 : ℕ.
537   (if match lookup_opt costlabel 16 program_counter'' cost_labels with 
538         [None ⇒ true
539         |Some _ ⇒ false
540         ] 
541    then
542      ∀start_status0:Status code_memory'.
543      ∀final_status0:Status code_memory'.
544      ∀trace_ends_flag0:trace_ends_with_ret.
545      ∀the_trace0:trace_any_label (ASM_abstract_status code_memory' cost_labels) trace_ends_flag0 start_status0 final_status0.
546        program_counter'' = program_counter (BitVectorTrie Byte 16) code_memory' start_status0 →
547        (∃n:ℕ.
548            trace_any_label_length code_memory' cost_labels
549              trace_ends_flag0 start_status0 final_status0 the_trace0
550                + S n
551                  = total_program_size) ∧
552                  pi1
553                    =compute_paid_trace_any_label code_memory' cost_labels
554                     trace_ends_flag0 start_status0 final_status0 the_trace0
555    else (pi1=O) )
556   →(∃n:ℕ.
557     trace_any_label_length code_memory' cost_labels trace_ends_flag
558      start_status final_status the_trace
559      +S n
560      =total_program_size)
561    ∧ticks+pi1
562     =compute_paid_trace_any_label code_memory' cost_labels trace_ends_flag
563      start_status final_status the_trace.
564  #code_memory' #program_counter' #total_program_size #cost_labels
565  #reachable_program_counter_witness #good_program_witness
566  #program_size' #recursive_case #ticks #instruction #program_counter'' #FETCH
567  #start_status #final_status
568  #trace_ends_flag #the_trace #program_counter_refl #classify_assm #recursive_block_cost
569  #recursive_assm
570  @(trace_any_label_inv_ind … the_trace)
571    [5:
572      #end_flag #status_pre #status_init #status_end #execute_assm #trace_any_label
573      #classifier_assm #costed_assm #trace_ends_refl #start_status_refl #final_status_refl
574      #the_trace_refl
575      destruct
576      whd in match (trace_any_label_length … (tal_step_default …));
577      whd in match (compute_paid_trace_any_label … (tal_step_default …));
578      whd in costed_assm:(?%);
579      generalize in match costed_assm;
580      generalize in match (refl … (lookup_opt … (program_counter … (execute_1 … status_pre)) cost_labels));
581      generalize in match (lookup_opt … (program_counter … (execute_1 … status_pre)) cost_labels)
582        in ⊢ (??%? → ?(match % with [ _ ⇒ ? | _ ⇒ ? ]) → ?);
583      #lookup_assm cases lookup_assm
584      [1:
585        #None_lookup_opt_assm normalize nodelta #ignore
586        generalize in match recursive_assm;
587        cut(program_counter'' = (program_counter (BitVectorTrie Byte 16) code_memory' (execute_1 code_memory' status_pre)))
588        [1:
589          <fetch_twice_fetch_execute_1
590          [1:
591            <FETCH %
592          |2:
593            >classifier_assm %
594          ]
595        |2:
596          #program_counter_assm >program_counter_assm <None_lookup_opt_assm
597          normalize nodelta #new_recursive_assm
598          cases(new_recursive_assm (execute_1 code_memory' status_pre) status_end
599            end_flag trace_any_label ?) try %
600          #exists_assm #recursive_block_cost_assm %
601          [1:
602            cases exists_assm #exists_n #total_program_size_refl
603            <total_program_size_refl
604            %{(exists_n - (nat_of_bitvector 16
605                (program_counter (BitVectorTrie Byte 16) code_memory'
606                  (execute_1 code_memory' status_pre))
607                - nat_of_bitvector 16
608                  (program_counter (BitVectorTrie Byte 16) code_memory' status_pre)))}
609            <plus_n_Sm <plus_n_Sm @eq_f >commutative_plus
610            cases daemon
611          |2:
612            >recursive_block_cost_assm
613            whd in match (current_instruction_cost … status_pre);
614            cut(ticks = \snd (fetch code_memory'
615               (program_counter (BitVectorTrie Byte 16) code_memory' status_pre)))
616            [1:
617              <FETCH %
618            |2:
619              #ticks_refl_assm >ticks_refl_assm %
620            ]
621          ]
622        ]
623      |2:
624        #costlabel #Some_lookup_opt_assm <Some_lookup_opt_assm normalize nodelta
625        #absurd cases absurd #absurd cases(absurd I)
626      ]
627    |1:
628      #status_start #status_final #execute_assm #classifier_assm #costed_assm
629      #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
630      destruct
631      whd in match (trace_any_label_length … (tal_base_not_return …));
632      whd in match (compute_paid_trace_any_label … (tal_base_not_return …));
633      whd in costed_assm;
634      generalize in match costed_assm;
635      generalize in match (refl … (lookup_opt … (program_counter … (execute_1 … status_start)) cost_labels));
636      generalize in match (lookup_opt … (program_counter … (execute_1 code_memory' status_start)) cost_labels)
637        in ⊢ (??%? → (match % with [ _ ⇒ ? | _ ⇒ ? ]) → ?);
638      #lookup_assm cases lookup_assm
639      [1:
640        #None_lookup_opt_assm normalize nodelta >None_lookup_opt_assm
641        #absurd cases absurd
642      |2:
643        #costlabel #Some_lookup_opt_assm normalize nodelta #ignore
644        generalize in match recursive_assm;
645        cut(program_counter'' = (program_counter (BitVectorTrie Byte 16) code_memory' (execute_1 … status_start)))
646        [1:
647          <fetch_twice_fetch_execute_1
648          [1:
649            <FETCH %
650          |2:
651            cases classifier_assm
652            [1:
653              whd in ⊢ (% → ?);
654              whd in ⊢ (??%? → ?);
655              whd in match (current_instruction code_memory' status_start);
656              <FETCH generalize in match classify_assm;
657              cases instruction
658              [8:
659                #preinstruction normalize nodelta
660                whd in match ASM_classify0; normalize nodelta
661                #contradiction >contradiction #absurd destruct(absurd)
662              ]
663              try(#addr1 #addr2 normalize nodelta #ignore #absurd destruct(absurd))
664              try(#addr normalize nodelta #ignore #absurd destruct(absurd))
665              normalize in ignore; destruct(ignore)
666            |2:
667              #classifier_assm' >classifier_assm' %
668            ]
669          ]
670        |2:
671          #program_counter_assm >program_counter_assm <Some_lookup_opt_assm
672          normalize nodelta #new_recursive_assm >new_recursive_assm %
673          [1:
674            %{(pred total_program_size)} whd in ⊢ (??%?);
675            @S_pred
676            @(reachable_program_counter_to_0_lt_total_program_size … reachable_program_counter_witness)
677          |2:
678            cut(ticks = \snd (fetch code_memory'
679               (program_counter (BitVectorTrie Byte 16) code_memory' status_start)))
680            [1:
681              <FETCH %
682            |2:
683              #ticks_refl_assm >ticks_refl_assm
684              <plus_n_O %
685            ]
686          ]
687        ]
688      ]
689    |2:
690      #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_assm
691      #start_status_refl #final_status_refl #the_trace_assm destruct @⊥
692    |3:
693      #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
694      #classifier_assm #after_return_assm #trace_label_return #costed_assm
695      #ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
696      destruct @⊥
697    |4:
698      #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
699      #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
700      #costed_assm #trace_any_label #trace_ends_flag_refl #start_status_refl
701      #final_status_refl #the_trace_refl destruct @⊥
702    ]
703  change with (ASM_classify0 ? = ?) in classifier_assm;
704  whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
705  whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
706  <FETCH in classifier_assm; >classify_assm #absurd destruct(absurd)
707qed.
708
709lemma trace_compute_paid_trace_cl_jump:
710  ∀code_memory': BitVectorTrie Byte 16.
711  ∀program_counter': Word.
712  ∀total_program_size: ℕ.
713  ∀cost_labels: BitVectorTrie costlabel 16.
714  ∀reachable_program_counter_witness: reachable_program_counter code_memory' total_program_size program_counter'.
715  ∀good_program_witness: good_program code_memory' total_program_size.
716  ∀first_time_around: bool.
717  ∀program_size': ℕ.
718  ∀recursive_case: total_program_size ≤ S program_size'+nat_of_bitvector 16 program_counter'.
719  ∀ticks: ℕ.
720  ∀instruction: instruction.
721  ∀program_counter'': Word.
722  ∀FETCH: 〈instruction,program_counter'',ticks〉 = fetch code_memory' program_counter'.
723  ∀start_status: (Status code_memory').
724  ∀final_status: (Status code_memory').
725  ∀trace_ends_flag: trace_ends_with_ret.
726  ∀the_trace: (trace_any_label (ASM_abstract_status code_memory' cost_labels) trace_ends_flag start_status final_status).
727  ∀program_counter_refl: (program_counter' = program_counter (BitVectorTrie Byte 16) code_memory' start_status).
728  ∀classify_assm: ASM_classify0 instruction = cl_jump.
729   (∃n:ℕ
730     .trace_any_label_length code_memory' cost_labels trace_ends_flag
731      start_status final_status the_trace
732      +S n
733      =total_program_size)
734    ∧ticks
735     =compute_paid_trace_any_label code_memory' cost_labels trace_ends_flag
736      start_status final_status the_trace.
737  #code_memory' #program_counter' #total_program_size #cost_labels
738  #reachable_program_counter_witness #good_program_witness #first_time_around
739  #program_size' #recursive_case #ticks #instruction #program_counter'' #FETCH
740  #start_status #final_status
741  #trace_ends_flag #the_trace #program_counter_refl #classify_assm
742  @(trace_any_label_inv_ind … the_trace)
743  [5:
744    #end_flag #status_pre #status_init #status_end #execute_assm #trace_any_label
745    #classifier_assm #costed_assm #trace_ends_refl #start_status_refl #final_status_refl
746    #the_trace_refl destruct @⊥
747  |1:
748    #status_start #status_final #execute_assm #classifier_assm #costed_assm
749    #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
750    destruct
751    whd in match (trace_any_label_length … (tal_base_not_return …));
752    whd in match (compute_paid_trace_any_label … (tal_base_not_return …)); %
753    [1:
754      %{(pred total_program_size)} whd in ⊢ (??%?);
755      @S_pred
756      @(reachable_program_counter_to_0_lt_total_program_size … reachable_program_counter_witness)
757    |2:
758      <FETCH %
759    ]
760  |2:
761    #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_assm
762    #start_status_refl #final_status_refl #the_trace_assm destruct @⊥
763  |3:
764    #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
765    #classifier_assm #after_return_assm #trace_label_return #costed_assm
766    #ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
767    destruct @⊥
768  |4:
769    #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
770    #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
771    #costed_assm #trace_any_label #trace_ends_flag_refl #start_status_refl
772    #final_status_refl #the_trace_refl destruct @⊥
773  ]
774  change with (ASM_classify0 ? = ?) in classifier_assm;
775  whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
776  whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
777  <FETCH in classifier_assm; >classify_assm #absurd destruct(absurd)
778qed.
779
780lemma trace_compute_paid_trace_cl_call:
781  ∀code_memory' : (BitVectorTrie Byte 16).
782  ∀program_counter' : Word.
783  ∀total_program_size : ℕ.
784  ∀cost_labels : (BitVectorTrie costlabel 16).
785  ∀reachable_program_counter_witness : (reachable_program_counter code_memory' total_program_size program_counter').
786  ∀good_program_witness : (good_program code_memory' total_program_size).
787  ∀program_size' : ℕ.
788  ∀recursive_case : (total_program_size≤S program_size'+nat_of_bitvector 16 program_counter').
789  ∀ticks : ℕ.
790  ∀instruction : instruction.
791  ∀program_counter'' : Word.
792  ∀FETCH : (〈instruction,program_counter'',ticks〉=fetch code_memory' program_counter').
793  ∀start_status : (Status code_memory').
794  ∀final_status : (Status code_memory').
795  ∀trace_ends_flag : trace_ends_with_ret.
796  ∀the_trace : (trace_any_label (ASM_abstract_status code_memory' cost_labels) trace_ends_flag start_status final_status).
797  ∀program_counter_refl : (program_counter' = program_counter (BitVectorTrie Byte 16) code_memory' start_status).
798  ∀classify_assm: ASM_classify0 instruction = cl_call.
799  (∀pi1:ℕ
800  .if match lookup_opt costlabel 16 program_counter'' cost_labels with 
801      [None ⇒ true | Some _ ⇒ false] 
802   then (∀start_status0:Status code_memory'
803             .∀final_status0:Status code_memory'
804              .∀trace_ends_flag0:trace_ends_with_ret
805               .∀the_trace0:trace_any_label
806                                        (ASM_abstract_status code_memory' cost_labels)
807                                        trace_ends_flag0 start_status0 final_status0
808                .program_counter''
809                 =program_counter (BitVectorTrie Byte 16) code_memory' start_status0
810                 →(∃n:ℕ
811                   .trace_any_label_length code_memory' cost_labels trace_ends_flag0
812                    start_status0 final_status0 the_trace0
813                    +S n
814                    =total_program_size)
815                  ∧pi1
816                   =compute_paid_trace_any_label code_memory' cost_labels
817                    trace_ends_flag0 start_status0 final_status0 the_trace0) 
818   else (pi1=O) 
819   →(∃n:ℕ
820     .trace_any_label_length code_memory' cost_labels trace_ends_flag
821      start_status final_status the_trace
822      +S n
823      =total_program_size)
824    ∧ticks+pi1
825     =compute_paid_trace_any_label code_memory' cost_labels trace_ends_flag
826      start_status final_status the_trace).
827  #code_memory' #program_counter' #total_program_size #cost_labels
828  #reachable_program_counter_witness #good_program_witness #program_size'
829  #recursive_case #ticks #instruction #program_counter'' #FETCH
830  #start_status #final_status #trace_ends_flag
831  #the_trace #program_counter_refl #classify_assm #recursive_block_cost #recursive_assm
832  @(trace_any_label_inv_ind … the_trace)
833  [5:
834    #end_flag #status_pre #status_init #status_end #execute_assm #trace_any_label
835    #classifier_assm #costed_assm #trace_ends_refl #start_status_refl #final_status_refl
836    #the_trace_refl destruct @⊥
837  |1:
838    #status_start #status_final #execute_assm #classifier_assm #costed_assm
839    #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
840    destruct @⊥
841  |2:
842    #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_assm
843    #start_status_refl #final_status_refl #the_trace_assm destruct @⊥
844  |3:
845    #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
846    #classifier_assm #after_return_assm #trace_label_return #costed_assm
847    #ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
848    destruct
849    whd in match (trace_any_label_length … (tal_base_call …));
850    whd in match (compute_paid_trace_any_label … (tal_base_call …));
851    whd in costed_assm;
852    generalize in match costed_assm;
853    generalize in match (refl … (lookup_opt … (program_counter … status_final) cost_labels));
854    generalize in match (lookup_opt … (program_counter … status_final) cost_labels)
855      in ⊢ (??%? → (match % with [ _ ⇒ ? | _ ⇒ ? ]) → ?);
856    #lookup_assm cases lookup_assm
857    [1:
858      #None_lookup_opt normalize nodelta #absurd cases absurd
859    |2:
860      #costlabel #Some_lookup_opt normalize nodelta #ignore
861      generalize in match recursive_assm;
862      cut(program_counter'' = (program_counter (BitVectorTrie Byte 16) code_memory' status_final))
863      [1:
864        generalize in match after_return_assm;
865        whd in ⊢ (% → ?); <FETCH normalize nodelta #relevant <relevant %
866      |2:
867        #program_counter_assm >program_counter_assm <Some_lookup_opt
868        normalize nodelta #new_recursive_assm >new_recursive_assm %
869        [1:
870          %{(pred total_program_size)} whd in ⊢ (??%?);
871          @S_pred
872          @(reachable_program_counter_to_0_lt_total_program_size … reachable_program_counter_witness)
873        |2:
874          cut(ticks = \snd (fetch code_memory' (program_counter (BitVectorTrie Byte 16) code_memory' status_pre_fun_call)))
875          [1:
876            <FETCH %
877          |2:
878            #ticks_refl_assm >ticks_refl_assm
879            <plus_n_O %
880          ]
881        ]
882      ]
883    ]
884  |4:
885    #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
886    #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
887    #costed_assm #trace_any_label #trace_ends_flag_refl #start_status_refl
888    #final_status_refl #the_trace_refl
889    generalize in match execute_assm; destruct #execute_assm
890    whd in match (trace_any_label_length … (tal_step_call …));
891    whd in match (compute_paid_trace_any_label … (tal_step_call …));
892    whd in costed_assm:(?%);
893    generalize in match costed_assm;
894    generalize in match (refl … (lookup_opt … (program_counter … status_after_fun_call) cost_labels));
895    generalize in match (lookup_opt … (program_counter … status_after_fun_call) cost_labels)
896      in ⊢ (??%? → ?(match % with [ _ ⇒ ? | _ ⇒ ? ]) → ?);
897    #lookup_assm cases lookup_assm
898    [1:
899      #None_lookup_opt_assm normalize nodelta #ignore
900      generalize in match recursive_assm;
901      cut(program_counter'' = program_counter … status_after_fun_call)
902      [1:
903        generalize in match after_return_assm;
904        whd in ⊢ (% → ?); <FETCH normalize nodelta #relevant >relevant %
905      |2:
906        #program_counter_refl >program_counter_refl <None_lookup_opt_assm
907        normalize nodelta #new_recursive_assm %
908        cases (new_recursive_assm … trace_any_label ?)
909        [1,3:
910          #exists_assm #recursive_block_cost_assm
911          [2:
912            >recursive_block_cost_assm
913            @plus_right_monotone
914            whd in ⊢ (???%); <FETCH %
915          |1:
916            cases exists_assm #exists_n #trace_total_program_size_assm
917            generalize in match execute_assm; destruct #execute_assm
918            %{(exists_n - ((nat_of_bitvector … (program_counter … status_after_fun_call)
919                - nat_of_bitvector … (program_counter … status_pre_fun_call))))}
920            >commutative_plus in ⊢ (??(?%?)?);
921            <plus_n_Sm <plus_n_Sm @eq_f >associative_plus in ⊢ (??%?);
922            @plus_left_monotone @sym_eq <commutative_plus in ⊢ (???%);
923            @plus_minus_m_m
924            cases daemon (* XXX: !!! *)
925          ]
926        |2,4:
927          %
928        ]
929      ]
930    |2:
931      #cost_label #Some_lookup_opt_assm normalize nodelta #absurd
932      cases absurd #absurd cases (absurd I)
933    ]
934  ]
935  try (change with (ASM_classify0 ? = ? ∨ ASM_classify0 ? = ?) in classifier_assm;)
936  try (change with (ASM_classify0 ? = ?) in classifier_assm;)
937  whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
938  whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
939  <FETCH in classifier_assm; >classify_assm #absurd destruct(absurd) cases absurd
940  #absurd destruct(absurd)
941qed.
942
943lemma trace_compute_paid_trace_cl_return:
944  ∀code_memory' : (BitVectorTrie Byte 16).
945  ∀program_counter' : Word.
946  ∀total_program_size : ℕ.
947  ∀cost_labels : (BitVectorTrie costlabel 16).
948  ∀reachable_program_counter_witness : (reachable_program_counter code_memory' total_program_size program_counter').
949  ∀good_program_witness : (good_program code_memory' total_program_size).
950  ∀program_size' : ℕ.
951  ∀recursive_case : (total_program_size≤S program_size'+nat_of_bitvector 16 program_counter').
952  ∀ticks : ℕ.
953  ∀instruction : instruction.
954  ∀program_counter'' : Word.
955  ∀FETCH : (〈instruction,program_counter'',ticks〉=fetch code_memory' program_counter').
956  ∀start_status : (Status code_memory').
957  ∀final_status : (Status code_memory').
958  ∀trace_ends_flag : trace_ends_with_ret.
959  ∀the_trace : (trace_any_label (ASM_abstract_status code_memory' cost_labels) trace_ends_flag start_status final_status).
960  ∀program_counter_refl : (program_counter' = program_counter (BitVectorTrie Byte 16) code_memory' start_status).
961  ∀classify_assm: ASM_classify0 instruction = cl_return.
962    (∃n:ℕ
963     .trace_any_label_length code_memory' cost_labels trace_ends_flag
964      start_status final_status the_trace
965      +S n
966      =total_program_size)
967    ∧ticks
968     =compute_paid_trace_any_label code_memory' cost_labels trace_ends_flag
969      start_status final_status the_trace.
970  #code_memory' #program_counter' #total_program_size #cost_labels
971  #reachable_program_counter_witness #good_program_witness #program_size'
972  #recursive_case #ticks #instruction #program_counter'' #FETCH
973  #start_status #final_status #trace_ends_flag
974  #the_trace #program_counter_refl #classify_assm
975  @(trace_any_label_inv_ind … the_trace)
976  [1:
977    #start_status' #final_status' #execute_assm #classifier_assm #costed_assm
978    #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
979    destruct @⊥
980  |2:
981    #start_status' #final_status' #execute_assm #classifier_assm #trace_ends_flag_refl
982    #start_status_refl #final_status_refl #the_trace_refl destruct
983    whd in match (trace_any_label_length … (tal_base_return …));
984    whd in match (compute_paid_trace_any_label … (tal_base_return …)); %
985    [1:
986      %{(pred total_program_size)} whd in ⊢ (??%?);
987      @S_pred
988      @(reachable_program_counter_to_0_lt_total_program_size … reachable_program_counter_witness)
989    |2:
990      <FETCH %
991    ]
992  |3:
993    #status_pre_fun_call #status_start_fun_call #status_final #execute_assm
994    #classifier_assm #after_return_assm #trace_label_return #costed_assm
995    #trace_ends_flag_refl #start_status_refl #final_status_refl #the_trace_refl
996    destruct @⊥
997  |4:
998    #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call
999    #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return
1000    #costed_assm #trace_any_label #trace_ends_flag_refl #start_status_refl
1001    #final_status_refl #the_trace_refl
1002    destruct @⊥
1003  |5:
1004    #end_flag #status_pre #status_init #status_end #execute_assm #trace_any_label
1005    #classifier_assm #costed_assm #trace_ends_flag_refl #start_status_refl
1006    #final_status_refl #the_trace_refl destruct @⊥
1007  ]
1008  try (change with (ASM_classify0 ? = ? ∨ ASM_classify0 ? = ?) in classifier_assm;)
1009  try (change with (ASM_classify0 ? = ?) in classifier_assm;)
1010  whd in match current_instruction in classifier_assm; normalize nodelta in classifier_assm;
1011  whd in match current_instruction0 in classifier_assm; normalize nodelta in classifier_assm;
1012  <FETCH in classifier_assm; >classify_assm
1013  #absurd try (destruct(absurd))
1014  cases absurd
1015  #absurd destruct(absurd)
1016qed.
1017     
1018let rec block_cost'
1019  (code_memory': BitVectorTrie Byte 16) (program_counter': Word)
1020    (program_size: nat) (total_program_size: nat) (cost_labels: BitVectorTrie costlabel 16)
1021      (reachable_program_counter_witness: reachable_program_counter code_memory' total_program_size program_counter')
1022        (good_program_witness: good_program code_memory' total_program_size) (first_time_around: bool)
1023          on program_size:
1024          total_program_size ≤ program_size + nat_of_bitvector … program_counter' →
1025          Σcost_of_block: nat.
1026          if (match lookup_opt … program_counter' cost_labels with [ None ⇒ true | _ ⇒ first_time_around ]) then
1027            ∀start_status: Status code_memory'.
1028            ∀final_status: Status code_memory'.
1029            ∀trace_ends_flag.
1030            ∀the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
1031              program_counter' = program_counter … start_status →
1032                (∃n: nat. trace_any_label_length … the_trace + (S n) = total_program_size) ∧
1033                  cost_of_block = compute_paid_trace_any_label code_memory' cost_labels trace_ends_flag start_status final_status the_trace
1034          else
1035            (cost_of_block = 0) ≝
1036  match program_size return λprogram_size: nat. total_program_size ≤ program_size + nat_of_bitvector … program_counter' → ? with
1037  [ O ⇒ λbase_case. ⊥
1038  | S program_size' ⇒ λrecursive_case.
1039    let 〈instruction, program_counter'', ticks〉 as FETCH ≝ fetch code_memory' program_counter' in
1040    let to_continue ≝
1041      match lookup_opt … program_counter' cost_labels with
1042      [ None ⇒ true
1043      | Some _ ⇒ first_time_around
1044      ]
1045    in
1046      ((if to_continue then
1047       pi1 … (match instruction return λx. x = instruction → ? with
1048        [ RealInstruction real_instruction ⇒ λreal_instruction_refl.
1049          match real_instruction return λx. x = real_instruction →
1050          Σcost_of_block: nat.
1051            ∀start_status: Status code_memory'.
1052            ∀final_status: Status code_memory'.
1053            ∀trace_ends_flag.
1054            ∀the_trace: trace_any_label (ASM_abstract_status code_memory' cost_labels) trace_ends_flag start_status final_status.
1055              program_counter' = program_counter … start_status →
1056                (∃n: nat. trace_any_label_length … the_trace + (S n) = total_program_size) ∧
1057                  cost_of_block = compute_paid_trace_any_label code_memory' cost_labels trace_ends_flag start_status final_status the_trace with
1058          [ RET                    ⇒ λinstr. ticks
1059          | RETI                   ⇒ λinstr. ticks
1060          | JC   relative          ⇒ λinstr. ticks
1061          | JNC  relative          ⇒ λinstr. ticks
1062          | JB   bit_addr relative ⇒ λinstr. ticks
1063          | JNB  bit_addr relative ⇒ λinstr. ticks
1064          | JBC  bit_addr relative ⇒ λinstr. ticks
1065          | JZ   relative          ⇒ λinstr. ticks
1066          | JNZ  relative          ⇒ λinstr. ticks
1067          | CJNE src_trgt relative ⇒ λinstr. ticks
1068          | DJNZ src_trgt relative ⇒ λinstr. ticks
1069          | _                      ⇒ λinstr.
1070         
1071              ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
1072          ] (refl …)
1073        | ACALL addr     ⇒ λinstr.
1074            ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
1075        | AJMP  addr     ⇒ λinstr. ticks
1076        | LCALL addr     ⇒ λinstr.
1077            ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
1078        | LJMP  addr     ⇒ λinstr. ticks
1079        | SJMP  addr     ⇒ λinstr. ticks
1080        | JMP   addr     ⇒ λinstr. (* XXX: actually a call due to use with fptrs *)
1081            ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
1082        | MOVC  src trgt ⇒ λinstr.
1083            ticks + block_cost' code_memory' program_counter'' program_size' total_program_size cost_labels ? good_program_witness false ?
1084        ] (refl …))
1085      else
1086        0)
1087      : Σcost_of_block: nat.
1088          match (match lookup_opt … program_counter' cost_labels with [ None ⇒ true | _ ⇒ first_time_around ]) with
1089          [ true ⇒
1090            ∀start_status: Status code_memory'.
1091            ∀final_status: Status code_memory'.
1092            ∀trace_ends_flag.
1093            ∀the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
1094              program_counter' = program_counter … start_status →
1095                (∃n: nat. trace_any_label_length … the_trace + (S n) = total_program_size) ∧
1096                  cost_of_block = compute_paid_trace_any_label code_memory' cost_labels trace_ends_flag start_status final_status the_trace
1097          | false ⇒
1098            (cost_of_block = 0)
1099          ])
1100  ].
1101  [1:
1102    cases reachable_program_counter_witness #_ #hyp
1103    @(absurd (total_program_size < total_program_size) … (not_le_Sn_n …))
1104    @(le_to_lt_to_lt … base_case hyp)
1105  |2:
1106    change with (if to_continue then ? else (? = 0))
1107    >p in ⊢ (match % return ? with [ _ ⇒ ? | _ ⇒ ? ]); normalize nodelta
1108    @pi2
1109  |3:
1110    change with (if to_continue then ? else (0 = 0))
1111    >p normalize nodelta %
1112  |7,8:
1113    #start_status #final_status #trace_ends_flag #the_trace #program_counter_refl
1114    @(trace_compute_paid_trace_cl_return … reachable_program_counter_witness good_program_witness … recursive_case … FETCH … the_trace program_counter_refl)
1115    destruct %
1116  |96,102,105:
1117    #start_status #final_status #trace_ends_flag #the_trace #program_counter_refl
1118    cases(block_cost' ?????????) -block_cost'
1119    @(trace_compute_paid_trace_cl_call … reachable_program_counter_witness good_program_witness … recursive_case … FETCH … program_counter_refl)
1120    destruct %
1121  |4,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,69,72,75,78,81,84,87,
1122   90,93:
1123    #start_status #final_status #trace_ends_flag #the_trace #program_counter_refl
1124    cases(block_cost' ?????????) -block_cost'
1125    @(trace_compute_paid_trace_cl_other … reachable_program_counter_witness good_program_witness … recursive_case … FETCH … the_trace program_counter_refl)
1126    destruct %
1127  |60,61,62,63,64,65,66,67,68,69,99,101,100,102:
1128    #start_status #final_status #trace_ends_flag #the_trace #program_counter_refl
1129    @(trace_compute_paid_trace_cl_jump … reachable_program_counter_witness good_program_witness … recursive_case … FETCH … the_trace program_counter_refl)
1130    destruct %
1131  |103:
1132    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1133    lapply(good_program_witness program_counter' reachable_program_counter_witness)
1134    <FETCH normalize nodelta <instr normalize nodelta
1135    @(subaddressing_mode_elim … [[addr16]] … [[addr16]]) [1: // ] #new_addr
1136    * * * * #n'
1137    #_ #_ #program_counter_lt' #program_counter_lt_tps'
1138    %
1139    [1:
1140      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
1141      <FETCH normalize nodelta whd in match ltb; normalize nodelta
1142      >(le_to_leb_true … program_counter_lt') %
1143    |2:
1144      assumption
1145    ]
1146  |104:
1147    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1148    lapply(good_program_witness program_counter' reachable_program_counter_witness)
1149    <FETCH normalize nodelta <instr normalize nodelta
1150    @(subaddressing_mode_elim … [[addr16]] … [[addr16]]) [1: // ] #new_addr
1151    * * * * #n'
1152    #_ #_ #program_counter_lt' #program_counter_lt_tps'
1153    @(transitive_le
1154      total_program_size
1155      ((S program_size') + nat_of_bitvector … program_counter')
1156      (program_size' + nat_of_bitvector … program_counter'') recursive_case)
1157    normalize in match (S program_size' + nat_of_bitvector … program_counter');
1158    >plus_n_Sm
1159    @monotonic_le_plus_r
1160    change with (
1161      nat_of_bitvector … program_counter' <
1162        nat_of_bitvector … program_counter'')
1163    assumption
1164  |106:
1165    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1166    lapply(good_program_witness program_counter' reachable_program_counter_witness)
1167    <FETCH normalize nodelta <instr normalize nodelta
1168    @(subaddressing_mode_elim … [[addr11]] … [[addr11]]) [1: // ] #new_addr
1169    cases (split … 8 8 program_counter'') #pc_bu #pc_bl normalize nodelta
1170    cases (split … 3 8 new_addr) #thr #eig normalize nodelta
1171    cases (split … 5 3 pc_bu) #fiv #thr' normalize nodelta * * * * #n'
1172    #_ #_ #program_counter_lt' #program_counter_lt_tps'
1173    %
1174    [1:
1175      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
1176      <FETCH normalize nodelta whd in match ltb; normalize nodelta
1177      >(le_to_leb_true … program_counter_lt') %
1178    |2:
1179      assumption
1180    ]
1181  |107:
1182    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1183    lapply(good_program_witness program_counter' reachable_program_counter_witness)
1184    <FETCH normalize nodelta <instr normalize nodelta
1185    @(subaddressing_mode_elim … [[addr11]] … [[addr11]]) [1: // ] #new_addr
1186    cases (split … 8 8 program_counter'') #pc_bu #pc_bl normalize nodelta
1187    cases (split … 3 8 new_addr) #thr #eig normalize nodelta
1188    cases (split … 5 3 pc_bu) #fiv #thr' normalize nodelta * * * * #n'
1189    #_ #_ #program_counter_lt' #program_counter_lt_tps'
1190    @(transitive_le
1191      total_program_size
1192      ((S program_size') + nat_of_bitvector … program_counter')
1193      (program_size' + nat_of_bitvector … program_counter'') recursive_case)
1194    normalize in match (S program_size' + nat_of_bitvector … program_counter');
1195    >plus_n_Sm
1196    @monotonic_le_plus_r
1197    change with (
1198      nat_of_bitvector … program_counter' <
1199        nat_of_bitvector … program_counter'')
1200    assumption
1201  |94,97:
1202    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1203    lapply(good_program_witness program_counter' reachable_program_counter_witness)
1204    <FETCH normalize nodelta <instr normalize nodelta *
1205    #program_counter_lt' #program_counter_lt_tps' %
1206    [1,3:
1207      %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
1208      <FETCH normalize nodelta whd in match ltb; normalize nodelta
1209      >(le_to_leb_true … program_counter_lt') %
1210    |2,4:
1211      assumption
1212    ]
1213  |5,10,12,13,16,18,19,22,25,28,31,34,37,40,43,46,49,52,55,58,70,73,
1214   76,79,82,85,88,91:
1215    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1216    lapply(good_program_witness program_counter' reachable_program_counter_witness)
1217    <FETCH normalize nodelta <real_instruction_refl <instr normalize nodelta *
1218    #program_counter_lt' #program_counter_lt_tps' %
1219    try assumption
1220    %{(S n)} whd in ⊢ (???%); <fetch_n_hyp normalize nodelta
1221    <FETCH normalize nodelta whd in match ltb; normalize nodelta
1222    >(le_to_leb_true … program_counter_lt') %
1223  |6,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,71,74,77,80,83,86,89,
1224   92:
1225    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1226    lapply(good_program_witness program_counter' reachable_program_counter_witness)
1227    <FETCH normalize nodelta
1228    <real_instruction_refl <instr normalize nodelta *
1229    #pc_pc_lt_hyp' #pc_tps_lt_hyp'
1230    @(transitive_le
1231      total_program_size
1232      ((S program_size') + nat_of_bitvector … program_counter')
1233      (program_size' + nat_of_bitvector … program_counter'') recursive_case)
1234    normalize in match (S program_size' + nat_of_bitvector … program_counter');
1235    >plus_n_Sm
1236    @monotonic_le_plus_r
1237    change with (
1238      nat_of_bitvector … program_counter' <
1239        nat_of_bitvector … program_counter'')
1240    assumption
1241  |95,98:
1242    cases reachable_program_counter_witness * #n #fetch_n_hyp #lt_hyp
1243    lapply(good_program_witness program_counter' reachable_program_counter_witness)
1244    <FETCH normalize nodelta <instr normalize nodelta
1245    try(<real_instruction_refl <instr normalize nodelta) *
1246    #pc_pc_lt_hyp' #pc_tps_lt_hyp'
1247    @(transitive_le
1248      total_program_size
1249      ((S program_size') + nat_of_bitvector … program_counter')
1250      (program_size' + nat_of_bitvector … program_counter'') recursive_case)
1251    normalize in match (S program_size' + nat_of_bitvector … program_counter');
1252    >plus_n_Sm
1253    @monotonic_le_plus_r
1254    change with (
1255      nat_of_bitvector … program_counter' <
1256        nat_of_bitvector … program_counter'')
1257    assumption
1258  ]
1259qed.
1260
1261definition block_cost:
1262    ∀code_memory': BitVectorTrie Byte 16.
1263    ∀program_counter': Word.
1264    ∀total_program_size: nat.
1265    ∀cost_labels: BitVectorTrie costlabel 16.
1266    ∀reachable_program_counter_witness: reachable_program_counter code_memory' total_program_size program_counter'.
1267    ∀good_program_witness: good_program code_memory' total_program_size.
1268      Σcost_of_block: nat.
1269        ∀start_status: Status code_memory'.
1270        ∀final_status: Status code_memory'.
1271        ∀trace_ends_flag.
1272        ∀the_trace: trace_any_label (ASM_abstract_status … cost_labels) trace_ends_flag start_status final_status.
1273          program_counter' = program_counter … start_status →
1274            (∃n: nat. trace_any_label_length … the_trace + (S n) = total_program_size) ∧
1275              cost_of_block = compute_paid_trace_any_label code_memory' cost_labels trace_ends_flag start_status final_status the_trace ≝
1276  λcode_memory: BitVectorTrie Byte 16.
1277  λprogram_counter: Word.
1278  λtotal_program_size: nat.
1279  λcost_labels: BitVectorTrie costlabel 16.
1280  λreachable_program_counter_witness: reachable_program_counter code_memory total_program_size program_counter.
1281  λgood_program_witness: good_program code_memory total_program_size. ?.
1282  cases(block_cost' code_memory program_counter total_program_size total_program_size cost_labels
1283    reachable_program_counter_witness good_program_witness true ?)
1284  [1:
1285    #cost_of_block #block_cost_hyp
1286    %{cost_of_block}
1287    cases(lookup_opt … cost_labels) in block_cost_hyp;
1288    [2: #cost_label] normalize nodelta
1289    #hyp assumption
1290  |2:
1291    @le_plus_n_r
1292  ]
1293qed.
1294
1295lemma fetch_program_counter_n_Sn:
1296  ∀instruction: instruction.
1297  ∀program_counter, program_counter': Word.
1298  ∀ticks, n: nat.
1299  ∀code_memory: BitVectorTrie Byte 16.
1300    Some … program_counter = fetch_program_counter_n n code_memory (zero 16) →
1301      〈instruction,program_counter',ticks〉 = fetch code_memory program_counter →
1302        nat_of_bitvector … program_counter < nat_of_bitvector … program_counter' →
1303          Some … program_counter' = fetch_program_counter_n (S n) code_memory (zero …).
1304  #instruction #program_counter #program_counter' #ticks #n #code_memory
1305  #fetch_program_counter_n_hyp #fetch_hyp #lt_hyp
1306  whd in match (fetch_program_counter_n (S n) code_memory (zero …));
1307  <fetch_program_counter_n_hyp normalize nodelta
1308  <fetch_hyp normalize nodelta
1309  change with (
1310    leb (S (nat_of_bitvector … program_counter)) (nat_of_bitvector … program_counter')
1311  ) in match (ltb (nat_of_bitvector … program_counter) (nat_of_bitvector … program_counter'));
1312  >(le_to_leb_true … lt_hyp) %
1313qed.
1314
1315(* XXX: to be moved into common/Identifiers.ma *)
1316lemma lookup_present_add_hit:
1317  ∀tag, A, map, k, v, k_pres.
1318    lookup_present tag A (add … map k v) k k_pres = v.
1319  #tag #a #map #k #v #k_pres
1320  lapply (lookup_lookup_present … (add … map k v) … k_pres)
1321  >lookup_add_hit #Some_assm destruct(Some_assm)
1322  <e0 %
1323qed.
1324
1325lemma lookup_present_add_miss:
1326  ∀tag, A, map, k, k', v, k_pres', k_pres''.
1327    k' ≠ k →
1328      lookup_present tag A (add … map k v) k' k_pres' = lookup_present tag A map k' k_pres''.
1329  #tag #A #map #k #k' #v #k_pres' #k_pres'' #neq_assm
1330  lapply (lookup_lookup_present … (add … map k v) ? k_pres')
1331  >lookup_add_miss try assumption
1332  #Some_assm
1333  lapply (lookup_lookup_present … map k') >Some_assm #Some_assm'
1334  lapply (Some_assm' k_pres'') #Some_assm'' destruct assumption
1335qed.
1336
1337(* XXX: to be moved into basics/types.ma *)
1338lemma not_None_to_Some:
1339  ∀A: Type[0].
1340  ∀o: option A.
1341    o ≠ None A → ∃v: A. o = Some A v.
1342  #A #o cases o
1343  [1:
1344    #absurd cases absurd #absurd' cases (absurd' (refl …))
1345  |2:
1346    #v' #ignore /2/
1347  ]
1348qed.
1349
1350lemma present_add_present:
1351  ∀tag, a, map, k, k', v.
1352    k' ≠ k →
1353      present tag a (add tag a map k v) k' →
1354        present tag a map k'.
1355  #tag #a #map #k #k' #v #neq_hyp #present_hyp
1356  whd in match present; normalize nodelta
1357  whd in match present in present_hyp; normalize nodelta in present_hyp;
1358  cases (not_None_to_Some a … present_hyp) #v' #Some_eq_hyp
1359  lapply (lookup_add_cases tag ?????? Some_eq_hyp) *
1360  [1:
1361    * #k_eq_hyp @⊥ /2/
1362  |2:
1363    #Some_eq_hyp' /2/
1364  ]
1365qed.
1366
1367lemma present_add_hit:
1368  ∀tag, a, map, k, v.
1369    present tag a (add tag a map k v) k.
1370  #tag #a #map #k #v
1371  whd >lookup_add_hit
1372  % #absurd destruct
1373qed.
1374
1375lemma present_add_miss:
1376  ∀tag, a, map, k, k', v.
1377    k' ≠ k → present tag a map k' → present tag a (add tag a map k v) k'.
1378  #tag #a #map #k #k' #v #neq_assm #present_assm
1379  whd >lookup_add_miss assumption
1380qed.
1381
1382lemma lt_to_le_to_le:
1383  ∀n, m, p: nat.
1384    n < m → m ≤ p → n ≤ p.
1385  #n #m #p #H #H1
1386  elim H
1387  [1:
1388    @(transitive_le n m p) /2/
1389  |2:
1390    /2/
1391  ]
1392qed.
1393
1394lemma eqb_decidable:
1395  ∀l, r: nat.
1396    (eqb l r = true) ∨ (eqb l r = false).
1397  #l #r //
1398qed.
1399
1400lemma r_Sr_and_l_r_to_Sl_r:
1401  ∀r, l: nat.
1402    (∃r': nat. r = S r' ∧ l = r') → S l = r.
1403  #r #l #exists_hyp
1404  cases exists_hyp #r'
1405  #and_hyp cases and_hyp
1406  #left_hyp #right_hyp
1407  destruct %
1408qed.
1409
1410lemma eqb_Sn_to_exists_n':
1411  ∀m, n: nat.
1412    eqb (S m) n = true → ∃n': nat. n = S n'.
1413  #m #n
1414  cases n
1415  [1:
1416    normalize #absurd
1417    destruct(absurd)
1418  |2:
1419    #n' #_ %{n'} %
1420  ]
1421qed.
1422
1423lemma eqb_true_to_eqb_S_S_true:
1424  ∀m, n: nat.
1425    eqb m n = true → eqb (S m) (S n) = true.
1426  #m #n normalize #assm assumption
1427qed.
1428
1429lemma eqb_S_S_true_to_eqb_true:
1430  ∀m, n: nat.
1431    eqb (S m) (S n) = true → eqb m n = true.
1432  #m #n normalize #assm assumption
1433qed.
1434
1435lemma eqb_true_to_refl:
1436  ∀l, r: nat.
1437    eqb l r = true → l = r.
1438  #l
1439  elim l
1440  [1:
1441    #r cases r
1442    [1:
1443      #_ %
1444    |2:
1445      #l' normalize
1446      #absurd destruct(absurd)
1447    ]
1448  |2:
1449    #l' #inductive_hypothesis #r
1450    #eqb_refl @r_Sr_and_l_r_to_Sl_r
1451    %{(pred r)} @conj
1452    [1:
1453      cases (eqb_Sn_to_exists_n' … eqb_refl)
1454      #r' #S_assm >S_assm %
1455    |2:
1456      cases (eqb_Sn_to_exists_n' … eqb_refl)
1457      #r' #refl_assm destruct normalize
1458      @inductive_hypothesis
1459      normalize in eqb_refl; assumption
1460    ]
1461  ]
1462qed.
1463
1464lemma r_O_or_exists_r_r_Sr_and_l_neq_r_to_Sl_neq_r:
1465  ∀r, l: nat.
1466    r = O ∨ (∃r': nat. r = S r' ∧ l ≠ r') → S l ≠ r.
1467  #r #l #disj_hyp
1468  cases disj_hyp
1469  [1:
1470    #r_O_refl destruct @nmk
1471    #absurd destruct(absurd)
1472  |2:
1473    #exists_hyp cases exists_hyp #r'
1474    #conj_hyp cases conj_hyp #left_conj #right_conj
1475    destruct @nmk #S_S_refl_hyp
1476    elim right_conj #hyp @hyp //
1477  ]
1478qed.
1479
1480lemma neq_l_r_to_neq_Sl_Sr:
1481  ∀l, r: nat.
1482    l ≠ r → S l ≠ S r.
1483  #l #r #l_neq_r_assm
1484  @nmk #Sl_Sr_assm cases l_neq_r_assm
1485  #assm @assm //
1486qed.
1487
1488lemma eqb_false_to_not_refl:
1489  ∀l, r: nat.
1490    eqb l r = false → l ≠ r.
1491  #l
1492  elim l
1493  [1:
1494    #r cases r
1495    [1:
1496      normalize #absurd destruct(absurd)
1497    |2:
1498      #r' #_ @nmk
1499      #absurd destruct(absurd)
1500    ]
1501  |2:
1502    #l' #inductive_hypothesis #r
1503    cases r
1504    [1:
1505      #eqb_false_assm
1506      @r_O_or_exists_r_r_Sr_and_l_neq_r_to_Sl_neq_r
1507      @or_introl %
1508    |2:
1509      #r' #eqb_false_assm
1510      @neq_l_r_to_neq_Sl_Sr
1511      @inductive_hypothesis
1512      assumption
1513    ]
1514  ]
1515qed.
1516
1517lemma le_to_lt_or_eq:
1518  ∀m, n: nat.
1519    m ≤ n → m = n ∨ m < n.
1520  #m #n #le_hyp
1521  cases le_hyp
1522  [1:
1523    @or_introl %
1524  |2:
1525    #m' #le_hyp'
1526    @or_intror
1527    normalize
1528    @le_S_S assumption
1529  ]
1530qed.
1531
1532lemma le_neq_to_lt:
1533  ∀m, n: nat.
1534    m ≤ n → m ≠ n → m < n.
1535  #m #n #le_hyp #neq_hyp
1536  cases neq_hyp
1537  #eq_absurd_hyp
1538  generalize in match (le_to_lt_or_eq m n le_hyp);
1539  #disj_assm cases disj_assm
1540  [1:
1541    #absurd cases (eq_absurd_hyp absurd)
1542  |2:
1543    #assm assumption
1544  ]
1545qed.
1546
1547inverter nat_jmdiscr for nat.
1548
1549(* XXX: this is false in the general case.  For instance, if n = 0 then the
1550        base case requires us prove 1 = 0, as it is the carry bit that holds
1551        the result of the addition. *)
1552axiom succ_nat_of_bitvector_half_add_1:
1553  ∀n: nat.
1554  ∀bv: BitVector n.
1555  ∀power_proof: nat_of_bitvector … bv < 2^n - 1.
1556    S (nat_of_bitvector … bv) = nat_of_bitvector …
1557      (\snd (half_add n (bitvector_of_nat … 1) bv)).
1558
1559lemma plus_lt_to_lt:
1560  ∀m, n, o: nat.
1561    m + n < o → m < o.
1562  #m #n #o
1563  elim n
1564  [1:
1565    <(plus_n_O m) in ⊢ (% → ?);
1566    #assumption assumption
1567  |2:
1568    #n' #inductive_hypothesis
1569    <(plus_n_Sm m n') in ⊢ (% → ?);
1570    #assm @inductive_hypothesis
1571    normalize in assm; normalize
1572    /2 by lt_S_to_lt/
1573  ]
1574qed.
1575
1576include "arithmetics/div_and_mod.ma".
1577
1578lemma n_plus_1_n_to_False:
1579  ∀n: nat.
1580    n + 1 = n → False.
1581  #n elim n
1582  [1:
1583    normalize #absurd destruct(absurd)
1584  |2:
1585    #n' #inductive_hypothesis normalize
1586    #absurd @inductive_hypothesis /2/
1587  ]
1588qed.
1589
1590lemma one_two_times_n_to_False:
1591  ∀n: nat.
1592    1=2*n→False.
1593  #n cases n
1594  [1:
1595    normalize #absurd destruct(absurd)
1596  |2:
1597    #n' normalize #absurd
1598    lapply (injective_S … absurd) -absurd #absurd
1599    /2/
1600  ]
1601qed.
1602
1603lemma generalized_nat_cases:
1604  ∀n: nat.
1605    n = 0 ∨ n = 1 ∨ ∃m: nat. n = S (S m).
1606  #n
1607  cases n
1608  [1:
1609    @or_introl @or_introl %
1610  |2:
1611    #n' cases n'
1612    [1:
1613      @or_introl @or_intror %
1614    |2:
1615      #n'' @or_intror %{n''} %
1616    ]
1617  ]
1618qed.
1619
1620let rec odd_p
1621  (n: nat)
1622    on n ≝
1623  match n with
1624  [ O ⇒ False
1625  | S n' ⇒ even_p n'
1626  ]
1627and even_p
1628  (n: nat)
1629    on n ≝
1630  match n with
1631  [ O ⇒ True
1632  | S n' ⇒ odd_p n'
1633  ].
1634
1635let rec n_even_p_to_n_plus_2_even_p
1636  (n: nat)
1637    on n: even_p n → even_p (n + 2) ≝
1638  match n with
1639  [ O ⇒ ?
1640  | S n' ⇒ ?
1641  ]
1642and n_odd_p_to_n_plus_2_odd_p
1643  (n: nat)
1644    on n: odd_p n → odd_p (n + 2) ≝
1645  match n with
1646  [ O ⇒ ?
1647  | S n' ⇒ ?
1648  ].
1649  [1,3:
1650    normalize #assm assumption
1651  |2:
1652    normalize @n_odd_p_to_n_plus_2_odd_p
1653  |4:
1654    normalize @n_even_p_to_n_plus_2_even_p
1655  ]
1656qed.
1657
1658let rec two_times_n_even_p
1659  (n: nat)
1660    on n: even_p (2 * n) ≝
1661  match n with
1662  [ O ⇒ ?
1663  | S n' ⇒ ?
1664  ]
1665and two_times_n_plus_one_odd_p
1666  (n: nat)
1667    on n: odd_p ((2 * n) + 1) ≝
1668  match n with
1669  [ O ⇒ ?
1670  | S n' ⇒ ?
1671  ].
1672  [1,3:
1673    normalize @I
1674  |2:
1675    normalize
1676    >plus_n_Sm
1677    <(associative_plus n' n' 1)
1678    >(plus_n_O (n' + n'))
1679    cut(n' + n' + 0 + 1 = 2 * n' + 1)
1680    [1:
1681      //
1682    |2:
1683      #refl_assm >refl_assm
1684      @two_times_n_plus_one_odd_p     
1685    ]
1686  |4:
1687    normalize
1688    >plus_n_Sm
1689    cut(n' + (n' + 1) + 1 = (2 * n') + 2)
1690    [1:
1691      normalize /2/
1692    |2:
1693      #refl_assm >refl_assm
1694      @n_even_p_to_n_plus_2_even_p
1695      @two_times_n_even_p
1696    ]
1697  ]
1698qed.
1699
1700let rec even_p_to_not_odd_p
1701  (n: nat)
1702    on n: even_p n → ¬ odd_p n ≝
1703  match n with
1704  [ O ⇒ ?
1705  | S n' ⇒ ?
1706  ]
1707and odd_p_to_not_even_p
1708  (n: nat)
1709    on n: odd_p n → ¬ even_p n ≝
1710  match n with
1711  [ O ⇒ ?
1712  | S n' ⇒ ?
1713  ].
1714  [1:
1715    normalize #_
1716    @nmk #assm assumption
1717  |3:
1718    normalize #absurd
1719    cases absurd
1720  |2:
1721    normalize
1722    @odd_p_to_not_even_p
1723  |4:
1724    normalize
1725    @even_p_to_not_odd_p
1726  ]
1727qed.
1728
1729lemma even_p_odd_p_cases:
1730  ∀n: nat.
1731    even_p n ∨ odd_p n.
1732  #n elim n
1733  [1:
1734    normalize @or_introl @I
1735  |2:
1736    #n' #inductive_hypothesis
1737    normalize
1738    cases inductive_hypothesis
1739    #assm
1740    try (@or_introl assumption)
1741    try (@or_intror assumption)
1742  ]
1743qed.
1744
1745lemma two_times_n_plus_one_refl_two_times_n_to_False:
1746  ∀m, n: nat.
1747    2 * m + 1 = 2 * n → False.
1748  #m #n
1749  #assm
1750  cut (even_p (2 * n) ∧ even_p ((2 * m) + 1))
1751  [1:
1752    >assm
1753    @conj
1754    @two_times_n_even_p
1755  |2:
1756    * #_ #absurd
1757    cases (even_p_to_not_odd_p … absurd)
1758    #assm @assm
1759    @two_times_n_plus_one_odd_p
1760  ]
1761qed.
1762
1763lemma nat_of_bitvector_aux_injective:
1764  ∀n: nat.
1765  ∀l, r: BitVector n.
1766  ∀acc_l, acc_r: nat.
1767    nat_of_bitvector_aux n acc_l l = nat_of_bitvector_aux n acc_r r →
1768      acc_l = acc_r ∧ l ≃ r.
1769  #n #l
1770  elim l #r
1771  [1:
1772    #acc_l #acc_r normalize
1773    >(BitVector_O r) normalize /2/
1774  |2:
1775    #hd #tl #inductive_hypothesis #r #acc_l #acc_r
1776    normalize normalize in inductive_hypothesis;
1777    cases (BitVector_Sn … r)
1778    #r_hd * #r_tl #r_refl destruct normalize
1779    cases hd cases r_hd normalize
1780    [1:
1781      #relevant
1782      cases (inductive_hypothesis … relevant)
1783      #acc_assm #tl_assm destruct % //
1784      lapply (injective_plus_l ? ? ? acc_assm)
1785      -acc_assm #acc_assm
1786      change with (2 * acc_l = 2 * acc_r) in acc_assm;
1787      lapply (injective_times_r ? ? ? ? acc_assm) /2/
1788    |4:
1789      #relevant
1790      cases (inductive_hypothesis … relevant)
1791      #acc_assm #tl_assm destruct % //
1792      change with (2 * acc_l = 2 * acc_r) in acc_assm;
1793      lapply(injective_times_r ? ? ? ? acc_assm) /2/
1794    |2:
1795      #relevant 
1796      change with ((nat_of_bitvector_aux r (2 * acc_l + 1) tl) =
1797        (nat_of_bitvector_aux r (2 * acc_r) r_tl)) in relevant;
1798      cases (eqb_decidable … (2 * acc_l + 1) (2 * acc_r))
1799      [1:
1800        #eqb_true_assm
1801        lapply (eqb_true_to_refl … eqb_true_assm)
1802        #refl_assm
1803        cases (two_times_n_plus_one_refl_two_times_n_to_False … refl_assm)
1804      |2:
1805        #eqb_false_assm
1806        lapply (eqb_false_to_not_refl … eqb_false_assm)
1807        #not_refl_assm cases not_refl_assm #absurd_assm
1808        cases (inductive_hypothesis … relevant) #absurd
1809        cases (absurd_assm absurd)
1810      ]
1811    |3:
1812      #relevant 
1813      change with ((nat_of_bitvector_aux r (2 * acc_l) tl) =
1814        (nat_of_bitvector_aux r (2 * acc_r + 1) r_tl)) in relevant;
1815      cases (eqb_decidable … (2 * acc_l) (2 * acc_r + 1))
1816      [1:
1817        #eqb_true_assm
1818        lapply (eqb_true_to_refl … eqb_true_assm)
1819        #refl_assm
1820        lapply (sym_eq ? (2 * acc_l) (2 * acc_r + 1) refl_assm)
1821        -refl_assm #refl_assm
1822        cases (two_times_n_plus_one_refl_two_times_n_to_False … refl_assm)
1823      |2:
1824        #eqb_false_assm
1825        lapply (eqb_false_to_not_refl … eqb_false_assm)
1826        #not_refl_assm cases not_refl_assm #absurd_assm
1827        cases (inductive_hypothesis … relevant) #absurd
1828        cases (absurd_assm absurd)
1829      ]
1830    ]
1831  ]
1832qed.
1833
1834lemma nat_of_bitvector_destruct:
1835  ∀n: nat.
1836  ∀l_hd, r_hd: bool.
1837  ∀l_tl, r_tl: BitVector n.
1838    nat_of_bitvector (S n) (l_hd:::l_tl) = nat_of_bitvector (S n) (r_hd:::r_tl) →
1839      l_hd = r_hd ∧ nat_of_bitvector n l_tl = nat_of_bitvector n r_tl.
1840  #n #l_hd #r_hd #l_tl #r_tl
1841  normalize
1842  cases l_hd cases r_hd
1843  normalize
1844  [4:
1845    /2/
1846  |1:
1847    #relevant
1848    cases (nat_of_bitvector_aux_injective … relevant)
1849    #_ #l_r_tl_refl destruct /2/
1850  |2,3:
1851    #relevant
1852    cases (nat_of_bitvector_aux_injective … relevant)
1853    #absurd destruct(absurd)
1854  ]
1855qed.
1856
1857lemma BitVector_cons_injective:
1858  ∀n: nat.
1859  ∀l_hd, r_hd: bool.
1860  ∀l_tl, r_tl: BitVector n.
1861    l_hd = r_hd → l_tl = r_tl → l_hd:::l_tl = r_hd:::r_tl.
1862  #l #l_hd #r_hd #l_tl #r_tl
1863  #l_refl #r_refl destruct %
1864qed.
1865
1866lemma refl_nat_of_bitvector_to_refl:
1867  ∀n: nat.
1868  ∀l, r: BitVector n.
1869    nat_of_bitvector n l = nat_of_bitvector n r → l = r.
1870  #n
1871  elim n
1872  [1:
1873    #l #r
1874    >(BitVector_O l)
1875    >(BitVector_O r)
1876    #_ %
1877  |2:
1878    #n' #inductive_hypothesis #l #r
1879    lapply (BitVector_Sn ? l) #l_hypothesis
1880    lapply (BitVector_Sn ? r) #r_hypothesis
1881    cases l_hypothesis #l_hd #l_tail_hypothesis
1882    cases r_hypothesis #r_hd #r_tail_hypothesis
1883    cases l_tail_hypothesis #l_tl #l_hd_tl_refl
1884    cases r_tail_hypothesis #r_tl #r_hd_tl_refl
1885    destruct #cons_refl
1886    cases (nat_of_bitvector_destruct n' l_hd r_hd l_tl r_tl cons_refl)
1887    #hd_refl #tl_refl
1888    @BitVector_cons_injective try assumption
1889    @inductive_hypothesis assumption
1890  ]
1891qed.
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