source: src/ASM/AssemblyProofSplit.ma @ 1967

Last change on this file since 1967 was 1967, checked in by sacerdot, 7 years ago

Mov case completed.

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1include "ASM/AssemblyProof.ma".
2include alias "arithmetics/nat.ma".
3include alias "basics/logic.ma".
4
5lemma set_arg_16_mk_Status_commutation:
6  ∀M: Type[0].
7  ∀cm: M.
8  ∀s: PreStatus M cm.
9  ∀w: Word.
10  ∀d: [[dptr]].
11    set_arg_16 M cm s w d =
12      mk_PreStatus M cm
13        (low_internal_ram … s)
14        (high_internal_ram … s)
15        (external_ram … s)
16        (program_counter … s)
17        (special_function_registers_8051 … (set_arg_16 M cm s w d))
18        (special_function_registers_8052 … s)
19        (p1_latch … s)
20        (p3_latch … s)
21        (clock … s).
22  #M #cm #s #w #d
23  whd in match set_arg_16; normalize nodelta
24  whd in match set_arg_16'; normalize nodelta
25  @(subaddressing_mode_elim … [[dptr]] … [[dptr]]) [1: // ]
26  cases (split bool 8 8 w) #bu #bl normalize nodelta
27  whd in match set_8051_sfr; normalize nodelta %
28qed.
29
30lemma set_program_counter_mk_Status_commutation:
31  ∀M: Type[0].
32  ∀cm: M.
33  ∀s: PreStatus M cm.
34  ∀w: Word.
35    set_program_counter M cm s w =
36      mk_PreStatus M cm
37        (low_internal_ram … s)
38        (high_internal_ram … s)
39        (external_ram … s)
40        w
41        (special_function_registers_8051 … s)
42        (special_function_registers_8052 … s)
43        (p1_latch … s)
44        (p3_latch … s)
45        (clock … s).
46  //
47qed.       
48
49lemma set_clock_mk_Status_commutation:
50  ∀M: Type[0].
51  ∀cm: M.
52  ∀s: PreStatus M cm.
53  ∀c: nat.
54    set_clock M cm s c =
55      mk_PreStatus M cm
56        (low_internal_ram … s)
57        (high_internal_ram … s)
58        (external_ram … s)
59        (program_counter … s)
60        (special_function_registers_8051 … s)
61        (special_function_registers_8052 … s)
62        (p1_latch … s)
63        (p3_latch … s)
64        c.
65  //
66qed.
67
68
69lemma special_function_registers_8051_set_arg_16:
70  ∀M, M': Type[0].
71  ∀cm: M.
72  ∀cm': M'.
73  ∀s: PreStatus M cm.
74  ∀s': PreStatus M' cm'.
75  ∀w, w': Word.
76  ∀d, d': [[dptr]].
77   special_function_registers_8051 ?? s = special_function_registers_8051 … s' →
78    w = w' →
79      special_function_registers_8051 ?? (set_arg_16 … s w d) =
80        special_function_registers_8051 ?? (set_arg_16 M' cm' s' w' d').
81  #M #M' #cm #cm' #s #s' #w #w'
82  @list_addressing_mode_tags_elim_prop try %
83  @list_addressing_mode_tags_elim_prop try %
84  #EQ1 #EQ2 <EQ2 normalize cases (split bool 8 8 w) #b1 #b2 normalize <EQ1 %
85qed.
86
87
88theorem main_thm:
89  ∀M.
90  ∀M'.
91  ∀program: pseudo_assembly_program.
92  ∀sigma: Word → Word.
93  ∀policy: Word → bool.
94  ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy.
95  ∀ps: PseudoStatus program.
96    next_internal_pseudo_address_map M program ps = Some … M' →
97      ∃n. execute n … (status_of_pseudo_status M … ps sigma policy) =
98        status_of_pseudo_status M' …
99          (execute_1_pseudo_instruction (ticks_of program sigma policy) program ps) sigma policy.
100  #M #M' * #preamble #instr_list #sigma #policy #sigma_policy_witness #ps
101  change with (next_internal_pseudo_address_map0 ????? = ? → ?)
102  @(pose … (program_counter ?? ps)) #ppc #EQppc
103  generalize in match (fetch_assembly_pseudo2 〈preamble,instr_list〉 sigma policy sigma_policy_witness ppc) in ⊢ ?;
104  @pair_elim #labels #costs #create_label_cost_refl normalize nodelta
105  @pair_elim #assembled #costs' #assembly_refl normalize nodelta
106  lapply (pair_destruct_1 ????? (sym_eq ??? assembly_refl)) #EQassembled
107  @pair_elim #pi #newppc #fetch_pseudo_refl normalize nodelta
108  @(pose … (λx. sigma (address_of_word_labels_code_mem instr_list x))) #lookup_labels #EQlookup_labels
109  @(pose … (λx. lookup_def … (construct_datalabels preamble) x (zero 16))) #lookup_datalabels #EQlookup_datalabels
110  whd in match execute_1_pseudo_instruction; normalize nodelta
111  whd in match ticks_of; normalize nodelta <EQppc >fetch_pseudo_refl normalize nodelta
112  lapply (snd_fetch_pseudo_instruction instr_list ppc) >fetch_pseudo_refl #EQnewppc >EQnewppc
113  lapply (snd_assembly_1_pseudoinstruction_ok 〈preamble,instr_list〉 … sigma policy sigma_policy_witness … ppc pi … EQlookup_labels EQlookup_datalabels ?)
114  [1: >fetch_pseudo_refl % ]
115  #assm1 #assm2 #assm3 generalize in match assm2; generalize in match assm3;
116  generalize in match assm1; -assm1 -assm2 -assm3
117  normalize nodelta
118  cases pi
119  [2,3:
120    #arg
121    (* XXX: we first work on sigma_increment_commutation *)
122    #sigma_increment_commutation
123    normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation;
124    (* XXX: we work on the maps *)
125    whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm
126    (* XXX: we assume the fetch_many hypothesis *)
127    #fetch_many_assm
128    (* XXX: we give the existential *)
129    %{0}
130    whd in match (execute_1_pseudo_instruction0 ?????);
131    (* XXX: work on the left hand side of the equality *)
132    whd in ⊢ (??%?);
133    @split_eq_status //
134    [1,2: /demod/ >EQnewppc >sigma_increment_commutation <add_zero % (*CSC: auto not working, why? *) ]
135  |6: (* Mov *)
136    #arg1 #arg2
137    (* XXX: we first work on sigma_increment_commutation *)
138    #sigma_increment_commutation
139    normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation;
140    (* XXX: we work on the maps *)
141    whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm
142    (* XXX: we assume the fetch_many hypothesis *)
143    #fetch_many_assm
144    (* XXX: we give the existential *)
145    %{1}
146    whd in match (execute_1_pseudo_instruction0 ?????);
147    (* XXX: work on the left hand side of the equality *)
148    whd in ⊢ (??%?); whd in match (program_counter ???); <EQppc
149    (* XXX: execute fetches some more *)
150    whd in match code_memory_of_pseudo_assembly_program; normalize nodelta
151    whd in fetch_many_assm;
152    >EQassembled in fetch_many_assm;
153    cases (fetch ??) * #instr #newpc #ticks normalize nodelta * *
154    #eq_instr #EQticks whd in EQticks:(???%); >EQticks
155    #fetch_many_assm whd in fetch_many_assm;
156    lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc
157    >(eq_instruction_to_eq … eq_instr) -instr whd in ⊢ (??%?);
158    (* XXX: now we start to work on the mk_PreStatus equality *)
159    change with (set_arg_16 ????? = ?)
160    >set_program_counter_mk_Status_commutation
161    >set_clock_mk_Status_commutation
162    >set_arg_16_mk_Status_commutation
163   
164    change with (? = status_of_pseudo_status ?? (set_arg_16 ?????) ??)
165    >set_program_counter_mk_Status_commutation
166    >set_clock_mk_Status_commutation
167    >set_arg_16_mk_Status_commutation in ⊢ (???%);
168   
169   
170    @split_eq_status //
171    [1:
172      assumption
173    |2:
174      @special_function_registers_8051_set_arg_16
175      [2: >EQlookup_datalabels %
176      |1: //
177      ]
178    ]
179
180  |4: (* Jmp *) #label whd in ⊢ (??%? → ???% → ?);
181   @Some_Some_elim #MAP cases (pol ?) normalize nodelta
182       [3: (* long *) #EQ3 @(Some_Some_elim ????? EQ3) #EQ3'
183         whd in match eject normalize nodelta >EQ3' in ⊢ (% → ?) whd in ⊢ (% → ?)
184         @pair_elim' * #instr #newppc' #ticks #EQ4       
185         * * #H2a #H2b whd in ⊢ (% → ?) #H2
186         >H2b >(eq_instruction_to_eq … H2a)
187         #EQ %[@1]
188         <MAP >(eq_bv_eq … H2) >EQ
189         whd in ⊢ (??%?) >EQ4 whd in ⊢ (??%?)
190         cases ps in EQ4; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXX >XXX %
191         whd in ⊢ (??%?)
192         whd in ⊢ (??(match ?%? with [_ ⇒ ?])?)
193         cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
194  |6: (* Mov *) #arg1 #arg2
195       #H1 #H2 #EQ %[@1]
196       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
197       change in ⊢ (? → ??%?) with (execute_1_0 ??)
198       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
199       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
200       >H2b >(eq_instruction_to_eq … H2a)
201       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
202       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?)
203       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
204       normalize nodelta;
205       [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
206       #result #flags
207       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
208  |5: (* Call *) #label #MAP
209      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP;
210      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
211       [ (* short *) #abs @⊥ destruct (abs)
212       |3: (* long *) #H1 #H2 #EQ %[@1]
213           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
214           change in ⊢ (? → ??%?) with (execute_1_0 ??)
215           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
216           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
217           >H2b >(eq_instruction_to_eq … H2a)
218           generalize in match EQ; -EQ;
219           whd in ⊢ (???% → ??%?);
220           generalize in match (refl … (half_add 8 (get_8051_sfr ? ps SFR_SP) (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry #new_sp #EQ1 normalize nodelta;
221           >(eq_bv_eq … H2c)
222           change with
223            ((?=let 〈ppc_bu,ppc_bl〉 ≝ split bool 8 8 newppc in ?) →
224                (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
225           generalize in match (refl … (split … 8 8 newppc)) cases (split bool 8 8 newppc) in ⊢ (??%? → %) #ppc_bu #ppc_bl #EQppc
226           generalize in match (refl … (split … 8 8 (sigma 〈preamble,instr_list〉 pol newppc))) cases (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)) in ⊢ (??%? → %) #pc_bu #pc_bl #EQpc normalize nodelta;
227           >get_8051_sfr_write_at_stack_pointer >get_8051_sfr_write_at_stack_pointer
228           >get_8051_sfr_set_8051_sfr >get_8051_sfr_set_8051_sfr
229           generalize in match (refl … (half_add ? new_sp (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry' #new_sp' #EQ2 normalize nodelta;
230           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
231           @split_eq_status;
232            [ >code_memory_write_at_stack_pointer whd in ⊢ (??%?)
233              >code_memory_write_at_stack_pointer %
234            | >set_program_counter_set_low_internal_ram
235              >set_clock_set_low_internal_ram
236              @low_internal_ram_write_at_stack_pointer
237               [ >EQ0 @pol | % | %
238               | @(  … EQ1)
239               | @(pair_destruct_2 … EQ2)
240               | >(pair_destruct_1 ????? EQpc)
241                 >(pair_destruct_2 ????? EQpc)
242                 @split_elim #x #y #H <H -x y H;
243                 >(pair_destruct_1 ????? EQppc)
244                 >(pair_destruct_2 ????? EQppc)
245                 @split_elim #x #y #H <H -x y H;
246                 >EQ0 % ]
247            | >set_low_internal_ram_set_high_internal_ram
248              >set_program_counter_set_high_internal_ram
249              >set_clock_set_high_internal_ram
250              @high_internal_ram_write_at_stack_pointer
251               [ >EQ0 @pol | % | %
252               | @(pair_destruct_2 … EQ1)
253               | @(pair_destruct_2 … EQ2)
254               | >(pair_destruct_1 ????? EQpc)
255                 >(pair_destruct_2 ????? EQpc)
256                 @split_elim #x #y #H <H -x y H;
257                 >(pair_destruct_1 ????? EQppc)
258                 >(pair_destruct_2 ????? EQppc)
259                 @split_elim #x #y #H <H -x y H;
260                 >EQ0 % ]           
261            | >external_ram_write_at_stack_pointer whd in ⊢ (??%?)
262              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
263              >external_ram_write_at_stack_pointer whd in ⊢ (???%)
264              >external_ram_write_at_stack_pointer %
265            | change with (? = sigma ?? (address_of_word_labels_code_mem (\snd (code_memory ? ps)) ?))
266              >EQ0 %
267            | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (??%?)
268              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
269              >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%)
270              >special_function_registers_8051_write_at_stack_pointer %
271            | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (??%?)
272              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
273              >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%)
274              >special_function_registers_8052_write_at_stack_pointer %
275            | >p1_latch_write_at_stack_pointer whd in ⊢ (??%?)
276              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
277              >p1_latch_write_at_stack_pointer whd in ⊢ (???%)
278              >p1_latch_write_at_stack_pointer %
279            | >p3_latch_write_at_stack_pointer whd in ⊢ (??%?)
280              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
281              >p3_latch_write_at_stack_pointer whd in ⊢ (???%)
282              >p3_latch_write_at_stack_pointer %
283            | >clock_write_at_stack_pointer whd in ⊢ (??%?)
284              >clock_write_at_stack_pointer whd in ⊢ (???%)
285              >clock_write_at_stack_pointer whd in ⊢ (???%)
286              >clock_write_at_stack_pointer %]
287       (*| (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
288         @pair_elim' #fst_5_addr #rest_addr #EQ1
289         @pair_elim' #fst_5_pc #rest_pc #EQ2
290         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
291         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
292         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
293         change in ⊢ (? →??%?) with (execute_1_0 ??)
294         @pair_elim' * #instr #newppc' #ticks #EQn
295          * * #H2a #H2b whd in ⊢ (% → ?) #H2c >H2b >(eq_instruction_to_eq … H2a) whd in ⊢ (??%?)
296          generalize in match EQ; -EQ; normalize nodelta; >(eq_bv_eq … H2c)
297          @pair_elim' #carry #new_sp change with (half_add ? (get_8051_sfr ? ps ?) ? = ? → ?) #EQ4
298          @split_elim' #pc_bu #pc_bl >program_counter_set_8051_sfr XXX change with (newppc = ?) #EQ5
299          @pair_elim' #carry' #new_sp' #EQ6 normalize nodelta; #EQx >EQx -EQx;
300          change in ⊢ (??(match ????% with [_ ⇒ ?])?) with (sigma … newppc)
301          @split_elim' #pc_bu' #pc_bl' #EQ7 change with (newppc' = ? → ?)
302          >get_8051_sfr_set_8051_sfr
303         
304          whd in EQ:(???%) ⊢ ? >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
305           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?)
306           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc)))
307           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
308           generalize in match (refl … (split bool 4 4 pc_bu))
309           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
310           generalize in match (refl … (split bool 3 8 rest_addr))
311           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
312           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
313           generalize in match
314            (refl …
315             (half_add 16 (sigma 〈preamble,instr_list〉 newppc)
316             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
317           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7
318           @split_eq_status try %
319            [ change with (? = sigma ? (address_of_word_labels ps label))
320              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
321            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
322              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] *)]
323  |4: (* Jmp *) #label #MAP
324      generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP >MAP -MAP;
325      whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta;
326       [3: (* long *) #H1 #H2 #EQ %[@1]
327           (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
328           change in ⊢ (? → ??%?) with (execute_1_0 ??)
329           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
330           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
331           >H2b >(eq_instruction_to_eq … H2a)
332           generalize in match EQ; -EQ;
333           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
334           cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX %
335       |1: (* short *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
336           generalize in match
337            (refl ?
338             (sub_16_with_carry
339              (sigma 〈preamble,instr_list〉 pol (program_counter … ps))
340              (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))
341              false))
342           cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta;
343           generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta;
344           generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta;
345           #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)]
346           generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
347           change in ⊢ (? → ??%?) with (execute_1_0 ??)
348           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
349           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
350           >H2b >(eq_instruction_to_eq … H2a)
351           generalize in match EQ; -EQ;
352           whd in ⊢ (???% → ?);
353           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c)
354           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ???) ? in ?) = ?)
355           generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) (sign_extension lower)))
356           cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4
357           @split_eq_status try % change with (newpc = sigma ?? (address_of_word_labels ps label))
358           (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
359       | (* medium *)  #H1 #H2 #EQ %[@1] generalize in match H1; -H1;
360         generalize in match
361          (refl …
362            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label))))
363         cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1
364         generalize in match
365          (refl …
366            (split … 5 11 (sigma 〈preamble,instr_list〉 pol (program_counter … ps))))
367         cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2
368         generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc))
369         cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)]
370         generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?)
371         change in ⊢ (? →??%?) with (execute_1_0 ??)
372           cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
373           * * #H2a #H2b whd in ⊢ (% → ?) #H2c
374           >H2b >(eq_instruction_to_eq … H2a)
375           generalize in match EQ; -EQ;
376           whd in ⊢ (???% → ?);
377           #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?)
378           change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?)
379           generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)))
380           cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4
381           generalize in match (refl … (split bool 4 4 pc_bu))
382           cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5
383           generalize in match (refl … (split bool 3 8 rest_addr))
384           cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6
385           change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?)
386           generalize in match
387            (refl …
388             (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc)
389             ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2)))
390           cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7   
391           @split_eq_status try %
392            [ change with (? = sigma ?? (address_of_word_labels ps label))
393              (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *)
394            | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?)
395              @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]]
396  | (* Instruction *) -pi;  whd in ⊢ (? → ??%? → ?) *; normalize nodelta;
397    [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1]
398       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
399       change in ⊢ (? → ??%?) with (execute_1_0 ??)
400       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
401       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
402       >H2b >(eq_instruction_to_eq … H2a)
403       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP;
404       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1;
405       @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2
406       normalize nodelta; #MAP;
407       [1: change in ⊢ (? → %) with
408        ((let 〈result,flags〉 ≝
409          add_8_with_carry
410           (get_arg_8 ? ps false ACC_A)
411           (get_arg_8 ?
412             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
413             false (DIRECT ARG2))
414           ? in ?) = ?)
415        [2,3: %]
416        change in ⊢ (???% → ?) with
417         (let 〈result,flags〉 ≝ add_8_with_carry ?(*(get_arg_8 ? ps false ACC_A)*) ?? in ?)
418        >get_arg_8_set_clock
419       [1,2: cases (addressing_mode_ok ???? ∧ addressing_mode_ok ????) in MAP ⊢ ?
420         [2,4: #abs @⊥ normalize in abs; destruct (abs)
421         |*:whd in ⊢ (??%? → ?) #H <(option_destruct_Some ??? H)]
422       [ change in ⊢ (? → %) with
423        ((let 〈result,flags〉 ≝
424          add_8_with_carry
425           (get_arg_8 ? ps false ACC_A)
426           (get_arg_8 ?
427             (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps)))
428             false (DIRECT ARG2))
429           ? in ?) = ?)
430          >get_arg_8_set_low_internal_ram
431       
432        cases (add_8_with_carry ???)
433         
434        [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)]
435       #result #flags
436       #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) %
437    | (* INC *) #arg1 #H1 #H2 #EQ %[@1]
438       normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?)
439       change in ⊢ (? → ??%?) with (execute_1_0 ??)
440       cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta;
441       * * #H2a #H2b whd in ⊢ (% → ?) #H2c
442       >H2b >(eq_instruction_to_eq … H2a)
443       generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?);
444       @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG]
445       [1,2,3,4: cases (half_add ???) #carry #result
446       | cases (half_add ???) #carry #bl normalize nodelta;
447         cases (full_add ????) #carry' #bu normalize nodelta ]
448        #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) -newppc';
449        [5: %
450        |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program
451      (set_program_counter pseudo_assembly_program ps newppc)
452      (\fst  (ticks_of0 〈preamble,instr_list〉
453                   (program_counter pseudo_assembly_program ps)
454                   (Instruction (INC Identifier (DIRECT ARG))))
455       +clock pseudo_assembly_program
456        (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG))
457        [2,3: // ]
458            <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://]
459            whd in ⊢ (??%%)
460            cases (split bool 4 4 ARG)
461            #nu' #nl'
462            normalize nodelta
463            cases (split bool 1 3 nu')
464            #bit_1' #ignore'
465            normalize nodelta
466            cases (get_index_v bool 4 nu' ? ?)
467            [ normalize nodelta (* HERE *) whd in ⊢ (??%%) %
468            |
469            ]
470cases daemon (* EASY CASES TO BE COMPLETED *)
471qed.
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