source: src/ASM/AssemblyProofSplitSplit.ma @ 2139

Last change on this file since 2139 was 2139, checked in by mulligan, 7 years ago

Changes to get the main lemma compiling again. Changes pushed into
the first half of the proof of the main theorem, too.

File size: 14.7 KB
Line 
1include "ASM/AssemblyProofSplit.ma".
2include "common/LabelledObjects.ma".
3
4lemma short_jump_cond_ok:
5  ∀v1, v2: Word.
6  ∀is_possible, offset.
7    〈is_possible, offset〉 = short_jump_cond v1 v2 →
8      is_possible → v2 = add 16 v1 (sign_extension offset).
9  #v1 #v2 #is_possible #offset
10  whd in match short_jump_cond; normalize nodelta
11  @pair_elim #result #flags #sub16_refl
12  @pair_elim #upper #lower #vsplit_refl
13  inversion (get_index' bool ???) #get_index_refl normalize nodelta
14  #relevant destruct(relevant) #relevant
15  [1:
16    @(sub_16_to_add_16_8_1 … flags)
17  |2:
18    @(sub_16_to_add_16_8_0 … flags)
19  ]
20  try assumption >sub16_refl
21  <(eq_bv_eq … relevant)
22  >(vsplit_ok … (sym_eq … vsplit_refl)) %
23qed.
24
25lemma absolute_jump_cond_ok:
26  ∀v1, v2: Word.
27  ∀is_possible, offset, v1_upper, v1_lower.
28    〈is_possible, offset〉 = absolute_jump_cond v1 v2 →
29      〈v1_upper, v1_lower〉 = vsplit ? 5 11 v1 →
30        is_possible → v2 = v1_upper @@ offset.
31  #v1 #v2 #is_possible #offset #v1_upper #v1_lower
32  whd in match absolute_jump_cond; normalize nodelta
33  @pair_elim #fst_5_addr #rest_addr #vsplit_v2_refl
34  @pair_elim #fst_5_pc #rest_pc #vsplit_v1_refl
35  #relevant destruct(relevant) normalize nodelta #relevant
36  destruct(relevant) #relevant
37  <(vsplit_ok … (sym_eq … vsplit_v2_refl))
38  >(eq_bv_eq … relevant) %
39qed.
40 
41theorem main_thm:
42  ∀M, M': internal_pseudo_address_map.
43  ∀program: pseudo_assembly_program.
44  ∀program_in_bounds: |\snd program| ≤ 2^16.
45  ∀addr_of: Identifier → PreStatus pseudo_assembly_program program → BitVector 16.
46  ∀is_well_labelled: is_well_labelled_p (\snd program).
47  ∀sigma: Word → Word.
48  ∀policy: Word → bool.
49  ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy.
50  ∀ps: PseudoStatus program.
51  ∀program_counter_in_bounds: nat_of_bitvector 16 (program_counter … ps) < |\snd program|.
52    next_internal_pseudo_address_map M program addr_of ps program_counter_in_bounds = Some … M' →
53      ∃n. execute n … (status_of_pseudo_status M … ps sigma policy) =
54        status_of_pseudo_status M' …
55          (execute_1_pseudo_instruction program (ticks_of program sigma policy) ps program_counter_in_bounds) sigma policy.
56    #M #M' * #preamble #instr_list #program_in_bounds #addr_of
57    #is_well_labelled_witness #sigma #policy #sigma_policy_witness #ps
58    letin ppc ≝ (program_counter pseudo_assembly_program ? ps) #ppc_in_bounds
59    change with (next_internal_pseudo_address_map0 ?????? = ? → ?)
60    generalize in match (fetch_assembly_pseudo2 〈preamble,instr_list〉 program_in_bounds sigma policy sigma_policy_witness ppc ppc_in_bounds) in ⊢ ?;
61    @pair_elim #labels #costs #create_label_cost_refl normalize nodelta
62    @pair_elim #assembled #costs' #assembly_refl normalize nodelta
63    lapply (pair_destruct_1 ????? (sym_eq ??? assembly_refl)) #EQassembled
64    @pair_elim #pi #newppc #fetch_pseudo_refl normalize nodelta
65    @(pose … (λx. bitvector_of_nat ? (lookup_def … labels x 0))) #lookup_labels #EQlookup_labels
66    @(pose … (λx. lookup_def … (construct_datalabels preamble) x (zero 16))) #lookup_datalabels #EQlookup_datalabels
67    whd in match execute_1_pseudo_instruction; normalize nodelta
68    whd in match ticks_of; normalize nodelta >fetch_pseudo_refl normalize nodelta
69    lapply (snd_fetch_pseudo_instruction instr_list ppc ppc_in_bounds) >fetch_pseudo_refl #EQnewppc >EQnewppc
70    lapply (snd_assembly_1_pseudoinstruction_ok 〈preamble,instr_list〉 … sigma policy sigma_policy_witness … ppc … pi … lookup_labels lookup_datalabels)
71    [1: assumption |2: assumption] >create_label_cost_refl
72    #X lapply (X EQlookup_labels EQlookup_datalabels ?) -X
73    [1: >fetch_pseudo_refl %]
74    #assm1 #assm2 #assm3 generalize in match assm2; generalize in match assm3;
75    generalize in match assm1; -assm1 -assm2 -assm3
76    normalize nodelta
77    inversion pi
78    [2,3:
79      #arg #_
80      (* XXX: we first work on sigma_increment_commutation *)
81      #sigma_increment_commutation
82      normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation;
83      (* XXX: we work on the maps *)
84      whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm
85      (* XXX: we assume the fetch_many hypothesis *)
86      #fetch_many_assm
87      (* XXX: we give the existential *)
88      %{0}
89      whd in match (execute_1_pseudo_instruction0 ?????);
90      (* XXX: work on the left hand side of the equality *)
91      whd in ⊢ (??%?);
92      @split_eq_status try %
93      /demod/ >add_commutative <add_zero % (*CSC: auto not working, why? *)
94    |6: (* Mov *)
95      #arg1 #arg2 #_
96      (* XXX: we first work on sigma_increment_commutation *)
97      #sigma_increment_commutation
98      normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation;
99      (* XXX: we work on the maps *)
100      whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm
101      (* XXX: we assume the fetch_many hypothesis *)
102      #fetch_many_assm
103      (* XXX: we give the existential *)
104      %{1}
105      whd in match (execute_1_pseudo_instruction0 ?????);
106      (* XXX: work on the left hand side of the equality *)
107      whd in ⊢ (??%?); whd in match (program_counter ???);
108      (* XXX: execute fetches some more *)
109      whd in match code_memory_of_pseudo_assembly_program; normalize nodelta
110      whd in fetch_many_assm;
111      >EQassembled in fetch_many_assm;
112      cases (fetch ??) * #instr #newpc #ticks normalize nodelta *
113      #eq_instr
114      #fetch_many_assm whd in fetch_many_assm;
115      lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc
116      destruct whd in ⊢ (??%?);
117      (* XXX: now we start to work on the mk_PreStatus equality *)
118      (* XXX: lhs *)
119      change with (set_arg_16 ????? = ?)
120      >set_program_counter_mk_Status_commutation
121      >set_clock_mk_Status_commutation
122      >set_arg_16_mk_Status_commutation
123      (* XXX: rhs *)
124      change with (status_of_pseudo_status ?? (set_arg_16 pseudo_assembly_program 〈preamble, instr_list〉 ?? arg1) ??) in ⊢ (???%);
125      >set_program_counter_mk_Status_commutation
126      >set_clock_mk_Status_commutation
127      >set_arg_16_mk_Status_commutation in ⊢ (???%);
128      (* here we are *)
129      @split_eq_status try %
130      [1:
131        assumption
132      |2:
133        @special_function_registers_8051_set_arg_16 %
134      ]
135    |1: (* Instruction *)
136      #instr #pi_refl #EQP #EQnext #fetch_many_assm
137      @(main_lemma_preinstruction M M' preamble instr_list 〈preamble, instr_list〉 (refl …) ? sigma policy sigma_policy_witness
138        ps ppc ? ? labels costs create_label_cost_refl newppc lookup_labels EQlookup_labels lookup_datalabels EQlookup_datalabels
139        EQnewppc instr ? (ticks_of0 〈preamble, instr_list〉 sigma policy ppc (Instruction instr)) (refl …)
140        (λx:Identifier. λy:PreStatus pseudo_assembly_program 〈preamble, instr_list〉. address_of_word_labels 〈preamble, instr_list〉 x) (refl …) (set_program_counter pseudo_assembly_program 〈preamble, instr_list〉 ps (add 16 ppc (bitvector_of_nat 16 1)))
141        (refl …) ? (refl …))
142      [1,2:
143        assumption
144      |3:
145        %
146      |4:
147        >fetch_pseudo_refl @pi_refl
148      |5:
149        <EQP %
150      |6:
151        >assembly_refl assumption
152      ]
153    |4: (* Jmp *)
154      #arg1 #pi_refl
155      (* XXX: we first work on sigma_increment_commutation *)
156      whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?);
157      whd in match (expand_pseudo_instruction ??????);
158      inversion (short_jump_cond ??) #sj_possible #offset #sjc_refl normalize nodelta
159      inversion (sj_possible ∧ ¬ policy ?) #sj_possible_refl normalize nodelta
160      [2:
161        inversion (absolute_jump_cond ??) #mj_possible #address #mjc_refl normalize nodelta
162        inversion (mj_possible ∧ ¬ policy ?) #mj_possible_refl normalize nodelta
163      ]
164      #sigma_increment_commutation
165      normalize in sigma_increment_commutation:(???(???(??%)));
166      (* XXX: we work on the maps *)
167      whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm
168      (* XXX: we assume the fetch_many hypothesis *)
169      * #fetch_refl #fetch_many_assm
170      (* XXX: we give the existential *)
171      %{1}
172      (* XXX: work on the left hand side of the equality *)
173      whd in ⊢ (??%?); whd in match (program_counter ???);
174      (* XXX: execute fetches some more *)
175      whd in match code_memory_of_pseudo_assembly_program; normalize nodelta
176      whd in fetch_many_assm;
177      >EQassembled in fetch_refl; #fetch_refl <fetch_refl
178      lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc
179      whd in ⊢ (??%%);
180      (* XXX: now we start to work on the mk_PreStatus equality *)
181      (* XXX: lhs *)
182      (* XXX: rhs *)
183      (* here we are *)
184      @split_eq_status try % /demod nohyps/
185      [1,3,4:
186        change with (add ???) in match (\snd (half_add ???));
187        whd in match execute_1_pseudo_instruction0; normalize nodelta
188        /demod nohyps/ >set_clock_set_program_counter
189        >program_counter_set_program_counter
190        whd in ⊢ (??%?); normalize nodelta whd in match address_of_word_labels; normalize nodelta
191        lapply (create_label_cost_map_ok 〈preamble, instr_list〉) >create_label_cost_refl
192        normalize nodelta #address_of_word_labels_assm <address_of_word_labels_assm
193        [1:
194          >EQnewpc
195          inversion (vsplit bool ???) #pc_bu #pc_bl #vsplit_refl normalize nodelta
196          @sym_eq
197          >EQlookup_labels in mjc_refl; #mjc_refl
198          @(absolute_jump_cond_ok ?????? (sym_eq … mjc_refl) (sym_eq … vsplit_refl))
199          >(andb_true_l … mj_possible_refl) @I
200        |3:
201          >EQlookup_labels normalize nodelta %
202        |5:
203          >EQnewpc
204          inversion (half_add ???) #carry #new_pc #half_add_refl normalize nodelta
205          @sym_eq
206          >EQlookup_labels in sjc_refl; #sjc_refl
207          >(pair_destruct_2 ????? (sym_eq … half_add_refl))
208          @(short_jump_cond_ok ???? (sym_eq … sjc_refl))
209          >(andb_true_l … sj_possible_refl) @I
210        ]
211        whd in is_well_labelled_witness;
212        @(is_well_labelled_witness ? ppc ppc_in_bounds pi)
213        try (>fetch_pseudo_refl %)
214        >pi_refl %
215      |2:
216        whd in ⊢ (??(?%%)%);
217        cases (bitvector_11_cases address)
218        * * * * * * #b4 * #b5
219        * #b6 * #b7 * #b8 * #b9 * #b10 * #b11 #address_refl >address_refl
220        normalize in match (fetch ??); <plus_n_Sm @eq_f
221        @commutative_plus
222      ]
223    |5: (* Call *)
224      #arg1 #pi_refl
225      (* XXX: we first work on sigma_increment_commutation *)
226      whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?);
227      whd in match (expand_pseudo_instruction ??????);
228      whd in match (execute_1_pseudo_instruction0 ?????);
229      inversion (absolute_jump_cond ??) #aj_possible #offset #ajc_refl normalize nodelta
230      inversion (aj_possible ∧ ¬ policy ?) #aj_possible_refl normalize nodelta
231      @pair_elim #carry #new_sp #carry_new_sp_refl lapply (refl_to_jmrefl ??? carry_new_sp_refl) -carry_new_sp_refl #carry_new_sp_refl
232      @pair_elim #pc_bu #pc_bl #pc_bu_bl_refl lapply (refl_to_jmrefl ??? pc_bu_bl_refl) -pc_bu_bl_refl #pc_bu_bl_refl
233      @pair_elim #carry' #new_sp' #carry_new_sp_refl' lapply (refl_to_jmrefl ??? carry_new_sp_refl') -carry_new_sp_refl' #carry_new_sp_refl'
234      #sigma_increment_commutation
235      normalize in sigma_increment_commutation:(???(???(??%)));
236      (* XXX: we work on the maps *)
237      whd in ⊢ (??%? → ?);
238      @pair_elim #callM #accM #Mrefl
239      @Some_Some_elim #map_refl_assm <map_refl_assm
240      (* XXX: we assume the fetch_many hypothesis *)
241      * #fetch_refl #fetch_many_assm
242      (* XXX: we give the existential *)
243      %{1}
244      (* XXX: work on the left hand side of the equality *)
245      whd in ⊢ (??%?); whd in match (program_counter ???);
246      (* XXX: execute fetches some more *)
247      whd in match code_memory_of_pseudo_assembly_program; normalize nodelta
248      whd in fetch_many_assm;
249      >EQassembled in fetch_refl; #fetch_refl <fetch_refl
250      lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc
251      whd in ⊢ (??(???%)?); normalize in match (length ? (assembly1 …));
252      normalize in match (length ? (assembly1 …)) in EQnewpc;
253      normalize nodelta
254      [1:
255        @(pair_replace ????? carry new_sp ??? carry_new_sp_refl)
256        try /demod nohyps/ [1: %]
257        @pair_elim #pc_bu' #pc_bl' #pc_bu_bl_refl'
258        @(pair_replace ????? carry' new_sp' ??? carry_new_sp_refl')
259        try /demod nohyps/ [1: %]
260        @pair_elim #fiv #thr' #fiv_thr_refl' whd in ⊢ (??%%);
261        @split_eq_status try /demod nohyps/ try %
262        [1,2:
263          (* XXX: need the commented out axioms from AssemblyProof.ma *)
264          cases daemon
265        |3:
266          whd in ajc_refl:(??%?); lapply ajc_refl -ajc_refl -map_refl_assm -Mrefl
267          >EQlookup_labels normalize nodelta
268          @vsplit_elim #fst_5_addr #rest_addr #fst_5_rest_refl normalize nodelta
269          <EQnewpc @vsplit_elim #fst_5_pc #rest_pc #fst_5_rest_pc_refl normalize nodelta
270          #relevant
271          lapply pc_bu_bl_refl' -pc_bu_bl_refl'
272          >program_counter_set_8051_sfr >set_clock_set_program_counter
273          >program_counter_set_program_counter >fst_5_rest_pc_refl
274          cut(fst_5_addr = fst_5_pc)
275          [1:
276            cases daemon (* XXX: !! *)
277          |2:
278            #relevant2 >relevant2
279            <(vsplit_ok ?????? (sym_eq … fiv_thr_refl'))
280            #relevant3
281            lapply(vsplit_ok ?????? (sym_eq … relevant3))
282            >(vector_associative_append bool 5 3 8) #relevant4
283            cut(fiv = fst_5_pc ∧ thr'@@pc_bl' = rest_pc)
284            [1:
285              cases daemon (* XXX: !! *)
286            |2:
287              * #relevant5 #relevant6 >relevant5
288              >(vector_associative_append bool 5 3 8) @eq_f
289              destruct(relevant)
290              <(vsplit_ok ?????? (sym_eq … thr_eig_refl)) @eq_f
291            ]
292          ] *)
293        |4:
294          >set_program_counter_mk_Status_commutation in ⊢ (???%);
295          >special_function_registers_8051_write_at_stack_pointer in ⊢ (???%);
296          (* XXX: require set_8051_sfr_write_at_stack_pointer and
297                          special_function_registers_8051_set_8051_sfr
298          *)
299          cases daemon
300        |5,6,7:
301          >set_program_counter_mk_Status_commutation in ⊢ (???%);
302          (* XXX: similar to above but for 8052
303          *)
304          cases daemon
305        |8:
306          whd in match (ticks_of_instruction ?);
307          cut(\snd  (fetch (load_code_memory (assembly1 (ACALL (ADDR11 offset)))) (zero 16)) = 2)
308          [1: //
309          |2:
310            normalize in match (assembly1 ?); >thr_eig_refl
311            @(bitvector_3_elim_prop … thr) %
312          |3:
313            #fetch_2_refl >fetch_2_refl >commutative_plus %
314          ]
315        |2:
316        ]
317      ]
318    ]
319  ]
320qed.
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