source: src/ASM/AssemblyProofSplitSplit.ma @ 2129

Last change on this file since 2129 was 2129, checked in by mulligan, 7 years ago

Large changes from today trying to complete the main theorem. Again :(

File size: 14.8 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. address_of_word_labels_code_mem instr_list x)) #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 … EQlookup_labels EQlookup_datalabels ?)
71    [1: >fetch_pseudo_refl % |2: skip |3: assumption ]
72    #assm1 #assm2 #assm3 generalize in match assm2; generalize in match assm3;
73    generalize in match assm1; -assm1 -assm2 -assm3
74    normalize nodelta
75    inversion pi
76    [(*2,3:
77      #arg #_
78      (* XXX: we first work on sigma_increment_commutation *)
79      #sigma_increment_commutation
80      normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation;
81      (* XXX: we work on the maps *)
82      whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm
83      (* XXX: we assume the fetch_many hypothesis *)
84      #fetch_many_assm
85      (* XXX: we give the existential *)
86      %{0}
87      whd in match (execute_1_pseudo_instruction0 ?????);
88      (* XXX: work on the left hand side of the equality *)
89      whd in ⊢ (??%?);
90      @split_eq_status try %
91      /demod/ >add_commutative <add_zero % (*CSC: auto not working, why? *)
92    |6: (* Mov *)
93      #arg1 #arg2 #_
94      (* XXX: we first work on sigma_increment_commutation *)
95      #sigma_increment_commutation
96      normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation;
97      (* XXX: we work on the maps *)
98      whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm
99      (* XXX: we assume the fetch_many hypothesis *)
100      #fetch_many_assm
101      (* XXX: we give the existential *)
102      %{1}
103      whd in match (execute_1_pseudo_instruction0 ?????);
104      (* XXX: work on the left hand side of the equality *)
105      whd in ⊢ (??%?); whd in match (program_counter ???);
106      (* XXX: execute fetches some more *)
107      whd in match code_memory_of_pseudo_assembly_program; normalize nodelta
108      whd in fetch_many_assm;
109      >EQassembled in fetch_many_assm;
110      cases (fetch ??) * #instr #newpc #ticks normalize nodelta *
111      #eq_instr
112      #fetch_many_assm whd in fetch_many_assm;
113      lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc
114      destruct whd in ⊢ (??%?);
115      (* XXX: now we start to work on the mk_PreStatus equality *)
116      (* XXX: lhs *)
117      change with (set_arg_16 ????? = ?)
118      >set_program_counter_mk_Status_commutation
119      >set_clock_mk_Status_commutation
120      >set_arg_16_mk_Status_commutation
121      (* XXX: rhs *)
122      change with (status_of_pseudo_status ?? (set_arg_16 pseudo_assembly_program 〈preamble, instr_list〉 ?? arg1) ??) in ⊢ (???%);
123      >set_program_counter_mk_Status_commutation
124      >set_clock_mk_Status_commutation
125      >set_arg_16_mk_Status_commutation in ⊢ (???%);
126      (* here we are *)
127      @split_eq_status try %
128      [1:
129        assumption
130      |2:
131        @special_function_registers_8051_set_arg_16 %
132      ]
133    |1: (* Instruction *)
134      #instr #_ #EQP #EQnext #fetch_many_assm
135      @(main_lemma_preinstruction M M' preamble instr_list 〈preamble, instr_list〉 (refl …) sigma policy sigma_policy_witness
136        ps ppc ? labels costs create_label_cost_refl newppc lookup_labels EQlookup_labels lookup_datalabels EQlookup_datalabels
137        EQnewppc instr (ticks_of0 〈preamble, instr_list〉 sigma policy ppc (Instruction instr)) (refl …)
138        (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)))
139        (refl …) ? (refl …))
140      try assumption try % >assembly_refl assumption
141    |4: (* Jmp *)
142      #arg1 #pi_refl
143      (* XXX: we first work on sigma_increment_commutation *)
144      whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?);
145      whd in match (expand_pseudo_instruction ??????);
146      inversion (short_jump_cond ??) #sj_possible #offset #sjc_refl normalize nodelta
147      inversion (sj_possible ∧ ¬ policy ?) #sj_possible_refl normalize nodelta
148      [2:
149        inversion (absolute_jump_cond ??) #mj_possible #address #mjc_refl normalize nodelta
150        inversion (mj_possible ∧ ¬ policy ?) #mj_possible_refl normalize nodelta
151      ]
152      #sigma_increment_commutation
153      normalize in sigma_increment_commutation:(???(???(??%)));
154      (* XXX: we work on the maps *)
155      whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm
156      (* XXX: we assume the fetch_many hypothesis *)
157      * #fetch_refl #fetch_many_assm
158      (* XXX: we give the existential *)
159      %{1}
160      (* XXX: work on the left hand side of the equality *)
161      whd in ⊢ (??%?); whd in match (program_counter ???);
162      (* XXX: execute fetches some more *)
163      whd in match code_memory_of_pseudo_assembly_program; normalize nodelta
164      whd in fetch_many_assm;
165      >EQassembled in fetch_refl; #fetch_refl <fetch_refl
166      lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc
167      whd in ⊢ (??%%);
168      (* XXX: now we start to work on the mk_PreStatus equality *)
169      (* XXX: lhs *)
170      (* XXX: rhs *)
171      (* here we are *)
172      @split_eq_status try % /demod nohyps/
173      [1,3,4:
174        change with (add ???) in match (\snd (half_add ???));
175        whd in match execute_1_pseudo_instruction0; normalize nodelta
176        /demod nohyps/ >set_clock_set_program_counter
177        >program_counter_set_program_counter
178        whd in ⊢ (??%?); normalize nodelta whd in match address_of_word_labels; normalize nodelta
179        lapply (create_label_cost_map_ok 〈preamble, instr_list〉) >create_label_cost_refl
180        normalize nodelta #address_of_word_labels_assm <address_of_word_labels_assm
181        [1:
182          >EQnewpc
183          inversion (vsplit bool ???) #pc_bu #pc_bl #vsplit_refl normalize nodelta
184          @sym_eq >address_of_word_labels_assm
185          [1:
186            >EQlookup_labels in mjc_refl; #mjc_refl
187            @(absolute_jump_cond_ok ?????? (sym_eq … mjc_refl) (sym_eq … vsplit_refl))
188            >(andb_true_l … mj_possible_refl) @I
189          ]
190        |3:
191          >EQlookup_labels normalize nodelta
192          >address_of_word_labels_assm try %
193        |5:
194          >EQnewpc
195          inversion (half_add ???) #carry #new_pc #half_add_refl normalize nodelta
196          @sym_eq >address_of_word_labels_assm
197          [1:
198            >EQlookup_labels in sjc_refl; #sjc_refl
199            >(pair_destruct_2 ????? (sym_eq … half_add_refl))
200            @(short_jump_cond_ok ???? (sym_eq … sjc_refl))
201            >(andb_true_l … sj_possible_refl) @I
202          ]
203        ]
204        @(is_well_labelled_witness … fetch_pseudo_refl)
205        >pi_refl %
206      |2:
207        whd in ⊢ (??(?%%)%);
208        cases (bitvector_11_cases address)
209        * * * * * * #b4 * #b5
210        * #b6 * #b7 * #b8 * #b9 * #b10 * #b11 #address_refl >address_refl
211        normalize in match (fetch ??); <plus_n_Sm @eq_f
212        @commutative_plus
213      ]
214    |*)5: (* Call *)
215      #arg1 #pi_refl
216      (* XXX: we first work on sigma_increment_commutation *)
217      whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?);
218      whd in match (expand_pseudo_instruction ??????);
219      whd in match (execute_1_pseudo_instruction0 ?????);
220      inversion (absolute_jump_cond ??) #aj_possible #offset #ajc_refl normalize nodelta
221      inversion (aj_possible ∧ ¬ policy ?) #aj_possible_refl normalize nodelta
222      @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
223      @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
224      @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'
225      #sigma_increment_commutation
226      normalize in sigma_increment_commutation:(???(???(??%)));
227      (* XXX: we work on the maps *)
228      whd in ⊢ (??%? → ?);
229      @pair_elim #callM #accM #Mrefl
230      @Some_Some_elim #map_refl_assm <map_refl_assm
231      (* XXX: we assume the fetch_many hypothesis *)
232      * #fetch_refl #fetch_many_assm
233      (* XXX: we give the existential *)
234      %{1}
235      (* XXX: work on the left hand side of the equality *)
236      whd in ⊢ (??%?); whd in match (program_counter ???);
237      (* XXX: execute fetches some more *)
238      whd in match code_memory_of_pseudo_assembly_program; normalize nodelta
239      whd in fetch_many_assm;
240      >EQassembled in fetch_refl; #fetch_refl <fetch_refl
241      lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc
242      whd in ⊢ (??(???%)?); normalize in match (length ? (assembly1 …));
243      normalize in match (length ? (assembly1 …)) in EQnewpc;
244      normalize nodelta
245      [1:
246        @(pair_replace ????? carry new_sp ??? carry_new_sp_refl)
247        try /demod nohyps/ [1: %]
248        @pair_elim #pc_bu' #pc_bl' #pc_bu_bl_refl'
249        @(pair_replace ????? carry' new_sp' ??? carry_new_sp_refl')
250        try /demod nohyps/ [1: %]
251        @pair_elim #thr #eig #thr_eig_refl
252        @pair_elim #fiv #thr' #fiv_thr_refl' whd in ⊢ (??%%);
253        @split_eq_status try /demod nohyps/ try %
254        [1,2:
255          (* XXX: need the commented out axioms from AssemblyProof.ma *)
256          cases daemon
257        |3:
258          >program_counter_set_program_counter
259          whd in ajc_refl:(??%?); lapply ajc_refl -ajc_refl -map_refl_assm -Mrefl
260          >EQlookup_labels normalize nodelta
261          @vsplit_elim #fst_5_addr #rest_addr #fst_5_rest_refl normalize nodelta
262          <EQnewpc @vsplit_elim #fst_5_pc #rest_pc #fst_5_rest_pc_refl normalize nodelta
263          #relevant >fst_5_rest_refl
264          check absolute_jump_cond
265          <(vsplit_ok ?????? (sym_eq … thr_eig_refl))
266          lapply pc_bu_bl_refl' -pc_bu_bl_refl'
267          >program_counter_set_8051_sfr >set_clock_set_program_counter
268          >program_counter_set_program_counter >fst_5_rest_pc_refl
269          cut(fst_5_addr = fst_5_pc)
270          [1:
271            cases daemon (* XXX: !! *)
272          |2:
273            #relevant2 >relevant2
274            <(vsplit_ok ?????? (sym_eq … fiv_thr_refl'))
275            #relevant3
276            lapply(vsplit_ok ?????? (sym_eq … relevant3))
277            >(vector_associative_append bool 5 3 8) #relevant4
278            cut(fiv = fst_5_pc ∧ thr'@@pc_bl' = rest_pc)
279            [1:
280              cases daemon (* XXX: !! *)
281            |2:
282              * #relevant5 #relevant6 >relevant5
283              >(vector_associative_append bool 5 3 8) @eq_f
284              destruct(relevant)
285              <(vsplit_ok ?????? (sym_eq … thr_eig_refl)) @eq_f
286            ]
287          ]
288        |4:
289          >set_program_counter_mk_Status_commutation in ⊢ (???%);
290          >special_function_registers_8051_write_at_stack_pointer in ⊢ (???%);
291          (* XXX: require set_8051_sfr_write_at_stack_pointer and
292                          special_function_registers_8051_set_8051_sfr
293          *)
294          cases daemon
295        |5,6,7:
296          >set_program_counter_mk_Status_commutation in ⊢ (???%);
297          (* XXX: similar to above but for 8052
298          *)
299          cases daemon
300        |8:
301          whd in match (ticks_of_instruction ?);
302          cut(\snd  (fetch (load_code_memory (assembly1 (ACALL (ADDR11 offset)))) (zero 16)) = 2)
303          [1: //
304          |2:
305            normalize in match (assembly1 ?); >thr_eig_refl
306            @(bitvector_3_elim_prop … thr) %
307          |3:
308            #fetch_2_refl >fetch_2_refl >commutative_plus %
309          ]
310        |2:
311        ]
312      ]
313    ]
314  ]
315qed.
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