source: src/ASM/AssemblyProofSplitSplit.ma @ 2131

Last change on this file since 2131 was 2131, checked in by sacerdot, 7 years ago

No more need for functional extensionality.

File size: 14.5 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 #_ #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      [1,2,3: try assumption try %] >assembly_refl assumption
143    |4: (* Jmp *)
144      #arg1 #pi_refl
145      (* XXX: we first work on sigma_increment_commutation *)
146      whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?);
147      whd in match (expand_pseudo_instruction ??????);
148      inversion (short_jump_cond ??) #sj_possible #offset #sjc_refl normalize nodelta
149      inversion (sj_possible ∧ ¬ policy ?) #sj_possible_refl normalize nodelta
150      [2:
151        inversion (absolute_jump_cond ??) #mj_possible #address #mjc_refl normalize nodelta
152        inversion (mj_possible ∧ ¬ policy ?) #mj_possible_refl normalize nodelta
153      ]
154      #sigma_increment_commutation
155      normalize in sigma_increment_commutation:(???(???(??%)));
156      (* XXX: we work on the maps *)
157      whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm
158      (* XXX: we assume the fetch_many hypothesis *)
159      * #fetch_refl #fetch_many_assm
160      (* XXX: we give the existential *)
161      %{1}
162      (* XXX: work on the left hand side of the equality *)
163      whd in ⊢ (??%?); whd in match (program_counter ???);
164      (* XXX: execute fetches some more *)
165      whd in match code_memory_of_pseudo_assembly_program; normalize nodelta
166      whd in fetch_many_assm;
167      >EQassembled in fetch_refl; #fetch_refl <fetch_refl
168      lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc
169      whd in ⊢ (??%%);
170      (* XXX: now we start to work on the mk_PreStatus equality *)
171      (* XXX: lhs *)
172      (* XXX: rhs *)
173      (* here we are *)
174      @split_eq_status try % /demod nohyps/
175      [1,3,4:
176        change with (add ???) in match (\snd (half_add ???));
177        whd in match execute_1_pseudo_instruction0; normalize nodelta
178        /demod nohyps/ >set_clock_set_program_counter
179        >program_counter_set_program_counter
180        whd in ⊢ (??%?); normalize nodelta whd in match address_of_word_labels; normalize nodelta
181        lapply (create_label_cost_map_ok 〈preamble, instr_list〉) >create_label_cost_refl
182        normalize nodelta #address_of_word_labels_assm <address_of_word_labels_assm
183        [1:
184          >EQnewpc
185          inversion (vsplit bool ???) #pc_bu #pc_bl #vsplit_refl normalize nodelta
186          @sym_eq
187          >EQlookup_labels in mjc_refl; #mjc_refl
188          @(absolute_jump_cond_ok ?????? (sym_eq … mjc_refl) (sym_eq … vsplit_refl))
189          >(andb_true_l … mj_possible_refl) @I
190        |3:
191          >EQlookup_labels normalize nodelta %
192        |5:
193          >EQnewpc
194          inversion (half_add ???) #carry #new_pc #half_add_refl normalize nodelta
195          @sym_eq
196          >EQlookup_labels in sjc_refl; #sjc_refl
197          >(pair_destruct_2 ????? (sym_eq … half_add_refl))
198          @(short_jump_cond_ok ???? (sym_eq … sjc_refl))
199          >(andb_true_l … sj_possible_refl) @I
200        ]
201        @(is_well_labelled_witness … fetch_pseudo_refl)
202        >pi_refl %
203      |2:
204        whd in ⊢ (??(?%%)%);
205        cases (bitvector_11_cases address)
206        * * * * * * #b4 * #b5
207        * #b6 * #b7 * #b8 * #b9 * #b10 * #b11 #address_refl >address_refl
208        normalize in match (fetch ??); <plus_n_Sm @eq_f
209        @commutative_plus
210      ]
211    |5: (* Call *)
212      #arg1 #pi_refl
213      (* XXX: we first work on sigma_increment_commutation *)
214      whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?);
215      whd in match (expand_pseudo_instruction ??????);
216      whd in match (execute_1_pseudo_instruction0 ?????);
217      inversion (absolute_jump_cond ??) #aj_possible #offset #ajc_refl normalize nodelta
218      inversion (aj_possible ∧ ¬ policy ?) #aj_possible_refl normalize nodelta
219      @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
220      @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
221      @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'
222      #sigma_increment_commutation
223      normalize in sigma_increment_commutation:(???(???(??%)));
224      (* XXX: we work on the maps *)
225      whd in ⊢ (??%? → ?);
226      @pair_elim #callM #accM #Mrefl
227      @Some_Some_elim #map_refl_assm <map_refl_assm
228      (* XXX: we assume the fetch_many hypothesis *)
229      * #fetch_refl #fetch_many_assm
230      (* XXX: we give the existential *)
231      %{1}
232      (* XXX: work on the left hand side of the equality *)
233      whd in ⊢ (??%?); whd in match (program_counter ???);
234      (* XXX: execute fetches some more *)
235      whd in match code_memory_of_pseudo_assembly_program; normalize nodelta
236      whd in fetch_many_assm;
237      >EQassembled in fetch_refl; #fetch_refl <fetch_refl
238      lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc
239      whd in ⊢ (??(???%)?); normalize in match (length ? (assembly1 …));
240      normalize in match (length ? (assembly1 …)) in EQnewpc;
241      normalize nodelta
242      [1:
243        @(pair_replace ????? carry new_sp ??? carry_new_sp_refl)
244        try /demod nohyps/ [1: %]
245        @pair_elim #pc_bu' #pc_bl' #pc_bu_bl_refl'
246        @(pair_replace ????? carry' new_sp' ??? carry_new_sp_refl')
247        try /demod nohyps/ [1: %]
248        @pair_elim #fiv #thr' #fiv_thr_refl' whd in ⊢ (??%%);
249        @split_eq_status try /demod nohyps/ try %
250        [1,2:
251          (* XXX: need the commented out axioms from AssemblyProof.ma *)
252          cases daemon
253        |3:
254          whd in ajc_refl:(??%?); lapply ajc_refl -ajc_refl -map_refl_assm -Mrefl
255          >EQlookup_labels normalize nodelta
256          @vsplit_elim #fst_5_addr #rest_addr #fst_5_rest_refl normalize nodelta
257          <EQnewpc @vsplit_elim #fst_5_pc #rest_pc #fst_5_rest_pc_refl normalize nodelta
258          #relevant >fst_5_rest_refl
259          lapply pc_bu_bl_refl' -pc_bu_bl_refl'
260          >program_counter_set_8051_sfr >set_clock_set_program_counter
261          >program_counter_set_program_counter >fst_5_rest_pc_refl
262          cut(fst_5_addr = fst_5_pc)
263          [1:
264            cases daemon (* XXX: !! *)
265          |2:
266            #relevant2 >relevant2
267            <(vsplit_ok ?????? (sym_eq … fiv_thr_refl'))
268            #relevant3
269            lapply(vsplit_ok ?????? (sym_eq … relevant3))
270            >(vector_associative_append bool 5 3 8) #relevant4
271            cut(fiv = fst_5_pc ∧ thr'@@pc_bl' = rest_pc)
272            [1:
273              cases daemon (* XXX: !! *)
274            |2:
275              * #relevant5 #relevant6 >relevant5
276              >(vector_associative_append bool 5 3 8) @eq_f
277              destruct(relevant)
278              <(vsplit_ok ?????? (sym_eq … thr_eig_refl)) @eq_f
279            ]
280          ]
281        |4:
282          >set_program_counter_mk_Status_commutation in ⊢ (???%);
283          >special_function_registers_8051_write_at_stack_pointer in ⊢ (???%);
284          (* XXX: require set_8051_sfr_write_at_stack_pointer and
285                          special_function_registers_8051_set_8051_sfr
286          *)
287          cases daemon
288        |5,6,7:
289          >set_program_counter_mk_Status_commutation in ⊢ (???%);
290          (* XXX: similar to above but for 8052
291          *)
292          cases daemon
293        |8:
294          whd in match (ticks_of_instruction ?);
295          cut(\snd  (fetch (load_code_memory (assembly1 (ACALL (ADDR11 offset)))) (zero 16)) = 2)
296          [1: //
297          |2:
298            normalize in match (assembly1 ?); >thr_eig_refl
299            @(bitvector_3_elim_prop … thr) %
300          |3:
301            #fetch_2_refl >fetch_2_refl >commutative_plus %
302          ]
303        |2:
304        ]
305      ]
306    ]
307  ]
308qed.
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