[2027] | 1 | include "ASM/AssemblyProofSplit.ma". |
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[2047] | 2 | include "common/LabelledObjects.ma". |
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[2026] | 3 | |
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[2051] | 4 | lemma short_jump_cond_ok: |
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| 5 | ∀v1, v2: Word. |
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| 6 | ∀is_possible, offset. |
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| 7 | 〈is_possible, offset〉 = short_jump_cond v1 v2 → |
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| 8 | is_possible → v2 = add 16 v1 (sign_extension offset). |
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| 9 | #v1 #v2 #is_possible #offset |
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| 10 | whd in match short_jump_cond; normalize nodelta |
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| 11 | @pair_elim #result #flags #sub16_refl |
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| 12 | @pair_elim #upper #lower #vsplit_refl |
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| 13 | inversion (get_index' bool ???) #get_index_refl normalize nodelta |
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| 14 | #relevant destruct(relevant) #relevant |
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| 15 | [1: |
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| 16 | @(sub_16_to_add_16_8_1 … flags) |
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| 17 | |2: |
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| 18 | @(sub_16_to_add_16_8_0 … flags) |
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| 19 | ] |
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| 20 | try assumption >sub16_refl |
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| 21 | <(eq_bv_eq … relevant) |
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| 22 | >(vsplit_ok … (sym_eq … vsplit_refl)) % |
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| 23 | qed. |
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| 24 | |
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[2110] | 25 | lemma absolute_jump_cond_ok: |
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[2051] | 26 | ∀v1, v2: Word. |
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| 27 | ∀is_possible, offset, v1_upper, v1_lower. |
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[2110] | 28 | 〈is_possible, offset〉 = absolute_jump_cond v1 v2 → |
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[2051] | 29 | 〈v1_upper, v1_lower〉 = vsplit ? 5 11 v1 → |
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| 30 | is_possible → v2 = v1_upper @@ offset. |
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| 31 | #v1 #v2 #is_possible #offset #v1_upper #v1_lower |
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[2110] | 32 | whd in match absolute_jump_cond; normalize nodelta |
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[2051] | 33 | @pair_elim #fst_5_addr #rest_addr #vsplit_v2_refl |
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| 34 | @pair_elim #fst_5_pc #rest_pc #vsplit_v1_refl |
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| 35 | #relevant destruct(relevant) normalize nodelta #relevant |
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| 36 | destruct(relevant) #relevant |
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| 37 | <(vsplit_ok … (sym_eq … vsplit_v2_refl)) |
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| 38 | >(eq_bv_eq … relevant) % |
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| 39 | qed. |
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[2129] | 40 | |
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[2026] | 41 | theorem main_thm: |
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| 42 | ∀M, M': internal_pseudo_address_map. |
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| 43 | ∀program: pseudo_assembly_program. |
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[2129] | 44 | ∀program_in_bounds: |\snd program| ≤ 2^16. |
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| 45 | ∀addr_of: Identifier → PreStatus pseudo_assembly_program program → BitVector 16. |
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[2047] | 46 | ∀is_well_labelled: is_well_labelled_p (\snd program). |
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[2026] | 47 | ∀sigma: Word → Word. |
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| 48 | ∀policy: Word → bool. |
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| 49 | ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy. |
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| 50 | ∀ps: PseudoStatus program. |
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[2062] | 51 | ∀program_counter_in_bounds: nat_of_bitvector 16 (program_counter … ps) < |\snd program|. |
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[2129] | 52 | next_internal_pseudo_address_map M program addr_of ps program_counter_in_bounds = Some … M' → |
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[2026] | 53 | ∃n. execute n … (status_of_pseudo_status M … ps sigma policy) = |
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| 54 | status_of_pseudo_status M' … |
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[2062] | 55 | (execute_1_pseudo_instruction program (ticks_of program sigma policy) ps program_counter_in_bounds) sigma policy. |
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[2129] | 56 | #M #M' * #preamble #instr_list #program_in_bounds #addr_of |
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| 57 | #is_well_labelled_witness #sigma #policy #sigma_policy_witness #ps |
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| 58 | letin ppc ≝ (program_counter pseudo_assembly_program ? ps) #ppc_in_bounds |
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| 59 | change with (next_internal_pseudo_address_map0 ?????? = ? → ?) |
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| 60 | generalize in match (fetch_assembly_pseudo2 〈preamble,instr_list〉 program_in_bounds sigma policy sigma_policy_witness ppc ppc_in_bounds) in ⊢ ?; |
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| 61 | @pair_elim #labels #costs #create_label_cost_refl normalize nodelta |
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| 62 | @pair_elim #assembled #costs' #assembly_refl normalize nodelta |
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| 63 | lapply (pair_destruct_1 ????? (sym_eq ??? assembly_refl)) #EQassembled |
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| 64 | @pair_elim #pi #newppc #fetch_pseudo_refl normalize nodelta |
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[2131] | 65 | @(pose … (λx. bitvector_of_nat ? (lookup_def … labels x 0))) #lookup_labels #EQlookup_labels |
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[2129] | 66 | @(pose … (λx. lookup_def … (construct_datalabels preamble) x (zero 16))) #lookup_datalabels #EQlookup_datalabels |
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| 67 | whd in match execute_1_pseudo_instruction; normalize nodelta |
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| 68 | whd in match ticks_of; normalize nodelta >fetch_pseudo_refl normalize nodelta |
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| 69 | lapply (snd_fetch_pseudo_instruction instr_list ppc ppc_in_bounds) >fetch_pseudo_refl #EQnewppc >EQnewppc |
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[2131] | 70 | lapply (snd_assembly_1_pseudoinstruction_ok 〈preamble,instr_list〉 … sigma policy sigma_policy_witness … ppc … pi … lookup_labels lookup_datalabels) |
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| 71 | [1: assumption |2: assumption] >create_label_cost_refl |
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| 72 | #X lapply (X EQlookup_labels EQlookup_datalabels ?) -X |
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| 73 | [1: >fetch_pseudo_refl %] |
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[2129] | 74 | #assm1 #assm2 #assm3 generalize in match assm2; generalize in match assm3; |
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| 75 | generalize in match assm1; -assm1 -assm2 -assm3 |
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| 76 | normalize nodelta |
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| 77 | inversion pi |
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[2131] | 78 | [2,3: |
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[2129] | 79 | #arg #_ |
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| 80 | (* XXX: we first work on sigma_increment_commutation *) |
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| 81 | #sigma_increment_commutation |
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| 82 | normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation; |
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| 83 | (* XXX: we work on the maps *) |
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| 84 | whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm |
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| 85 | (* XXX: we assume the fetch_many hypothesis *) |
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| 86 | #fetch_many_assm |
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| 87 | (* XXX: we give the existential *) |
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| 88 | %{0} |
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| 89 | whd in match (execute_1_pseudo_instruction0 ?????); |
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| 90 | (* XXX: work on the left hand side of the equality *) |
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| 91 | whd in ⊢ (??%?); |
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| 92 | @split_eq_status try % |
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| 93 | /demod/ >add_commutative <add_zero % (*CSC: auto not working, why? *) |
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| 94 | |6: (* Mov *) |
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| 95 | #arg1 #arg2 #_ |
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| 96 | (* XXX: we first work on sigma_increment_commutation *) |
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| 97 | #sigma_increment_commutation |
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| 98 | normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation; |
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| 99 | (* XXX: we work on the maps *) |
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| 100 | whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm |
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| 101 | (* XXX: we assume the fetch_many hypothesis *) |
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| 102 | #fetch_many_assm |
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| 103 | (* XXX: we give the existential *) |
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| 104 | %{1} |
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| 105 | whd in match (execute_1_pseudo_instruction0 ?????); |
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| 106 | (* XXX: work on the left hand side of the equality *) |
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| 107 | whd in ⊢ (??%?); whd in match (program_counter ???); |
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| 108 | (* XXX: execute fetches some more *) |
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| 109 | whd in match code_memory_of_pseudo_assembly_program; normalize nodelta |
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| 110 | whd in fetch_many_assm; |
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| 111 | >EQassembled in fetch_many_assm; |
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| 112 | cases (fetch ??) * #instr #newpc #ticks normalize nodelta * |
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| 113 | #eq_instr |
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| 114 | #fetch_many_assm whd in fetch_many_assm; |
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| 115 | lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc |
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| 116 | destruct whd in ⊢ (??%?); |
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| 117 | (* XXX: now we start to work on the mk_PreStatus equality *) |
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| 118 | (* XXX: lhs *) |
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| 119 | change with (set_arg_16 ????? = ?) |
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| 120 | >set_program_counter_mk_Status_commutation |
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| 121 | >set_clock_mk_Status_commutation |
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| 122 | >set_arg_16_mk_Status_commutation |
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| 123 | (* XXX: rhs *) |
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| 124 | change with (status_of_pseudo_status ?? (set_arg_16 pseudo_assembly_program 〈preamble, instr_list〉 ?? arg1) ??) in ⊢ (???%); |
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| 125 | >set_program_counter_mk_Status_commutation |
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| 126 | >set_clock_mk_Status_commutation |
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| 127 | >set_arg_16_mk_Status_commutation in ⊢ (???%); |
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| 128 | (* here we are *) |
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| 129 | @split_eq_status try % |
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[2047] | 130 | [1: |
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[2129] | 131 | assumption |
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| 132 | |2: |
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| 133 | @special_function_registers_8051_set_arg_16 % |
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| 134 | ] |
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| 135 | |1: (* Instruction *) |
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[2139] | 136 | #instr #pi_refl #EQP #EQnext #fetch_many_assm |
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| 137 | @(main_lemma_preinstruction M M' preamble instr_list 〈preamble, instr_list〉 (refl …) ? sigma policy sigma_policy_witness |
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| 138 | ps ppc ? ? labels costs create_label_cost_refl newppc lookup_labels EQlookup_labels lookup_datalabels EQlookup_datalabels |
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| 139 | EQnewppc instr ? (ticks_of0 〈preamble, instr_list〉 sigma policy ppc (Instruction instr)) (refl …) |
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[2129] | 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))) |
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| 141 | (refl …) ? (refl …)) |
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[2139] | 142 | [1,2: |
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| 143 | assumption |
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| 144 | |3: |
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| 145 | % |
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| 146 | |4: |
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| 147 | >fetch_pseudo_refl @pi_refl |
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| 148 | |5: |
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| 149 | <EQP % |
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| 150 | |6: |
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| 151 | >assembly_refl assumption |
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[2140] | 152 | |7: |
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| 153 | <EQassembled assumption |
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[2139] | 154 | ] |
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[2129] | 155 | |4: (* Jmp *) |
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| 156 | #arg1 #pi_refl |
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| 157 | (* XXX: we first work on sigma_increment_commutation *) |
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| 158 | whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?); |
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| 159 | whd in match (expand_pseudo_instruction ??????); |
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| 160 | inversion (short_jump_cond ??) #sj_possible #offset #sjc_refl normalize nodelta |
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| 161 | inversion (sj_possible ∧ ¬ policy ?) #sj_possible_refl normalize nodelta |
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| 162 | [2: |
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| 163 | inversion (absolute_jump_cond ??) #mj_possible #address #mjc_refl normalize nodelta |
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| 164 | inversion (mj_possible ∧ ¬ policy ?) #mj_possible_refl normalize nodelta |
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| 165 | ] |
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| 166 | #sigma_increment_commutation |
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| 167 | normalize in sigma_increment_commutation:(???(???(??%))); |
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| 168 | (* XXX: we work on the maps *) |
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| 169 | whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm |
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| 170 | (* XXX: we assume the fetch_many hypothesis *) |
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| 171 | * #fetch_refl #fetch_many_assm |
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| 172 | (* XXX: we give the existential *) |
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| 173 | %{1} |
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| 174 | (* XXX: work on the left hand side of the equality *) |
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| 175 | whd in ⊢ (??%?); whd in match (program_counter ???); |
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| 176 | (* XXX: execute fetches some more *) |
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| 177 | whd in match code_memory_of_pseudo_assembly_program; normalize nodelta |
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| 178 | whd in fetch_many_assm; |
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| 179 | >EQassembled in fetch_refl; #fetch_refl <fetch_refl |
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| 180 | lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc |
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| 181 | whd in ⊢ (??%%); |
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| 182 | (* XXX: now we start to work on the mk_PreStatus equality *) |
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| 183 | (* XXX: lhs *) |
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| 184 | (* XXX: rhs *) |
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| 185 | (* here we are *) |
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| 186 | @split_eq_status try % /demod nohyps/ |
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| 187 | [1,3,4: |
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| 188 | change with (add ???) in match (\snd (half_add ???)); |
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| 189 | whd in match execute_1_pseudo_instruction0; normalize nodelta |
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| 190 | /demod nohyps/ >set_clock_set_program_counter |
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| 191 | >program_counter_set_program_counter |
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| 192 | whd in ⊢ (??%?); normalize nodelta whd in match address_of_word_labels; normalize nodelta |
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| 193 | lapply (create_label_cost_map_ok 〈preamble, instr_list〉) >create_label_cost_refl |
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| 194 | normalize nodelta #address_of_word_labels_assm <address_of_word_labels_assm |
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[2051] | 195 | [1: |
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[2129] | 196 | >EQnewpc |
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| 197 | inversion (vsplit bool ???) #pc_bu #pc_bl #vsplit_refl normalize nodelta |
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[2131] | 198 | @sym_eq |
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| 199 | >EQlookup_labels in mjc_refl; #mjc_refl |
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| 200 | @(absolute_jump_cond_ok ?????? (sym_eq … mjc_refl) (sym_eq … vsplit_refl)) |
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| 201 | >(andb_true_l … mj_possible_refl) @I |
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[2129] | 202 | |3: |
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[2131] | 203 | >EQlookup_labels normalize nodelta % |
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[2129] | 204 | |5: |
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| 205 | >EQnewpc |
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| 206 | inversion (half_add ???) #carry #new_pc #half_add_refl normalize nodelta |
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[2131] | 207 | @sym_eq |
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| 208 | >EQlookup_labels in sjc_refl; #sjc_refl |
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| 209 | >(pair_destruct_2 ????? (sym_eq … half_add_refl)) |
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| 210 | @(short_jump_cond_ok ???? (sym_eq … sjc_refl)) |
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| 211 | >(andb_true_l … sj_possible_refl) @I |
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[2051] | 212 | ] |
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[2139] | 213 | whd in is_well_labelled_witness; |
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| 214 | @(is_well_labelled_witness ? ppc ppc_in_bounds pi) |
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| 215 | try (>fetch_pseudo_refl %) |
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[2129] | 216 | >pi_refl % |
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| 217 | |2: |
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| 218 | whd in ⊢ (??(?%%)%); |
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| 219 | cases (bitvector_11_cases address) |
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| 220 | * * * * * * #b4 * #b5 |
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| 221 | * #b6 * #b7 * #b8 * #b9 * #b10 * #b11 #address_refl >address_refl |
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| 222 | normalize in match (fetch ??); <plus_n_Sm @eq_f |
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| 223 | @commutative_plus |
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| 224 | ] |
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[2131] | 225 | |5: (* Call *) |
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[2129] | 226 | #arg1 #pi_refl |
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| 227 | (* XXX: we first work on sigma_increment_commutation *) |
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| 228 | whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?); |
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| 229 | whd in match (expand_pseudo_instruction ??????); |
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| 230 | whd in match (execute_1_pseudo_instruction0 ?????); |
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| 231 | inversion (absolute_jump_cond ??) #aj_possible #offset #ajc_refl normalize nodelta |
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| 232 | inversion (aj_possible ∧ ¬ policy ?) #aj_possible_refl normalize nodelta |
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| 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 |
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| 234 | @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 |
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| 235 | @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' |
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| 236 | #sigma_increment_commutation |
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| 237 | normalize in sigma_increment_commutation:(???(???(??%))); |
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| 238 | (* XXX: we work on the maps *) |
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| 239 | whd in ⊢ (??%? → ?); |
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| 240 | @pair_elim #callM #accM #Mrefl |
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| 241 | @Some_Some_elim #map_refl_assm <map_refl_assm |
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| 242 | (* XXX: we assume the fetch_many hypothesis *) |
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| 243 | * #fetch_refl #fetch_many_assm |
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| 244 | (* XXX: we give the existential *) |
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| 245 | %{1} |
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| 246 | (* XXX: work on the left hand side of the equality *) |
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| 247 | whd in ⊢ (??%?); whd in match (program_counter ???); |
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| 248 | (* XXX: execute fetches some more *) |
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| 249 | whd in match code_memory_of_pseudo_assembly_program; normalize nodelta |
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| 250 | whd in fetch_many_assm; |
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| 251 | >EQassembled in fetch_refl; #fetch_refl <fetch_refl |
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| 252 | lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc |
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| 253 | whd in ⊢ (??(???%)?); normalize in match (length ? (assembly1 …)); |
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| 254 | normalize in match (length ? (assembly1 …)) in EQnewpc; |
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| 255 | normalize nodelta |
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| 256 | [1: |
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| 257 | @(pair_replace ????? carry new_sp ??? carry_new_sp_refl) |
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| 258 | try /demod nohyps/ [1: %] |
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| 259 | @pair_elim #pc_bu' #pc_bl' #pc_bu_bl_refl' |
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| 260 | @(pair_replace ????? carry' new_sp' ??? carry_new_sp_refl') |
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| 261 | try /demod nohyps/ [1: %] |
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| 262 | @pair_elim #fiv #thr' #fiv_thr_refl' whd in ⊢ (??%%); |
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| 263 | @split_eq_status try /demod nohyps/ try % |
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| 264 | [1,2: |
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| 265 | (* XXX: need the commented out axioms from AssemblyProof.ma *) |
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| 266 | cases daemon |
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| 267 | |3: |
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[2140] | 268 | lapply (create_label_cost_map_ok 〈preamble, instr_list〉) >create_label_cost_refl normalize nodelta |
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| 269 | change with (address_of_word_labels_code_mem ??) in match (address_of_word_labels ??); |
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| 270 | #relevant <(relevant arg1) |
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| 271 | [2: |
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| 272 | @(is_well_labelled_witness … ppc_in_bounds pi) |
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| 273 | [1: |
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| 274 | >fetch_pseudo_refl % |
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| 275 | |2: |
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| 276 | >pi_refl % |
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| 277 | ] |
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| 278 | |1: |
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| 279 | whd in ajc_refl:(??%?); lapply ajc_refl -ajc_refl -map_refl_assm -Mrefl |
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| 280 | >EQlookup_labels normalize nodelta |
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| 281 | @vsplit_elim #fst_5_addr #rest_addr #fst_5_rest_refl |
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| 282 | >fst_5_rest_refl normalize nodelta |
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| 283 | @vsplit_elim #fst_5_pc #rest_pc #fst_5_rest_pc_refl normalize nodelta |
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| 284 | #pair_true_refl destruct(pair_true_refl) |
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| 285 | <EQnewpc in fst_5_rest_pc_refl; |
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| 286 | lapply pc_bu_bl_refl' -pc_bu_bl_refl' |
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| 287 | >program_counter_set_8051_sfr >set_clock_set_program_counter |
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| 288 | >program_counter_set_program_counter #relevant2 |
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| 289 | <(vsplit_ok ?????? (sym_eq … relevant2)) |
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[2129] | 290 | <(vsplit_ok ?????? (sym_eq … fiv_thr_refl')) |
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[2140] | 291 | >(vector_associative_append bool 5 3 8) #relevant3 |
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| 292 | >(? : fiv = fst_5_addr) |
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[2129] | 293 | [1: |
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[2140] | 294 | % |
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[2129] | 295 | |2: |
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[2140] | 296 | cases daemon |
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[2129] | 297 | ] |
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[2140] | 298 | ] |
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[2129] | 299 | |4: |
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| 300 | >set_program_counter_mk_Status_commutation in ⊢ (???%); |
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| 301 | >special_function_registers_8051_write_at_stack_pointer in ⊢ (???%); |
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| 302 | (* XXX: require set_8051_sfr_write_at_stack_pointer and |
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| 303 | special_function_registers_8051_set_8051_sfr |
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| 304 | *) |
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| 305 | cases daemon |
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| 306 | |5,6,7: |
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| 307 | >set_program_counter_mk_Status_commutation in ⊢ (???%); |
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| 308 | (* XXX: similar to above but for 8052 |
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| 309 | *) |
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| 310 | cases daemon |
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| 311 | |8: |
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| 312 | whd in match (ticks_of_instruction ?); |
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| 313 | cut(\snd (fetch (load_code_memory (assembly1 (ACALL (ADDR11 offset)))) (zero 16)) = 2) |
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| 314 | [1: // |
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| 315 | |2: |
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[2140] | 316 | normalize in match (assembly1 ?); |
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| 317 | @(bitvector_3_elim_prop … (\fst (vsplit bool 3 8 offset))) % |
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[2129] | 318 | |3: |
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| 319 | #fetch_2_refl >fetch_2_refl >commutative_plus % |
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| 320 | ] |
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[2140] | 321 | ] |
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| 322 | |2: |
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| 323 | @(pair_replace ????? carry new_sp ??? carry_new_sp_refl) |
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| 324 | try /demod nohyps/ [1: %] |
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| 325 | @pair_elim #pc_bu' #pc_bl' #pc_bu_bl_refl' |
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| 326 | @(pair_replace ????? carry' new_sp' ??? carry_new_sp_refl') |
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| 327 | try /demod nohyps/ [1: %] |
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| 328 | @split_eq_status try /demod nohyps/ try % |
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| 329 | [1: |
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| 330 | cut(pc_bu' @@ pc_bl' = sigma (pc_bu @@ pc_bl)) |
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| 331 | [1: |
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| 332 | >(vsplit_ok ?????? (sym_eq … pc_bu_bl_refl')) /demod nohyps/ |
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| 333 | >set_clock_set_program_counter >program_counter_set_program_counter |
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| 334 | >add_commutative >EQnewpc @eq_f @sym_eq @(vsplit_ok ?????? (sym_eq … pc_bu_bl_refl)) |
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| 335 | |2: |
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| 336 | #relevant cases daemon |
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| 337 | (* XXX: need axioms *) |
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| 338 | ] |
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[2129] | 339 | |2: |
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[2140] | 340 | cases daemon (* XXX: need axioms from assembly.ma *) |
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| 341 | |3: |
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| 342 | change with (address_of_word_labels_code_mem ??) in match (address_of_word_labels ??); |
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| 343 | lapply (create_label_cost_map_ok 〈preamble, instr_list〉) >create_label_cost_refl normalize nodelta |
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| 344 | #relevant <(relevant arg1) |
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| 345 | [1: |
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| 346 | >EQlookup_labels % |
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| 347 | |2: |
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| 348 | @(is_well_labelled_witness … ppc_in_bounds pi) |
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| 349 | [1: |
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| 350 | >fetch_pseudo_refl % |
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| 351 | |2: |
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| 352 | >pi_refl % |
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| 353 | ] |
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| 354 | ] |
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| 355 | |4: |
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| 356 | >set_program_counter_mk_Status_commutation in ⊢ (???%); |
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| 357 | >special_function_registers_8051_write_at_stack_pointer in ⊢ (???%); |
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| 358 | (* XXX: require set_8051_sfr_write_at_stack_pointer and |
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| 359 | special_function_registers_8051_set_8051_sfr |
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| 360 | *) |
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| 361 | cases daemon |
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| 362 | |5,6,7: |
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| 363 | >set_program_counter_mk_Status_commutation in ⊢ (???%); |
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| 364 | (* XXX: similar to above but for 8052 |
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| 365 | *) |
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| 366 | cases daemon |
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[2051] | 367 | ] |
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[2047] | 368 | ] |
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| 369 | ] |
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[2110] | 370 | qed. |
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