| 1 | include "ASM/AssemblyProofSplit.ma". |
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| 2 | include "common/LabelledObjects.ma". |
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| 3 | |
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| 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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| 25 | lemma absolute_jump_cond_ok: |
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| 26 | ∀v1, v2: Word. |
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| 27 | ∀is_possible, offset, v1_upper, v1_lower. |
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| 28 | 〈is_possible, offset〉 = absolute_jump_cond v1 v2 → |
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| 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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| 32 | whd in match absolute_jump_cond; normalize nodelta |
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| 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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| 40 | |
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| 41 | lemma ticks_of_instruction_AJMP: |
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| 42 | ∀address. ticks_of_instruction (AJMP (ADDR11 address)) = 2. |
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| 43 | try % #addr @(vsplit_elim … 3 8 addr) #vl #vm #EQ >EQ |
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| 44 | @(bitvector_3_elim_prop … vl) % |
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| 45 | qed. |
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| 46 | |
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| 47 | lemma ticks_of_instruction_ACALL: |
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| 48 | ∀address. ticks_of_instruction (ACALL (ADDR11 address)) = 2. |
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| 49 | try % #addr @(vsplit_elim … 3 8 addr) #vl #vm #EQ >EQ |
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| 50 | @(bitvector_3_elim_prop … vl) % |
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| 51 | qed. |
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| 52 | |
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| 53 | theorem main_thm: |
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| 54 | ∀M, M': internal_pseudo_address_map. |
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| 55 | ∀program: pseudo_assembly_program. |
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| 56 | ∀program_in_bounds: |\snd program| ≤ 2^16. |
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| 57 | let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in |
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| 58 | let addr_of ≝ λid.λ_.bitvector_of_nat 16 (lookup_def ASMTag ℕ labels id O) in |
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| 59 | ∀is_well_labelled: is_well_labelled_p (\snd program). |
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| 60 | ∀sigma: Word → Word. |
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| 61 | ∀policy: Word → bool. |
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| 62 | ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy. |
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| 63 | ∀ps: PseudoStatus program. |
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| 64 | ∀program_counter_in_bounds: nat_of_bitvector 16 (program_counter … ps) < |\snd program|. |
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| 65 | next_internal_pseudo_address_map M program addr_of ps program_counter_in_bounds = Some … M' → |
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| 66 | ∃n. execute n … (status_of_pseudo_status M … ps sigma policy) = |
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| 67 | status_of_pseudo_status M' … |
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| 68 | (execute_1_pseudo_instruction program (ticks_of program (λid. addr_of id ps) sigma policy) ps program_counter_in_bounds) sigma policy. |
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| 69 | #M #M' * #preamble #instr_list #program_in_bounds |
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| 70 | @pair_elim #labels #cost #create_label_cost_refl |
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| 71 | #is_well_labelled_witness #sigma #policy #sigma_policy_witness #ps |
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| 72 | letin ppc ≝ (program_counter pseudo_assembly_program ? ps) #ppc_in_bounds |
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| 73 | change with (next_internal_pseudo_address_map0 ?????? = ? → ?) |
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| 74 | 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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| 75 | >create_label_cost_refl normalize nodelta |
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| 76 | @pair_elim #assembled #costs' #assembly_refl normalize nodelta |
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| 77 | lapply (pair_destruct_1 ????? (sym_eq ??? assembly_refl)) #EQassembled |
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| 78 | @pair_elim #pi #newppc #fetch_pseudo_refl normalize nodelta |
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| 79 | @(pose … (λx. bitvector_of_nat ? (lookup_def … labels x 0))) #lookup_labels #EQlookup_labels |
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| 80 | @(pose … (λx. lookup_def … (construct_datalabels preamble) x (zero 16))) #lookup_datalabels #EQlookup_datalabels |
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| 81 | whd in match execute_1_pseudo_instruction; normalize nodelta |
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| 82 | whd in match ticks_of; normalize nodelta >fetch_pseudo_refl normalize nodelta |
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| 83 | lapply (snd_fetch_pseudo_instruction instr_list ppc ppc_in_bounds) >fetch_pseudo_refl #EQnewppc >EQnewppc |
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| 84 | lapply (snd_assembly_1_pseudoinstruction_ok 〈preamble,instr_list〉 … sigma policy sigma_policy_witness … ppc … pi … lookup_labels lookup_datalabels) |
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| 85 | [1: assumption |2: assumption] >create_label_cost_refl |
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| 86 | #X lapply (X EQlookup_labels EQlookup_datalabels ?) -X |
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| 87 | [1: >fetch_pseudo_refl %] |
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| 88 | #assm1 #assm2 #assm3 generalize in match assm2; generalize in match assm3; |
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| 89 | generalize in match assm1; -assm1 -assm2 -assm3 |
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| 90 | normalize nodelta |
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| 91 | inversion pi |
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| 92 | [2,3: |
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| 93 | #arg #_ |
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| 94 | (* XXX: we first work on sigma_increment_commutation *) |
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| 95 | #sigma_increment_commutation |
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| 96 | normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation; |
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| 97 | (* XXX: we work on the maps *) |
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| 98 | whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm |
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| 99 | (* XXX: we assume the fetch_many hypothesis *) |
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| 100 | #fetch_many_assm |
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| 101 | (* XXX: we give the existential *) |
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| 102 | %{0} |
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| 103 | whd in match execute_1_pseudo_instruction0; normalize nodelta |
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| 104 | change with (status_of_pseudo_status ????? = ?) |
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| 105 | whd in ⊢ (??%%); @split_eq_status try % |
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| 106 | [ >set_clock_set_program_counter ] |
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| 107 | >program_counter_set_program_counter >sigma_increment_commutation @add_zero |
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| 108 | |6: (* Mov *) |
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| 109 | #arg1 #arg2 #_ |
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| 110 | (* XXX: we first work on sigma_increment_commutation *) |
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| 111 | #sigma_increment_commutation |
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| 112 | normalize in match (assembly_1_pseudoinstruction ??????) in sigma_increment_commutation; |
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| 113 | (* XXX: we work on the maps *) |
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| 114 | whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm |
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| 115 | (* XXX: we assume the fetch_many hypothesis *) |
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| 116 | #fetch_many_assm |
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| 117 | (* XXX: we give the existential *) |
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| 118 | %{1} |
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| 119 | whd in match (execute_1_pseudo_instruction0 ?????); |
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| 120 | (* XXX: work on the left hand side of the equality *) |
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| 121 | whd in ⊢ (??%?); whd in match (program_counter ???); |
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| 122 | (* XXX: execute fetches some more *) |
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| 123 | whd in match code_memory_of_pseudo_assembly_program; normalize nodelta |
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| 124 | whd in fetch_many_assm; |
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| 125 | >EQassembled in fetch_many_assm; |
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| 126 | cases (fetch ??) * #instr #newpc #ticks normalize nodelta * |
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| 127 | #eq_instr |
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| 128 | #fetch_many_assm whd in fetch_many_assm; |
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| 129 | lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc |
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| 130 | destruct |
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| 131 | (* XXX: now we start to work on the mk_PreStatus equality *) |
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| 132 | (* XXX: lhs *) |
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| 133 | change with (set_arg_16 ????? = ?) |
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| 134 | @set_arg_16_status_of_pseudo_status |
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| 135 | [3: @(subaddressing_mode_elim … arg1) % |
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| 136 | |2: % |
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| 137 | | @sym_eq @set_clock_status_of_pseudo_status |
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| 138 | [ @sym_eq @set_program_counter_status_of_pseudo_status [<EQnewpc % | %] |
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| 139 | | % ]] |
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| 140 | |1: (* Instruction *) |
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| 141 | #instr #pi_refl |
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| 142 | change with (execute_1_preinstruction ???????) in match (execute_1_pseudo_instruction0 ?????); |
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| 143 | >EQassembled whd in match address_of_word_labels; normalize nodelta |
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| 144 | >create_label_cost_refl in ⊢ (? → ? → ? → %); |
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| 145 | @(main_lemma_preinstruction M M' preamble instr_list 〈preamble, instr_list〉 (refl …) ? sigma policy sigma_policy_witness |
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| 146 | ps ppc ? ? labels cost create_label_cost_refl newppc lookup_labels EQlookup_labels lookup_datalabels EQlookup_datalabels |
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| 147 | EQnewppc instr ? ? (refl …) ? (refl …) |
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| 148 | (set_program_counter pseudo_assembly_program 〈preamble, instr_list〉 ps (add 16 ppc (bitvector_of_nat 16 1))) |
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| 149 | (refl …) ? (refl …)) |
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| 150 | try % try assumption >fetch_pseudo_refl assumption |
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| 151 | |4: (* Jmp *) |
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| 152 | #arg1 #pi_refl |
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| 153 | (* XXX: we first work on sigma_increment_commutation *) |
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| 154 | whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?); |
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| 155 | whd in match (expand_pseudo_instruction ??????); |
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| 156 | whd in match (ticks_of0 ??????); |
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| 157 | inversion (short_jump_cond ??) #sj_possible #offset #sjc_refl normalize nodelta |
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| 158 | inversion (sj_possible ∧ ¬ policy ?) #sj_possible_refl normalize nodelta |
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| 159 | [2: |
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| 160 | inversion (absolute_jump_cond ??) #mj_possible #address #mjc_refl normalize nodelta |
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| 161 | inversion (mj_possible ∧ ¬ policy ?) #mj_possible_refl normalize nodelta |
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| 162 | ] |
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| 163 | #sigma_increment_commutation |
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| 164 | normalize in sigma_increment_commutation:(???(???(??%))); |
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| 165 | (* XXX: we work on the maps *) |
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| 166 | whd in ⊢ (??%? → ?); @Some_Some_elim #map_refl_assm <map_refl_assm |
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| 167 | (* XXX: we assume the fetch_many hypothesis *) |
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| 168 | * #fetch_refl #fetch_many_assm |
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| 169 | (* XXX: we give the existential *) |
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| 170 | %{1} |
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| 171 | (* XXX: work on the left hand side of the equality *) |
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| 172 | whd in ⊢ (??%?); whd in match (program_counter ???); |
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| 173 | (* XXX: execute fetches some more *) |
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| 174 | whd in match code_memory_of_pseudo_assembly_program; normalize nodelta |
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| 175 | whd in fetch_many_assm; |
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| 176 | >EQassembled in fetch_refl; #fetch_refl <fetch_refl |
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| 177 | lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc |
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| 178 | change with (set_program_counter ???? = |
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| 179 | status_of_pseudo_status ?? (set_program_counter ????) ??) |
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| 180 | @set_program_counter_status_of_pseudo_status |
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| 181 | [2,4,6: @sym_eq @set_clock_status_of_pseudo_status try % |
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| 182 | [1,3,4: @sym_eq @set_program_counter_status_of_pseudo_status try % |
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| 183 | @sigma_increment_commutation |
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| 184 | | @eq_f2 try % @ticks_of_instruction_AJMP ]] |
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| 185 | whd in ⊢ (??%%); normalize nodelta whd in match address_of_word_labels; |
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| 186 | normalize nodelta |
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| 187 | [ inversion (vsplit bool ???) #pc_bu #pc_bl #vsplit_refl normalize nodelta |
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| 188 | >EQlookup_labels in mjc_refl; >create_label_cost_refl #mjc_refl |
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| 189 | @(absolute_jump_cond_ok ????? pc_bl (sym_eq … mjc_refl)) |
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| 190 | [2: >(andb_true_l … mj_possible_refl) % |
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| 191 | | <vsplit_refl @eq_f <EQnewpc % ] |
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| 192 | | >EQlookup_labels >create_label_cost_refl % |
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| 193 | | inversion (half_add ???) #carry #new_pc #half_add_refl normalize nodelta |
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| 194 | >create_label_cost_refl |
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| 195 | >EQlookup_labels in sjc_refl; #sjc_refl |
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| 196 | >(pair_destruct_2 ????? (sym_eq … half_add_refl)) |
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| 197 | >(short_jump_cond_ok ???? (sym_eq … sjc_refl)) |
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| 198 | [2: >(andb_true_l … sj_possible_refl) % |
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| 199 | | >EQnewpc % ]] |
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| 200 | |5: (* Call *) |
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| 201 | #arg1 #pi_refl |
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| 202 | (* XXX: we first work on sigma_increment_commutation *) |
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| 203 | whd in match (assembly_1_pseudoinstruction ??????) in ⊢ (% → ?); |
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| 204 | whd in match (expand_pseudo_instruction ??????); |
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| 205 | whd in match (execute_1_pseudo_instruction0 ?????); |
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| 206 | whd in match (ticks_of0 ??????); |
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| 207 | inversion (absolute_jump_cond ??) #aj_possible #offset #ajc_refl normalize nodelta |
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| 208 | inversion (aj_possible ∧ ¬ policy ?) #aj_possible_refl normalize nodelta |
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| 209 | @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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| 210 | @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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| 211 | @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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| 212 | #sigma_increment_commutation |
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| 213 | normalize in sigma_increment_commutation:(???(???(??%))); |
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| 214 | (* XXX: we work on the maps *) |
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| 215 | whd in ⊢ (??%? → ?); |
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| 216 | @pair_elim #callM #accM #Mrefl |
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| 217 | @Some_Some_elim #map_refl_assm <map_refl_assm |
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| 218 | (* XXX: we assume the fetch_many hypothesis *) |
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| 219 | * #fetch_refl #fetch_many_assm |
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| 220 | (* XXX: we give the existential *) |
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| 221 | %{1} |
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| 222 | (* XXX: work on the left hand side of the equality *) |
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| 223 | whd in ⊢ (??%?); whd in match (program_counter ???); |
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| 224 | (* XXX: execute fetches some more *) |
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| 225 | whd in match code_memory_of_pseudo_assembly_program; normalize nodelta |
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| 226 | whd in fetch_many_assm; |
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| 227 | >EQassembled in fetch_refl; #fetch_refl <fetch_refl |
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| 228 | lapply (eq_bv_eq … fetch_many_assm) -fetch_many_assm #EQnewpc |
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| 229 | whd in ⊢ (??%?); |
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| 230 | [1: |
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| 231 | @(pair_replace ????? carry new_sp ??? carry_new_sp_refl) |
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| 232 | [ @eq_f2 try % @sym_eq |
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| 233 | @(pose … (set_clock ????)) #status #status_refl @sym_eq >status_refl |
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| 234 | /demod nohyps/ |
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| 235 | (*CSC: mess with get_8051_sfr_set_program_counter + missing high level lemmas*) |
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| 236 | cut (∀A,B:Type[0]. ∀f,g:A → B. ∀a:A. f=g → f a = g a) [#A #B #f #f #a * %] #eq_fun |
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| 237 | >(eq_fun ???? ? (get_8051_sfr_set_program_counter (BitVectorTrie Byte 16) … SFR_SP …)) |
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| 238 | >(eq_fun ???? ? (get_8051_sfr_set_program_counter pseudo_assembly_program … SFR_SP …)) |
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| 239 | whd in match get_8051_sfr; normalize nodelta whd in match status_of_pseudo_status; |
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| 240 | normalize nodelta whd in match sfr_8051_of_pseudo_sfr_8051; normalize nodelta |
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| 241 | cases accM try % normalize nodelta #ul #addr cases (vsplit bool ???) |
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| 242 | normalize nodelta #v1 #v2 cases (eq_upper_lower ul upper) normalize nodelta |
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| 243 | >get_index_v_set_index_miss try % #abs normalize in abs; destruct(abs) |
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| 244 | ] whd in ⊢ (??%?); |
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| 245 | @pair_elim #pc_bu' #pc_bl' #pc_bu_bl_refl' |
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| 246 | @(pair_replace ????? ?? ??? carry_new_sp_refl') |
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| 247 | [ @eq_f2 try % @sym_eq |
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| 248 | @(pose … (write_at_stack_pointer ????)) #status #status_refl @sym_eq |
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| 249 | @(get_8051_sfr_status_of_pseudo_status … 〈callM,accM〉 … status) >status_refl -status_refl try % |
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| 250 | @sym_eq cases daemon (*CSC: write_at_stack_pointer_status_of_pseudo_status*) |
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| 251 | ] |
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| 252 | whd in ⊢ (??%?); |
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| 253 | @pair_elim #fiv #thr' #fiv_thr_refl' |
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| 254 | change with (set_program_counter ???? = ?) |
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| 255 | @set_program_counter_status_of_pseudo_status |
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| 256 | [2: @sym_eq cases daemon (*CSC: missing @write_at_stack_pointer_status_of_pseudo_status try % |
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| 257 | [1,3,4: @sym_eq @set_program_counter_status_of_pseudo_status try % |
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| 258 | @sigma_increment_commutation |
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| 259 | | @eq_f2 try % @ticks_of_instruction_AJMP ]*)] |
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| 260 | whd in ajc_refl:(??%?); lapply ajc_refl -ajc_refl -map_refl_assm -Mrefl |
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| 261 | >EQlookup_labels normalize nodelta |
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| 262 | @vsplit_elim #fst_5_addr #rest_addr #fst_5_rest_refl |
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| 263 | >fst_5_rest_refl normalize nodelta |
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| 264 | @vsplit_elim #fst_5_pc #rest_pc #fst_5_rest_pc_refl normalize nodelta |
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| 265 | #pair_true_refl destruct(pair_true_refl) |
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| 266 | <EQnewpc in fst_5_rest_pc_refl; |
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| 267 | lapply pc_bu_bl_refl' -pc_bu_bl_refl' |
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| 268 | >program_counter_set_8051_sfr >set_clock_set_program_counter |
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| 269 | >program_counter_set_program_counter #relevant2 |
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| 270 | <(vsplit_ok ?????? (sym_eq … relevant2)) |
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| 271 | <(vsplit_ok ?????? (sym_eq … fiv_thr_refl')) |
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| 272 | >(vector_associative_append bool 5 3 8) #relevant3 |
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| 273 | >(? : fiv = fst_5_addr) |
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| 274 | [1: <fst_5_rest_refl whd in match address_of_word_labels; normalize nodelta |
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| 275 | >create_label_cost_refl % |
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| 276 | |2: cases (bv_append_eq_to_eq … relevant3) #H #_ >H |
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| 277 | cases (conjunction_true … aj_possible_refl) #K #_ @sym_eq |
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| 278 | @eq_bv_eq assumption ] |
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| 279 | | @(pair_replace ????? carry new_sp ??? carry_new_sp_refl) |
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| 280 | [ @eq_f2 try % |
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| 281 | @sym_eq @(pose … (set_clock ????)) #status #status_refl @sym_eq |
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| 282 | @(get_8051_sfr_status_of_pseudo_status … 〈callM,accM〉 … status) |
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| 283 | >status_refl -status_refl try % |
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| 284 | @sym_eq @set_clock_status_of_pseudo_status try % |
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| 285 | @sym_eq @set_program_counter_status_of_pseudo_status try % assumption ] |
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| 286 | @pair_elim #pc_bu' #pc_bl' #pc_bu_bl_refl' |
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| 287 | @(pair_replace ????? carry' new_sp' ??? carry_new_sp_refl') |
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| 288 | [ @eq_f2 try % @sym_eq |
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| 289 | @(pose … (write_at_stack_pointer ????)) #status #status_refl @sym_eq |
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| 290 | @(get_8051_sfr_status_of_pseudo_status … 〈callM,accM〉 … status) >status_refl -status_refl try % |
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| 291 | @sym_eq cases daemon (*CSC: write_at_stack_pointer_status_of_pseudo_status*) ] |
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| 292 | change with (set_program_counter ???? = ?) |
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| 293 | @set_program_counter_status_of_pseudo_status |
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| 294 | [ >EQlookup_labels whd in match address_of_word_labels; normalize nodelta |
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| 295 | >create_label_cost_refl % ] |
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| 296 | cut(pc_bu' @@ pc_bl' = sigma (pc_bu @@ pc_bl)) |
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| 297 | [1: |
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| 298 | >(vsplit_ok ?????? (sym_eq … pc_bu_bl_refl')) |
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| 299 | >(vsplit_ok ?????? (sym_eq … pc_bu_bl_refl)) |
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| 300 | >add_commutative |
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| 301 | >program_counter_set_8051_sfr >set_clock_set_program_counter |
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| 302 | >program_counter_set_program_counter >add_commutative |
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| 303 | >program_counter_set_8051_sfr >set_clock_set_program_counter |
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| 304 | >program_counter_set_program_counter assumption ] #sigma_pc_bu_pc_bl_refl |
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| 305 | @sym_eq (*CSC: write_at_stack_pointer_status_of_pseudo_status*) |
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| 306 | cases daemon |
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| 307 | ] |
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| 308 | ] |
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| 309 | qed. |
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