[1996] | 1 | |
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| 2 | include "compiler.ma". |
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| 3 | |
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| 4 | include "ASM/Interpret2.ma". |
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| 5 | |
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[2150] | 6 | include "Clight/labelSimulation.ma". |
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| 7 | |
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| 8 | theorem correct : |
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[2852] | 9 | ∀observe,input_program,output. |
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[2774] | 10 | (* ∀lobject_code,labelled,cost_map. *) |
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[2852] | 11 | compile observe input_program = return output → |
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[2325] | 12 | |
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| 13 | not_wrong … (exec_inf … clight_fullexec input_program) → |
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[2150] | 14 | |
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[2774] | 15 | sim_with_labels |
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| 16 | (exec_inf … clight_fullexec input_program) |
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| 17 | (exec_inf … clight_fullexec (c_labelled_clight … output)) |
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[2150] | 18 | ∧ |
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| 19 | True (* TODO *). |
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| 20 | |
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[2852] | 21 | #observe #input_program #output |
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[2150] | 22 | #COMPILE |
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[2325] | 23 | #NOT_WRONG |
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[2320] | 24 | cases (bind_inversion ????? COMPILE) -COMPILE * * #init_cost #labelled' #rtlabs_program * #FRONTEND #COMPILE |
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[2765] | 25 | cases (bind_inversion ????? COMPILE) -COMPILE #lobject_code' * #ASSEMBLER #COMPILE |
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[2150] | 26 | whd in COMPILE:(??%%); destruct |
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| 27 | cases (bind_inversion ????? FRONTEND) -FRONTEND #cminor_program * #CMINOR #FRONTEND |
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| 28 | whd in FRONTEND:(??%%); destruct |
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| 29 | |
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| 30 | % |
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[2475] | 31 | [ (* Needs switch removal too, now |
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| 32 | @labelling_sim @NOT_WRONG |
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| 33 | *) cases daemon |
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[2150] | 34 | | @I |
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| 35 | ] qed. |
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| 36 | |
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[2322] | 37 | |
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[2802] | 38 | include "Clight/Clight_classified_system.ma". |
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[2322] | 39 | |
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[2323] | 40 | (* From measurable on Clight, we will end up with an RTLabs flat trace where |
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| 41 | we know that there are some m' and n' such that the prefix in Clight matches |
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| 42 | the prefix in RTLabs given by m', the next n steps in Clight are equivalent |
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| 43 | to the n' steps in RTLabs, and we have a suitable "will_return" for RTLabs |
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| 44 | for those n' steps so that we can build a corresponding structured trace. |
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| 45 | |
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| 46 | "Equivalent" here means, in particular, that the observables will be the same, |
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| 47 | and those observables will include the stack space costs. |
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| 48 | *) |
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| 49 | |
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[2506] | 50 | definition in_execution_prefix : execution_prefix Clight_state → costlabel → Prop ≝ |
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[2399] | 51 | λx,l. Exists … (λtrs. Exists … (λev. ev = EVcost l) (\fst trs)) x. |
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| 52 | |
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| 53 | let rec foldl_exists_aux (A,B:Type[0]) (l,l':list B) (f:A → ∀b:B. Exists … (λx.x=b) l → A) (a:A) on l' : (∀b. Exists … (λx.x=b) l' → Exists … (λx.x=b) l) → A ≝ |
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| 54 | match l' return λl'. (∀b. Exists … (λx.x=b) l' → Exists … (λx.x=b) l) → A with |
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| 55 | [ nil ⇒ λ_. a |
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| 56 | | cons h t ⇒ λH. foldl_exists_aux A B l t f (f a h (H …)) ? |
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| 57 | ]. |
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| 58 | [ %1 % |
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| 59 | | #b #H' @H %2 @H' |
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| 60 | ] qed. |
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| 61 | |
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| 62 | definition foldl_exists : ∀A,B:Type[0]. ∀l:list B. (A → ∀b:B. Exists … (λx. x = b ) l → A) → A → A ≝ |
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| 63 | λA,B,l,f,a. foldl_exists_aux A B l l f a (λb,H. H). |
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| 64 | |
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| 65 | lemma Exists_lift : ∀A,P,Q,l. |
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| 66 | (∀x. P x → Q x) → |
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| 67 | Exists A P l → |
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| 68 | Exists A Q l. |
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| 69 | #A #P #Q #l elim l |
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| 70 | [ // |
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| 71 | | #h #t #IH #H * [ #H' %1 @H @H' | #H' %2 @IH /2/ ] |
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| 72 | ] qed. |
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| 73 | |
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[2506] | 74 | definition measure_clock : ∀x:execution_prefix Clight_state. ((Σl:costlabel.in_execution_prefix x l)→ℕ) → nat ≝ |
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[2399] | 75 | λx,costmap. foldl_exists … x |
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| 76 | (λclock,trs,H. |
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| 77 | foldl_exists … (\fst trs) (λclock,ev. match ev return λev. Exists … (λx. x=ev) ? → nat with [ EVcost l ⇒ λH'. clock + costmap «l,?» | _ ⇒ λ_. clock ]) clock) |
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| 78 | 0. |
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| 79 | whd @(Exists_lift … H) * #tr1 #s1 #E destruct @(Exists_lift … H') #ev1 #E @E |
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| 80 | qed. |
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| 81 | |
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| 82 | definition clight_clock_after : ∀p:clight_program. nat → ((Σl:costlabel.in_clight_program p l)→ℕ) → option nat ≝ |
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| 83 | λp,n,costmap. |
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| 84 | let x ≝ exec_inf … clight_fullexec p in |
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[2506] | 85 | match split_trace … x n with |
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[2399] | 86 | [ Some traces ⇒ |
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| 87 | Some ? (measure_clock (\fst traces) (λl.costmap «l,?»)) |
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| 88 | | None ⇒ None ? |
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| 89 | ]. |
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| 90 | cases daemon |
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| 91 | qed. |
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| 92 | |
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[2774] | 93 | include "common/AssocList.ma". |
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| 94 | |
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| 95 | definition lookup_stack_cost : stack_cost_model → ident → nat ≝ |
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| 96 | λstack_cost,id. |
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| 97 | match assoc_list_lookup ?? id (eq_identifier …) stack_cost with |
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| 98 | [ None ⇒ 0 | Some n ⇒ n ]. |
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| 99 | |
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[2322] | 100 | definition simulates ≝ |
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[2774] | 101 | λp: compiler_output. |
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| 102 | let initial_status ≝ initialise_status … (load_code_memory (oc (c_labelled_object_code … p))) in |
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| 103 | ∀m1,m2. |
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| 104 | measurable Clight_pcs (c_labelled_clight … p) m1 m2 |
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| 105 | (lookup_stack_cost (c_stack_cost … p)) (c_max_stack … p) → |
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| 106 | ∀c1,c2. |
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| 107 | clight_clock_after (c_labelled_clight … p) m1 (c_clight_cost_map … p) = Some ? c1 → |
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| 108 | clight_clock_after (c_labelled_clight … p) m2 (c_clight_cost_map … p) = Some ? c2 → |
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| 109 | ∃n1,n2. |
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| 110 | observables Clight_pcs (c_labelled_clight … p) m1 m2 = |
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[2875] | 111 | observables (OC_preclassified_system (c_labelled_object_code … p)) |
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| 112 | (c_labelled_object_code … p) n1 n2 |
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[2774] | 113 | ∧ |
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[2765] | 114 | minus c2 c1 = clock … (execute n2 ? initial_status) - clock … (execute n1 ? initial_status). |
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[2322] | 115 | |
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[2928] | 116 | include "common/ExtraMonads.ma". |
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| 117 | |
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[2322] | 118 | theorem correct' : |
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[2852] | 119 | ∀observe. |
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[2774] | 120 | ∀input_program,output. |
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[2852] | 121 | compile observe input_program = return output → |
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[2322] | 122 | not_wrong … (exec_inf … clight_fullexec input_program) → |
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[2774] | 123 | sim_with_labels |
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| 124 | (exec_inf … clight_fullexec input_program) |
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| 125 | (exec_inf … clight_fullexec (c_labelled_clight … output)) |
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[2322] | 126 | ∧ |
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[2774] | 127 | simulates output. |
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[2928] | 128 | #observe #p_in #out |
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| 129 | #H @('bind_inversion H) -H |
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| 130 | ** #init_cost #labelled #p_rtlabs #EQ_front_end |
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| 131 | #H @('bind_inversion H) -H |
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| 132 | ** #p_asm #stack_costs #globals_size |
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| 133 | #H @('bind_inversion H) -H |
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| 134 | #p_asm' |
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| 135 | #H lapply (opt_eq_from_res ???? H) #EQ_lin_to_asm |
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| 136 | whd in ⊢ (??%%→?); #EQ lapply (sym_eq ??? EQ) -EQ #EQ destruct(EQ) |
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| 137 | #H @('bind_inversion H) -H |
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| 138 | #p_loc #EQ_assembler |
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| 139 | whd in ⊢ (??%%→?); #EQ lapply (sym_eq ??? EQ) -EQ #EQ destruct(EQ) |
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| 140 | #NOT_WRONG % |
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| 141 | [ cases daemon (* TODO *) |
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| 142 | | #m1 #m2 #m1_m2_meas #c1 #c2 #c1_prf #c2_prf |
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| 143 | |
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| 144 | cut (∀n,s1,s2,obs1,obs2. |
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| 145 | exec_with_observables_n n (RTLabs_init_state p_rtlabs) = return 〈obs1, s1〉 → |
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| 146 | ∀tlr : trace_label_return (RTLabs_status p_rtlabs) s1 s2. |
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| 147 | tlr_observables … tlr (function_of … s1) = obs2 → |
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| 148 | (* maybe instead of function_of the called id can rather be obtained from execution? *) |
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| 149 | ∃m,p,s_fin. |
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| 150 | observables (OC_preclassified_system (c_labelled_object_code … p)) |
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| 151 | (c_labelled_object_code … p) m p = return 〈obs1, obs2〉) |
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| 152 | |
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| 153 | |
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| 154 | |
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| 155 | |
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[2322] | 156 | |
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| 157 | (* start of old simulates |
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| 158 | |
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[2325] | 159 | (* [nth_state_of_with_stack state stack_cost stack_bound exec n] returns [Some s] iff after |
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| 160 | [n] steps of [exec] we have reached [s] without exceeding the [stack_bound] |
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| 161 | according to the [stack_cost] function. *) |
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| 162 | axiom nth_state_of_with_stack : ∀state. (state → nat) → nat → execution state io_out io_in → nat → option state. |
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| 163 | axiom nth_state_of : ∀state. execution state io_out io_in → nat → option state. |
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| 164 | |
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| 165 | |
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[2322] | 166 | let cl_trace ≝ exec_inf … clight_fullexec labelled in |
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| 167 | let asm_trace ≝ exec_inf … ASM_fullexec object_code in |
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| 168 | not_wrong ? cl_trace → |
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| 169 | ∀n,s. nth_state_of_with_stack ? stack_cost stack_bound cl_trace n = Some ? s → |
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| 170 | 𝚺m,s'. nth_state_of ? asm_trace m = Some ? s' ∧ s ≃ s' |
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| 171 | |
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| 172 | *) |
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| 173 | |
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[1996] | 174 | (* TODO |
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| 175 | |
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[2322] | 176 | |
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[1996] | 177 | ∀input_program. |
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[2003] | 178 | ! 〈object_code,costlabel_map,labelled,cost_map〉 ← compile input_program |
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[1996] | 179 | |
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[2001] | 180 | exec_inf … clight_fullexec input_program ≃l exec_inf … clight_fullexec labelled |
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[1996] | 181 | |
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| 182 | ∧ |
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| 183 | |
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[2004] | 184 | exec_inf … clight_fullexec labelled ≈ exec_inf … ASM_fullexec object_code |
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| 185 | (* Should we be lifting labels in some way here? *) |
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[1996] | 186 | |
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[2001] | 187 | ∧ |
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| 188 | |
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[2322] | 189 | ∀i,f : clight_status. |
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| 190 | Clight_labelled i → |
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| 191 | Clight_labelled f → |
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| 192 | ∀mx,time. |
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| 193 | let trace ≝ exec_inf_aux … clight_fullexec labelled i in |
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| 194 | will_return O O mx time f trace → |
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| 195 | mx < max_allowed_stack → |
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[2004] | 196 | ∃!i',f'. i ≃ i' ∧ f ≃ f' ∧ i' 8051~> f' ∧ |
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[2322] | 197 | time = clock f' - clock i'. |
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[1996] | 198 | |
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[2001] | 199 | |
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[2003] | 200 | ∀s,flat. |
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| 201 | let ge ≝ (globalenvs … labelled) in |
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| 202 | subtrace_of (exec_inf … RTLabs_fullexec labelled) flat → |
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| 203 | RTLabs_cost s = true → |
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| 204 | ∀WR : will_return ge 0 s flat. |
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| 205 | let structured_trace_rtlabs ≝ make_label_return' ge 0 s flat ??? WR in |
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| 206 | let labels_rtlabs ≝ flat_label_trace … flat WR in |
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| 207 | ∃!initial,final,structured_trace_asm. |
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| 208 | structured_trace_rtlabs ≈ structured_trace_asm ∧ |
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| 209 | clock … code_memory … final = clock … code_memory … initial + |
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| 210 | (Σ_{i < |labels_rtlabs|} (cost_map (match nth i labels_rtlabs with [ Some k ⇒ k | None ⇒ 0 ])). |
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[2001] | 211 | |
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[2003] | 212 | |
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| 213 | |
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[2001] | 214 | What is ≃l? Must show that "labelled" does everything that |
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| 215 | "input_program" does, without getting lost in some |
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| 216 | non-terminating loop part way. |
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| 217 | |
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[1996] | 218 | *) |
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| 219 | |
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