[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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[2802] | 8 | include "Clight/Clight_classified_system.ma". |
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[2322] | 9 | |
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[2323] | 10 | (* From measurable on Clight, we will end up with an RTLabs flat trace where |
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| 11 | we know that there are some m' and n' such that the prefix in Clight matches |
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| 12 | the prefix in RTLabs given by m', the next n steps in Clight are equivalent |
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| 13 | to the n' steps in RTLabs, and we have a suitable "will_return" for RTLabs |
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| 14 | for those n' steps so that we can build a corresponding structured trace. |
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| 15 | |
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| 16 | "Equivalent" here means, in particular, that the observables will be the same, |
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| 17 | and those observables will include the stack space costs. |
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| 18 | *) |
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| 19 | |
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[2506] | 20 | definition in_execution_prefix : execution_prefix Clight_state → costlabel → Prop ≝ |
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[2399] | 21 | λx,l. Exists … (λtrs. Exists … (λev. ev = EVcost l) (\fst trs)) x. |
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| 22 | |
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| 23 | 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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| 24 | match l' return λl'. (∀b. Exists … (λx.x=b) l' → Exists … (λx.x=b) l) → A with |
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| 25 | [ nil ⇒ λ_. a |
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| 26 | | cons h t ⇒ λH. foldl_exists_aux A B l t f (f a h (H …)) ? |
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| 27 | ]. |
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| 28 | [ %1 % |
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| 29 | | #b #H' @H %2 @H' |
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| 30 | ] qed. |
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| 31 | |
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| 32 | 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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| 33 | λA,B,l,f,a. foldl_exists_aux A B l l f a (λb,H. H). |
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| 34 | |
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| 35 | lemma Exists_lift : ∀A,P,Q,l. |
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| 36 | (∀x. P x → Q x) → |
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| 37 | Exists A P l → |
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| 38 | Exists A Q l. |
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| 39 | #A #P #Q #l elim l |
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| 40 | [ // |
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| 41 | | #h #t #IH #H * [ #H' %1 @H @H' | #H' %2 @IH /2/ ] |
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| 42 | ] qed. |
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| 43 | |
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[3021] | 44 | include "RTLabs/MeasurableToStructured.ma". |
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[2399] | 45 | |
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[3021] | 46 | definition clight_clock_after : ∀p:clight_program. nat → (costlabel→ℕ) → res nat ≝ |
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[2399] | 47 | λp,n,costmap. |
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[3021] | 48 | ! 〈s,itrace〉 ← exec_steps_with_obs Clight_pcs p n; |
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| 49 | let ctrace ≝ filter_map … (λi. match i with [IEVcost cl ⇒ Some … cl | _ ⇒ None ? ]) itrace in |
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| 50 | return Σ_{l ∈ ctrace}(costmap l). |
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[2399] | 51 | |
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[2774] | 52 | include "common/AssocList.ma". |
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| 53 | |
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[3003] | 54 | definition lookup_stack_cost : stack_cost_model → ident → option nat ≝ |
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[2774] | 55 | λstack_cost,id. |
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[3003] | 56 | assoc_list_lookup ?? id (eq_identifier …) stack_cost. |
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[2774] | 57 | |
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[2322] | 58 | definition simulates ≝ |
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[2774] | 59 | λp: compiler_output. |
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[3003] | 60 | let initial_status ≝ initialise_status … (cm (c_labelled_object_code … p)) in |
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[2774] | 61 | ∀m1,m2. |
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| 62 | measurable Clight_pcs (c_labelled_clight … p) m1 m2 |
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| 63 | (lookup_stack_cost (c_stack_cost … p)) (c_max_stack … p) → |
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| 64 | ∀c1,c2. |
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[3021] | 65 | clight_clock_after (c_labelled_clight … p) m1 (c_clight_cost_map … p) = OK … c1 → |
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| 66 | clight_clock_after (c_labelled_clight … p) m2 (c_clight_cost_map … p) = OK … c2 → |
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[2774] | 67 | ∃n1,n2. |
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| 68 | observables Clight_pcs (c_labelled_clight … p) m1 m2 = |
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[2875] | 69 | observables (OC_preclassified_system (c_labelled_object_code … p)) |
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| 70 | (c_labelled_object_code … p) n1 n2 |
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[2774] | 71 | ∧ |
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[2765] | 72 | minus c2 c1 = clock … (execute n2 ? initial_status) - clock … (execute n1 ? initial_status). |
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[2322] | 73 | |
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[2928] | 74 | include "common/ExtraMonads.ma". |
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| 75 | |
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[3030] | 76 | theorem correct : |
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[2852] | 77 | ∀observe. |
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[2774] | 78 | ∀input_program,output. |
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[2852] | 79 | compile observe input_program = return output → |
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[2322] | 80 | not_wrong … (exec_inf … clight_fullexec input_program) → |
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[2774] | 81 | sim_with_labels |
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| 82 | (exec_inf … clight_fullexec input_program) |
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| 83 | (exec_inf … clight_fullexec (c_labelled_clight … output)) |
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[2322] | 84 | ∧ |
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[2774] | 85 | simulates output. |
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[2928] | 86 | #observe #p_in #out |
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| 87 | #H @('bind_inversion H) -H |
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| 88 | ** #init_cost #labelled #p_rtlabs #EQ_front_end |
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| 89 | #H @('bind_inversion H) -H |
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| 90 | ** #p_asm #stack_costs #globals_size |
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| 91 | #H @('bind_inversion H) -H |
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| 92 | #p_asm' |
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| 93 | #H lapply (opt_eq_from_res ???? H) #EQ_lin_to_asm |
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| 94 | whd in ⊢ (??%%→?); #EQ lapply (sym_eq ??? EQ) -EQ #EQ destruct(EQ) |
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| 95 | #H @('bind_inversion H) -H |
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| 96 | #p_loc #EQ_assembler |
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| 97 | whd in ⊢ (??%%→?); #EQ lapply (sym_eq ??? EQ) -EQ #EQ destruct(EQ) |
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[3022] | 98 | |
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[3030] | 99 | whd in EQ_front_end:(??%?); |
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| 100 | letin cl_switch_removed ≝ (program_switch_removal p_in) in EQ_front_end; |
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| 101 | lapply (refl ? (clight_label cl_switch_removed)) |
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| 102 | cases (clight_label ?) in ⊢ (???% → %); #cl_labelled #init_cost' #CL_LABELLED |
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| 103 | whd in ⊢ (??%? → ?); |
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| 104 | letin cl_simplified ≝ (simplify_program ?) |
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| 105 | #H @('bind_inversion H) -H #cminor #CMINOR |
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[3022] | 106 | #H @('bind_inversion H) -H #WCL #_ |
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| 107 | #H @('bind_inversion H) -H #INJ #_ |
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[3030] | 108 | letin rtlabs ≝ (cminor_to_rtlabs cminor) |
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| 109 | #EQ_front_end |
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[3022] | 110 | |
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[2928] | 111 | #NOT_WRONG % |
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[3030] | 112 | [ cases daemon (* TODO *) |
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[2928] | 113 | | #m1 #m2 #m1_m2_meas #c1 #c2 #c1_prf #c2_prf |
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| 114 | |
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| 115 | cut (∀n,s1,s2,obs1,obs2. |
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| 116 | exec_with_observables_n n (RTLabs_init_state p_rtlabs) = return 〈obs1, s1〉 → |
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| 117 | ∀tlr : trace_label_return (RTLabs_status p_rtlabs) s1 s2. |
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| 118 | tlr_observables … tlr (function_of … s1) = obs2 → |
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| 119 | (* maybe instead of function_of the called id can rather be obtained from execution? *) |
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| 120 | ∃m,p,s_fin. |
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| 121 | observables (OC_preclassified_system (c_labelled_object_code … p)) |
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| 122 | (c_labelled_object_code … p) m p = return 〈obs1, obs2〉) |
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| 123 | |
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| 124 | |
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| 125 | |
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| 126 | |
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[2322] | 127 | |
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| 128 | (* start of old simulates |
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| 129 | |
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[2325] | 130 | (* [nth_state_of_with_stack state stack_cost stack_bound exec n] returns [Some s] iff after |
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| 131 | [n] steps of [exec] we have reached [s] without exceeding the [stack_bound] |
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| 132 | according to the [stack_cost] function. *) |
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| 133 | axiom nth_state_of_with_stack : ∀state. (state → nat) → nat → execution state io_out io_in → nat → option state. |
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| 134 | axiom nth_state_of : ∀state. execution state io_out io_in → nat → option state. |
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| 135 | |
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| 136 | |
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[2322] | 137 | let cl_trace ≝ exec_inf … clight_fullexec labelled in |
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| 138 | let asm_trace ≝ exec_inf … ASM_fullexec object_code in |
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| 139 | not_wrong ? cl_trace → |
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| 140 | ∀n,s. nth_state_of_with_stack ? stack_cost stack_bound cl_trace n = Some ? s → |
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| 141 | 𝚺m,s'. nth_state_of ? asm_trace m = Some ? s' ∧ s ≃ s' |
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| 142 | |
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| 143 | *) |
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| 144 | |
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[1996] | 145 | (* TODO |
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| 146 | |
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[2322] | 147 | |
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[1996] | 148 | ∀input_program. |
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[2003] | 149 | ! 〈object_code,costlabel_map,labelled,cost_map〉 ← compile input_program |
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[1996] | 150 | |
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[2001] | 151 | exec_inf … clight_fullexec input_program ≃l exec_inf … clight_fullexec labelled |
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[1996] | 152 | |
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| 153 | ∧ |
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| 154 | |
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[2004] | 155 | exec_inf … clight_fullexec labelled ≈ exec_inf … ASM_fullexec object_code |
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| 156 | (* Should we be lifting labels in some way here? *) |
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[1996] | 157 | |
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[2001] | 158 | ∧ |
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| 159 | |
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[2322] | 160 | ∀i,f : clight_status. |
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| 161 | Clight_labelled i → |
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| 162 | Clight_labelled f → |
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| 163 | ∀mx,time. |
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| 164 | let trace ≝ exec_inf_aux … clight_fullexec labelled i in |
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| 165 | will_return O O mx time f trace → |
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| 166 | mx < max_allowed_stack → |
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[2004] | 167 | ∃!i',f'. i ≃ i' ∧ f ≃ f' ∧ i' 8051~> f' ∧ |
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[2322] | 168 | time = clock f' - clock i'. |
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[1996] | 169 | |
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[2001] | 170 | |
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[2003] | 171 | ∀s,flat. |
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| 172 | let ge ≝ (globalenvs … labelled) in |
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| 173 | subtrace_of (exec_inf … RTLabs_fullexec labelled) flat → |
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| 174 | RTLabs_cost s = true → |
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| 175 | ∀WR : will_return ge 0 s flat. |
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| 176 | let structured_trace_rtlabs ≝ make_label_return' ge 0 s flat ??? WR in |
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| 177 | let labels_rtlabs ≝ flat_label_trace … flat WR in |
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| 178 | ∃!initial,final,structured_trace_asm. |
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| 179 | structured_trace_rtlabs ≈ structured_trace_asm ∧ |
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| 180 | clock … code_memory … final = clock … code_memory … initial + |
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| 181 | (Σ_{i < |labels_rtlabs|} (cost_map (match nth i labels_rtlabs with [ Some k ⇒ k | None ⇒ 0 ])). |
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[2001] | 182 | |
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[2003] | 183 | |
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| 184 | |
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[2001] | 185 | What is ≃l? Must show that "labelled" does everything that |
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| 186 | "input_program" does, without getting lost in some |
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| 187 | non-terminating loop part way. |
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| 188 | |
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[1996] | 189 | *) |
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| 190 | |
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