[1996] | 1 | |
---|
| 2 | include "compiler.ma". |
---|
| 3 | |
---|
| 4 | include "common/SmallstepExec.ma". |
---|
| 5 | include "Clight/Cexec.ma". |
---|
| 6 | include "ASM/Interpret2.ma". |
---|
| 7 | |
---|
[2150] | 8 | include "Clight/labelSimulation.ma". |
---|
| 9 | |
---|
| 10 | theorem correct : |
---|
| 11 | ∀input_program. |
---|
| 12 | |
---|
| 13 | ∀object_code,costlabel_map,labelled,cost_map. |
---|
[2399] | 14 | compile input_program = OK ? 〈〈object_code,costlabel_map〉,❬labelled,cost_map❭〉 → |
---|
[2325] | 15 | |
---|
| 16 | not_wrong … (exec_inf … clight_fullexec input_program) → |
---|
[2150] | 17 | |
---|
| 18 | sim_with_labels (exec_inf … clight_fullexec input_program) (exec_inf … clight_fullexec labelled) |
---|
| 19 | ∧ |
---|
| 20 | True (* TODO *). |
---|
| 21 | |
---|
| 22 | #input_program |
---|
| 23 | #object_code #costlabel_map #labelled #cost_map |
---|
| 24 | #COMPILE |
---|
[2325] | 25 | #NOT_WRONG |
---|
[2320] | 26 | cases (bind_inversion ????? COMPILE) -COMPILE * * #init_cost #labelled' #rtlabs_program * #FRONTEND #COMPILE |
---|
[2150] | 27 | cases (bind_inversion ????? COMPILE) -COMPILE * #object_code' #costlabel_map' * #ASSEMBLER #COMPILE |
---|
| 28 | whd in COMPILE:(??%%); destruct |
---|
| 29 | cases (bind_inversion ????? FRONTEND) -FRONTEND #cminor_program * #CMINOR #FRONTEND |
---|
| 30 | whd in FRONTEND:(??%%); destruct |
---|
| 31 | |
---|
| 32 | % |
---|
[2475] | 33 | [ (* Needs switch removal too, now |
---|
| 34 | @labelling_sim @NOT_WRONG |
---|
| 35 | *) cases daemon |
---|
[2150] | 36 | | @I |
---|
| 37 | ] qed. |
---|
| 38 | |
---|
[2322] | 39 | |
---|
[2325] | 40 | include "Clight/abstract.ma". |
---|
[2322] | 41 | |
---|
[2325] | 42 | definition Clight_stack_T ≝ ∀s:Clight_state. match Clight_classify s with [ cl_call ⇒ True | cl_return ⇒ True | _ ⇒ False ] → nat. |
---|
[2322] | 43 | |
---|
[2325] | 44 | definition execution_prefix : Type[0] ≝ list (trace × Clight_state). |
---|
[2399] | 45 | let rec split_trace (x:execution Clight_state io_out io_in) (n:nat) on n : option (execution_prefix × (execution Clight_state io_out io_in)) ≝ |
---|
| 46 | match n with |
---|
| 47 | [ O ⇒ Some ? 〈[ ], x〉 |
---|
| 48 | | S n' ⇒ |
---|
| 49 | match x with |
---|
| 50 | [ e_step tr s x' ⇒ |
---|
| 51 | ! 〈pre,x''〉 ← split_trace x' n'; |
---|
| 52 | Some ? 〈〈tr,s〉::pre,x''〉 |
---|
| 53 | | _ ⇒ None ? |
---|
| 54 | ] |
---|
| 55 | ]. |
---|
[2322] | 56 | |
---|
[2399] | 57 | definition trace_labelled : execution_prefix → Prop ≝ |
---|
| 58 | λx. ∃tr1,s1,x',tr2,s2. x = 〈tr1,s1〉::(x'@[〈tr2,s2〉]) ∧ bool_to_Prop (Clight_labelled s1) ∧ bool_to_Prop (Clight_labelled s2). |
---|
| 59 | |
---|
| 60 | definition measure_stack : Clight_stack_T → execution_prefix → nat × nat ≝ |
---|
| 61 | λstack_cost. |
---|
| 62 | foldl … (λx, trs. |
---|
| 63 | let 〈current,max_stack〉 ≝ x in |
---|
| 64 | let 〈tr,s〉 ≝ trs in |
---|
| 65 | let new ≝ |
---|
| 66 | match Clight_classify s return λcl. (match cl in status_class with [_⇒?] → ?) → ? with |
---|
| 67 | [ cl_call ⇒ λsc. current + sc I |
---|
| 68 | | cl_return ⇒ λsc. current - sc I |
---|
| 69 | | _ ⇒ λ_. current |
---|
| 70 | ] (stack_cost s) in |
---|
| 71 | 〈new, max max_stack new〉) 〈0,0〉. |
---|
| 72 | |
---|
| 73 | definition stack_after : Clight_stack_T → execution_prefix → nat ≝ |
---|
| 74 | λsc,x. \fst (measure_stack sc x). |
---|
| 75 | |
---|
| 76 | definition max_stack : Clight_stack_T → execution_prefix → nat ≝ |
---|
| 77 | λsc,x. \snd (measure_stack sc x). |
---|
| 78 | |
---|
[2412] | 79 | let rec will_return_aux (depth:nat) |
---|
| 80 | (trace:execution_prefix) on trace : bool ≝ |
---|
[2325] | 81 | match trace with |
---|
[2412] | 82 | [ nil ⇒ match depth with [ O ⇒ true | _ ⇒ false ] |
---|
[2325] | 83 | | cons h tl ⇒ |
---|
| 84 | let 〈tr,s〉 ≝ h in |
---|
[2412] | 85 | match Clight_classify s with |
---|
| 86 | [ cl_call ⇒ will_return_aux (S depth) tl |
---|
| 87 | | cl_return ⇒ |
---|
[2325] | 88 | match depth with |
---|
[2412] | 89 | [ O ⇒ false |
---|
| 90 | | S d ⇒ will_return_aux d tl |
---|
[2325] | 91 | ] |
---|
[2412] | 92 | | _ ⇒ will_return_aux depth tl |
---|
| 93 | ] |
---|
[2325] | 94 | ]. |
---|
[2412] | 95 | definition will_return' : execution_prefix → bool ≝ will_return_aux O. |
---|
[2325] | 96 | |
---|
[2322] | 97 | definition measurable : clight_program → nat → nat → Clight_stack_T → nat → Prop ≝ |
---|
[2412] | 98 | λp,m,n,stack_cost,max_allowed_stack. ∀prefix,suffix,interesting,remainder. |
---|
[2322] | 99 | let cl_trace ≝ exec_inf … clight_fullexec p in |
---|
| 100 | split_trace cl_trace m = Some ? 〈prefix,suffix〉 ∧ |
---|
| 101 | split_trace suffix n = Some ? 〈interesting,remainder〉 ∧ |
---|
[2412] | 102 | trace_labelled interesting ∧ |
---|
| 103 | bool_to_Prop (will_return' interesting) ∧ |
---|
| 104 | max_stack stack_cost (prefix@interesting) < max_allowed_stack. |
---|
[2322] | 105 | |
---|
[2323] | 106 | (* From measurable on Clight, we will end up with an RTLabs flat trace where |
---|
| 107 | we know that there are some m' and n' such that the prefix in Clight matches |
---|
| 108 | the prefix in RTLabs given by m', the next n steps in Clight are equivalent |
---|
| 109 | to the n' steps in RTLabs, and we have a suitable "will_return" for RTLabs |
---|
| 110 | for those n' steps so that we can build a corresponding structured trace. |
---|
| 111 | |
---|
| 112 | "Equivalent" here means, in particular, that the observables will be the same, |
---|
| 113 | and those observables will include the stack space costs. |
---|
| 114 | *) |
---|
| 115 | |
---|
[2399] | 116 | definition observables : clight_program → nat → nat → option ((list trace) × (list trace)) ≝ |
---|
| 117 | λp,m,n. |
---|
| 118 | let cl_trace ≝ exec_inf … clight_fullexec p in |
---|
| 119 | match split_trace cl_trace m with |
---|
| 120 | [ Some x ⇒ |
---|
| 121 | let 〈prefix,suffix〉 ≝ x in |
---|
| 122 | match split_trace suffix n with |
---|
| 123 | [ Some y ⇒ |
---|
| 124 | let interesting ≝ \fst y in |
---|
| 125 | Some ? 〈map … (fst ??) prefix, map … (fst ??) interesting〉 |
---|
| 126 | | _ ⇒ None ? |
---|
| 127 | ] |
---|
| 128 | | _ ⇒ None ? |
---|
| 129 | ]. |
---|
| 130 | |
---|
[2322] | 131 | axiom observables_8051 : object_code → nat → nat → option ((list trace) × (list trace)). |
---|
[2399] | 132 | |
---|
| 133 | definition in_execution_prefix : execution_prefix → costlabel → Prop ≝ |
---|
| 134 | λx,l. Exists … (λtrs. Exists … (λev. ev = EVcost l) (\fst trs)) x. |
---|
| 135 | |
---|
| 136 | 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 ≝ |
---|
| 137 | match l' return λl'. (∀b. Exists … (λx.x=b) l' → Exists … (λx.x=b) l) → A with |
---|
| 138 | [ nil ⇒ λ_. a |
---|
| 139 | | cons h t ⇒ λH. foldl_exists_aux A B l t f (f a h (H …)) ? |
---|
| 140 | ]. |
---|
| 141 | [ %1 % |
---|
| 142 | | #b #H' @H %2 @H' |
---|
| 143 | ] qed. |
---|
| 144 | |
---|
| 145 | definition foldl_exists : ∀A,B:Type[0]. ∀l:list B. (A → ∀b:B. Exists … (λx. x = b ) l → A) → A → A ≝ |
---|
| 146 | λA,B,l,f,a. foldl_exists_aux A B l l f a (λb,H. H). |
---|
| 147 | |
---|
| 148 | lemma Exists_lift : ∀A,P,Q,l. |
---|
| 149 | (∀x. P x → Q x) → |
---|
| 150 | Exists A P l → |
---|
| 151 | Exists A Q l. |
---|
| 152 | #A #P #Q #l elim l |
---|
| 153 | [ // |
---|
| 154 | | #h #t #IH #H * [ #H' %1 @H @H' | #H' %2 @IH /2/ ] |
---|
| 155 | ] qed. |
---|
| 156 | |
---|
| 157 | definition measure_clock : ∀x:execution_prefix. ((Σl:costlabel.in_execution_prefix x l)→ℕ) → nat ≝ |
---|
| 158 | λx,costmap. foldl_exists … x |
---|
| 159 | (λclock,trs,H. |
---|
| 160 | 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) |
---|
| 161 | 0. |
---|
| 162 | whd @(Exists_lift … H) * #tr1 #s1 #E destruct @(Exists_lift … H') #ev1 #E @E |
---|
| 163 | qed. |
---|
| 164 | |
---|
| 165 | definition clight_clock_after : ∀p:clight_program. nat → ((Σl:costlabel.in_clight_program p l)→ℕ) → option nat ≝ |
---|
| 166 | λp,n,costmap. |
---|
| 167 | let x ≝ exec_inf … clight_fullexec p in |
---|
| 168 | match split_trace x n with |
---|
| 169 | [ Some traces ⇒ |
---|
| 170 | Some ? (measure_clock (\fst traces) (λl.costmap «l,?»)) |
---|
| 171 | | None ⇒ None ? |
---|
| 172 | ]. |
---|
| 173 | cases daemon |
---|
| 174 | qed. |
---|
| 175 | |
---|
[2322] | 176 | axiom initial_8051_status : ∀oc. Status oc. |
---|
| 177 | |
---|
| 178 | definition simulates ≝ |
---|
[2399] | 179 | λstack_cost, stack_bound, labelled, object_code, cost_map. |
---|
[2322] | 180 | let initial_status ≝ initial_8051_status (load_code_memory object_code) in |
---|
| 181 | ∀m1,m2. measurable labelled m1 m2 stack_cost stack_bound → |
---|
[2399] | 182 | ∀c1,c2. clight_clock_after labelled m1 cost_map = Some ? c1 → clight_clock_after labelled m2 cost_map = Some ? c2 → |
---|
[2322] | 183 | ∃n1,n2. observables labelled m1 m2 = observables_8051 object_code n1 n2 ∧ |
---|
| 184 | c2 - c1 = clock … (execute n2 ? initial_status) - clock … (execute n1 ? initial_status). |
---|
| 185 | |
---|
[2399] | 186 | axiom compile' : clight_program → res (object_code × costlabel_map × |
---|
| 187 | (𝚺labelled:clight_program. ((Σl:costlabel.in_clight_program labelled l)→ℕ)) × Clight_stack_T × nat). |
---|
[2322] | 188 | |
---|
| 189 | theorem correct' : |
---|
| 190 | ∀input_program. |
---|
| 191 | |
---|
| 192 | not_wrong … (exec_inf … clight_fullexec input_program) → |
---|
| 193 | |
---|
| 194 | ∀object_code,costlabel_map,labelled,cost_map,stack_cost,stack_bound. |
---|
[2399] | 195 | compile' input_program = OK ? 〈〈〈object_code,costlabel_map〉,❬labelled,cost_map❭〉,stack_cost,stack_bound〉 → |
---|
[2322] | 196 | |
---|
| 197 | sim_with_labels (exec_inf … clight_fullexec input_program) (exec_inf … clight_fullexec labelled) |
---|
| 198 | ∧ |
---|
| 199 | |
---|
[2399] | 200 | simulates stack_cost stack_bound labelled object_code cost_map. |
---|
[2322] | 201 | |
---|
| 202 | |
---|
| 203 | |
---|
| 204 | (* start of old simulates |
---|
| 205 | |
---|
[2325] | 206 | (* [nth_state_of_with_stack state stack_cost stack_bound exec n] returns [Some s] iff after |
---|
| 207 | [n] steps of [exec] we have reached [s] without exceeding the [stack_bound] |
---|
| 208 | according to the [stack_cost] function. *) |
---|
| 209 | axiom nth_state_of_with_stack : ∀state. (state → nat) → nat → execution state io_out io_in → nat → option state. |
---|
| 210 | axiom nth_state_of : ∀state. execution state io_out io_in → nat → option state. |
---|
| 211 | |
---|
| 212 | |
---|
[2322] | 213 | let cl_trace ≝ exec_inf … clight_fullexec labelled in |
---|
| 214 | let asm_trace ≝ exec_inf … ASM_fullexec object_code in |
---|
| 215 | not_wrong ? cl_trace → |
---|
| 216 | ∀n,s. nth_state_of_with_stack ? stack_cost stack_bound cl_trace n = Some ? s → |
---|
| 217 | 𝚺m,s'. nth_state_of ? asm_trace m = Some ? s' ∧ s ≃ s' |
---|
| 218 | |
---|
| 219 | *) |
---|
| 220 | |
---|
[1996] | 221 | (* TODO |
---|
| 222 | |
---|
[2322] | 223 | |
---|
[1996] | 224 | ∀input_program. |
---|
[2003] | 225 | ! 〈object_code,costlabel_map,labelled,cost_map〉 ← compile input_program |
---|
[1996] | 226 | |
---|
[2001] | 227 | exec_inf … clight_fullexec input_program ≃l exec_inf … clight_fullexec labelled |
---|
[1996] | 228 | |
---|
| 229 | ∧ |
---|
| 230 | |
---|
[2004] | 231 | exec_inf … clight_fullexec labelled ≈ exec_inf … ASM_fullexec object_code |
---|
| 232 | (* Should we be lifting labels in some way here? *) |
---|
[1996] | 233 | |
---|
[2001] | 234 | ∧ |
---|
| 235 | |
---|
[2322] | 236 | ∀i,f : clight_status. |
---|
| 237 | Clight_labelled i → |
---|
| 238 | Clight_labelled f → |
---|
| 239 | ∀mx,time. |
---|
| 240 | let trace ≝ exec_inf_aux … clight_fullexec labelled i in |
---|
| 241 | will_return O O mx time f trace → |
---|
| 242 | mx < max_allowed_stack → |
---|
[2004] | 243 | ∃!i',f'. i ≃ i' ∧ f ≃ f' ∧ i' 8051~> f' ∧ |
---|
[2322] | 244 | time = clock f' - clock i'. |
---|
[1996] | 245 | |
---|
[2001] | 246 | |
---|
[2003] | 247 | ∀s,flat. |
---|
| 248 | let ge ≝ (globalenvs … labelled) in |
---|
| 249 | subtrace_of (exec_inf … RTLabs_fullexec labelled) flat → |
---|
| 250 | RTLabs_cost s = true → |
---|
| 251 | ∀WR : will_return ge 0 s flat. |
---|
| 252 | let structured_trace_rtlabs ≝ make_label_return' ge 0 s flat ??? WR in |
---|
| 253 | let labels_rtlabs ≝ flat_label_trace … flat WR in |
---|
| 254 | ∃!initial,final,structured_trace_asm. |
---|
| 255 | structured_trace_rtlabs ≈ structured_trace_asm ∧ |
---|
| 256 | clock … code_memory … final = clock … code_memory … initial + |
---|
| 257 | (Σ_{i < |labels_rtlabs|} (cost_map (match nth i labels_rtlabs with [ Some k ⇒ k | None ⇒ 0 ])). |
---|
[2001] | 258 | |
---|
[2003] | 259 | |
---|
| 260 | |
---|
[2001] | 261 | What is ≃l? Must show that "labelled" does everything that |
---|
| 262 | "input_program" does, without getting lost in some |
---|
| 263 | non-terminating loop part way. |
---|
| 264 | |
---|
[1996] | 265 | *) |
---|
| 266 | |
---|