Changeset 2685 for src/common/Measurable.ma
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 Feb 21, 2013, 11:38:36 AM (8 years ago)
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src/common/Measurable.ma
r2670 r2685 11 11 cs_labelled : state … cs_exec cs_global → bool; 12 12 cs_classify : state … cs_exec cs_global → status_class; 13 cs_ stack : ∀s. match cs_classify … s with [ cl_call ⇒ True  cl_return ⇒ True  _ ⇒ False ] → nat13 cs_callee : ∀s. match cs_classify … s with [ cl_call ⇒ True  _ ⇒ False ] → ident 14 14 }. 15 15 … … 25 25 λC,x. ∃tr1,s1,x',tr2,s2. x = 〈s1,tr1〉::(x'@[〈s2,tr2〉]) ∧ bool_to_Prop (cs_labelled C s1) ∧ cs_classify C s2 = cl_return. 26 26 27 definition measure_stack_aux ≝ 28 λC. (λx. λstr:cs_state … C × trace. 29 let 〈current,max_stack〉 ≝ x in 30 let 〈s,tr〉 ≝ str in 31 let new ≝ 32 match cs_classify C s return λcl. (match cl in status_class with [_⇒?] → ?) → ? with 33 [ cl_call ⇒ λsc. current + sc I 34  cl_return ⇒ λsc. current  sc I 35  _ ⇒ λ_. current 36 ] (cs_stack C s) in 37 〈new, max max_stack new〉). 38 39 definition measure_stack : ∀C. nat → list (cs_state … C × trace) → nat × nat ≝ 40 λC,current. 41 foldl … (measure_stack_aux C) 〈current,0〉. 42 43 definition stack_after : ∀C. nat → list (cs_state … C × trace) → nat ≝ 44 λC,current,x. \fst (measure_stack C current x). 45 46 definition max_stack : ∀C. nat → list (cs_state … C × trace) → nat ≝ 47 λC,current,x. \snd (measure_stack C current x). 27 (* For intensional_event; should separate out definition *) 28 include "common/StatusSimulation.ma". 29 30 definition intensional_event_of_event : event → list intensional_event ≝ 31 λev. match ev with 32 [ EVcost l ⇒ [ IEVcost l ] 33  EVextcall _ _ _ ⇒ [ ] (* No equivalent, but there shouldn't be any for now *) 34 ]. 35 36 definition intensional_events_of_events : trace → list intensional_event ≝ 37 λtr. flatten ? (map ?? intensional_event_of_event tr). 38 39 let rec intensional_trace_of_trace C (callstack:list ident) (trace:list (cs_state … C × trace)) on trace : list ident × (list intensional_event) ≝ 40 match trace with 41 [ nil ⇒ 〈callstack, [ ]〉 42  cons str tl ⇒ 43 let 〈s,tr〉 ≝ str in 44 let 〈callstack, call_event〉 ≝ 45 match cs_classify C s return λx. (match x with [ cl_call ⇒ True  _ ⇒ False ] → ident) → list ident × (list intensional_event) with 46 [ cl_call ⇒ λcallee. let id ≝ callee I in 〈id::callstack, [IEVcall id]〉 47  cl_return ⇒ λ_. match callstack with [ nil ⇒ 〈[ ], [ ]〉  cons id tl ⇒ 〈tl, [IEVret id]〉 ] 48  _ ⇒ λ_. 〈callstack, [ ]〉 49 ] (cs_callee C s) in 50 let other_events ≝ intensional_events_of_events tr in 51 let 〈callstack,rem〉 ≝ intensional_trace_of_trace C callstack tl in 52 〈callstack, call_event@other_events@rem〉 53 ]. 54 55 definition normal_state : ∀C:classified_system. cs_state … C → bool ≝ 56 λC,s. match cs_classify C s with [ cl_other ⇒ true  cl_jump ⇒ true  _ ⇒ false ]. 57 58 lemma normal_state_inv : ∀C,s. 59 normal_state C s → 60 cs_classify C s = cl_other ∨ cs_classify C s = cl_jump. 61 #C #s whd in ⊢ (?% → ?); cases (cs_classify C s) /2/ * 62 qed. 63 64 lemma int_trace_of_normal : ∀C,callstack,s,tr,trace. 65 normal_state C s → 66 intensional_trace_of_trace C callstack (〈s,tr〉::trace) = 67 (let 〈stk',tl〉 ≝ intensional_trace_of_trace C callstack trace in 68 〈stk', (intensional_events_of_events tr)@tl〉). 69 #C #callstack #s #tr #trace #NORMAL 70 whd in ⊢ (??%?); 71 generalize in match (cs_callee C s); 72 cases (normal_state_inv … NORMAL) 73 #CL >CL normalize nodelta #_ 74 cases (intensional_trace_of_trace C callstack trace) 75 #callstack' #tl normalize nodelta 76 % 77 qed. 78 79 lemma flatten_append : ∀A,l1,l2. 80 flatten A (l1@l2) = (flatten A l1)@(flatten A l2). 81 #A #l1 #l2 82 elim l1 83 [ % 84  #h #t #IH whd in ⊢ (??%(??%?)); 85 change with (flatten ??) in match (foldr ?????); >IH 86 change with (flatten ??) in match (foldr ?????); 87 >associative_append % 88 ] qed. 89 90 91 lemma int_events_append : ∀tr1,tr2. 92 intensional_events_of_events (tr1@tr2) = (intensional_events_of_events tr1)@(intensional_events_of_events tr2). 93 #tr1 #tr2 94 change with (flatten ??) in ⊢ (??%(??%%)); 95 <map_append >flatten_append % 96 qed. 97 98 99 lemma int_trace_irr : ∀C,callstack,s,trace. 100 normal_state C s → 101 intensional_trace_of_trace C callstack (〈s,E0〉::trace) = intensional_trace_of_trace C callstack trace. 102 #C #callstate #s #trace #NORMAL >(int_trace_of_normal … NORMAL) 103 cases (intensional_trace_of_trace ???) // 104 qed. 105 106 lemma int_trace_append : ∀C,callstack,s,t1,t2,trace. 107 normal_state C s → 108 intensional_trace_of_trace C callstack (〈s,t1@t2〉::trace) = intensional_trace_of_trace C callstack (〈s,t1〉::〈s,t2〉::trace). 109 #C #callstack #s #t1 #t2 #trace #NORMAL 110 >(int_trace_of_normal … NORMAL) 111 >(int_trace_of_normal … NORMAL) 112 >(int_trace_of_normal … NORMAL) 113 cases (intensional_trace_of_trace ???) #callstack' #trace' 114 normalize nodelta 115 >int_events_append 116 >associative_append % 117 qed. 118 119 lemma build_eq_trace : ∀C,C',callstack,s,trace,rem,rem'. 120 normal_state C s → 121 all … (λstr. normal_state C' (\fst str)) trace → 122 intensional_trace_of_trace C callstack rem = intensional_trace_of_trace C' callstack rem' → 123 intensional_trace_of_trace C callstack (〈s,gather_trace … trace〉::rem) = intensional_trace_of_trace C' callstack (trace@rem'). 124 #C #C' #callstack #s #trace #rem #rem' #NORMAL #NORMAL' 125 >(int_trace_of_normal … NORMAL) 126 cases (intensional_trace_of_trace C callstack rem) #callstack' #trace' 127 #REM whd in ⊢ (??%?); 128 elim trace in NORMAL' ⊢ %; 129 [ #_ @REM 130  * #s' #tr' #tl #IH #NORMAL' 131 cases (andb_Prop_true … NORMAL') #NORMALs #NORMALtl 132 >int_trace_of_normal 133 [ <(IH NORMALtl) whd in match (gather_trace ??); whd in ⊢ (???%); 134 >int_events_append >associative_append % 135  @NORMALs 136 ] 137 ] qed. 138 139 lemma int_trace_append' : ∀C,t1,t2,callstack. 140 intensional_trace_of_trace C callstack (t1@t2) = 141 (let 〈cs',t1'〉 ≝ intensional_trace_of_trace C callstack t1 in 142 let 〈cs'',t2'〉 ≝ intensional_trace_of_trace C cs' t2 in 143 〈cs'', t1'@t2'〉). 144 #C #t1 #t2 elim t1 145 [ #callstack whd in match ([ ]@t2); whd in ⊢ (???%); 146 cases (intensional_trace_of_trace ???) #cs' #trace' % 147  * #s #tr #tl #IH 148 #callstack 149 whd in match (intensional_trace_of_trace ???); 150 whd in match (intensional_trace_of_trace ???); 151 generalize in match (cs_callee C s); 152 cases (cs_classify C s) 153 normalize nodelta #callee 154 [ cases callstack [2: #cshd #cdtl] normalize nodelta ] 155 >IH cases (intensional_trace_of_trace C ? tl) #cs' #tl' 156 normalize nodelta 157 cases (intensional_trace_of_trace C ? t2) #cs'' #tl'' 158 normalize nodelta >associative_append >associative_append 159 % 160 ] qed. 161 162 lemma int_trace_normal_cs : ∀C,callstack,trace. 163 all ? (λstr. normal_state C (\fst str)) trace → 164 callstack = \fst (intensional_trace_of_trace C callstack trace). 165 #C #callstack #trace elim trace 166 [ // 167  * #s #tr #tl #IH #N cases (andb_Prop_true … N) #N1 #Ntl 168 >(int_trace_of_normal … N1) 169 >(IH Ntl) in ⊢ (??%?); 170 cases (intensional_trace_of_trace ???) /2/ 171 ] qed. 172 173 lemma int_trace_append_normal : ∀C,t1,t2,callstack. 174 all ? (λstr. normal_state C (\fst str)) t1 → 175 intensional_trace_of_trace C callstack (t1@t2) = 176 (let t1' ≝ \snd (intensional_trace_of_trace C callstack t1) in 177 let 〈cs'',t2'〉 ≝ intensional_trace_of_trace C callstack t2 in 178 〈cs'', t1'@t2'〉). 179 #C #t1 #t2 #callstack #NORMAL lapply (int_trace_append' C t1 t2 callstack) 180 lapply (int_trace_normal_cs C callstack t1 NORMAL) 181 cases (intensional_trace_of_trace ?? t1) #cs #tl #E destruct // 182 qed. 183 184 lemma build_return_trace : ∀C,C',callstack,s,s',tr,trace',rem,rem'. 185 cs_classify C s = cl_return → 186 cs_classify C' s' = cl_return → 187 all ? (λstr. normal_state C' (\fst str)) trace' → 188 intensional_trace_of_trace C (tail … callstack) rem = intensional_trace_of_trace C' (tail … callstack) rem' → 189 let trace ≝ 〈s',tr〉::trace' in 190 intensional_trace_of_trace C callstack (〈s,gather_trace … trace〉::rem) = intensional_trace_of_trace C' callstack (trace@rem'). 191 #C #C' #callstack #s #s' #tr #trace #rem #rem' #CL #CL' #NORMAL #E 192 whd 193 whd in ⊢ (??%%); normalize nodelta 194 generalize in match (cs_callee C s); generalize in match (cs_callee C' s'); 195 >CL >CL' normalize nodelta #_ #_ 196 cases callstack in E ⊢ %; [2: #stkhd #stktl] 197 normalize nodelta 198 cases (intensional_trace_of_trace ?? rem) #cs_rem #ev_rem normalize nodelta 199 >(int_trace_append_normal … NORMAL) normalize nodelta 200 cases (intensional_trace_of_trace ?? rem') #cs_rem' #ev_rem' normalize nodelta #E 201 destruct @eq_f @eq_f 202 whd in match (gather_trace ??); >int_events_append 203 >associative_append @eq_f 204 elim trace in NORMAL ⊢ %; 205 [ 1,3: #_ % 206  2,4: 207 * #s1 #tr1 #tl #IH 208 #N cases (andb_Prop_true … N) #N1 #Ntl 209 whd in match (gather_trace ??); >int_events_append 210 >associative_append >(IH Ntl) 211 >(int_trace_of_normal … N1) 212 cases (intensional_trace_of_trace ?? tl) 213 #cs' #tl' >associative_append % 214 ] qed. 215 216 lemma generalize_callee : ∀C,s,H. ∀P: ? → ? → Prop. 217 (∀f. P f (f H)) → 218 P (cs_callee C s) (cs_callee C s H). 219 #C #s #H #P #f @f 220 qed. 221 222 lemma build_call_trace : ∀C,C',callstack,s,s',tr,trace',rem,rem',H,H'. 223 cs_classify C s = cl_call → 224 cs_classify C' s' = cl_call → 225 all ? (λstr. normal_state C' (\fst str)) trace' → 226 intensional_trace_of_trace C (cs_callee C s H::callstack) rem = intensional_trace_of_trace C' (cs_callee C s H::callstack) rem' → 227 cs_callee C s H = cs_callee C' s' H' → 228 let trace ≝ 〈s',tr〉::trace' in 229 intensional_trace_of_trace C callstack (〈s,gather_trace … trace〉::rem) = intensional_trace_of_trace C' callstack (trace@rem'). 230 #C #C' #callstack #s #s' #tr #trace #rem #rem' #H #H' #CL #CL' #NORMAL 231 whd in ⊢ (? → ? → %); 232 whd in ⊢ (? → ? → ??%%); normalize nodelta 233 @generalize_callee 234 @generalize_callee 235 >CL in H ⊢ %; * >CL' in H' ⊢ %; * normalize nodelta #calleef #calleef' #E #CALLEE <CALLEE 236 cases (intensional_trace_of_trace ?? rem) in E ⊢ %; #cs_rem #ev_rem normalize nodelta 237 >(int_trace_append_normal … NORMAL) normalize nodelta 238 cases (intensional_trace_of_trace ?? rem') #cs_rem' #ev_rem' normalize nodelta #E 239 destruct @eq_f @eq_f 240 whd in match (gather_trace ??); >int_events_append 241 >associative_append @eq_f 242 elim trace in NORMAL ⊢ %; 243 [ 1,3: #_ % 244  2,4: 245 * #s1 #tr1 #tl #IH 246 #N cases (andb_Prop_true … N) #N1 #Ntl 247 whd in match (gather_trace ??); >int_events_append 248 >associative_append >(IH Ntl) 249 >(int_trace_of_normal … N1) 250 cases (intensional_trace_of_trace ?? tl) 251 #cs' #tl' >associative_append % 252 ] qed. 253 254 255 definition measure_stack : (ident → nat) → nat → list intensional_event → nat × nat ≝ 256 λcosts,start. 257 foldl ?? (λx. λev. 258 match ev with 259 [ IEVcall id ⇒ 260 let 〈current_stack,max_stack〉 ≝ x in 261 let new_stack ≝ current_stack + costs id in 262 〈new_stack, max new_stack max_stack〉 263  IEVret id ⇒ 264 let 〈current_stack,max_stack〉 ≝ x in 265 〈current_stack  costs id, max_stack〉 266  _ ⇒ x 267 ]) 〈start,start〉. 268 269 definition max_stack : (ident → nat) → nat → list intensional_event → nat ≝ 270 λcosts, start, trace. \snd (measure_stack costs start trace). 48 271 49 272 lemma foldl_inv : ∀A,B. ∀P:A → Prop. ∀f. … … 55 278  #h #t #IH' #acc #H @IH' @IH @H 56 279 ] qed. 57 280 (* 58 281 lemma max_stack_step : ∀C,a,m,a',m',tr,s. 59 282 measure_stack_aux C 〈a,m〉 〈s,tr〉 = 〈a',m'〉 → … … 107 330 >IH % 108 331 ] qed. 109 332 *) 110 333 111 334 (* Check that the trace ends with the return from the starting function and one … … 140 363 pcs_labelled : ∀g. state … pcs_exec g → bool; 141 364 pcs_classify : ∀g. state … pcs_exec g → status_class; 142 pcs_ stack : (ident → nat) → ∀g,s. match pcs_classify g s with [ cl_call ⇒ True  cl_return ⇒ True  _ ⇒ False ] → nat365 pcs_callee : ∀g,s. match pcs_classify g s with [ cl_call ⇒ True  _ ⇒ False ] → ident 143 366 }. 144 367 145 definition pcs_to_cs : ∀C:preclassified_system. global … C → (ident → nat) →classified_system ≝146 λC,g ,stack_cost.147 mk_classified_system (pcs_exec C) g (pcs_labelled C ?) (pcs_classify C ?) (pcs_ stack C stack_cost?).368 definition pcs_to_cs : ∀C:preclassified_system. global … C → classified_system ≝ 369 λC,g. 370 mk_classified_system (pcs_exec C) g (pcs_labelled C ?) (pcs_classify C ?) (pcs_callee C ?). 148 371 149 372 (* FIXME: this definition is unreasonable because it presumes we can easily … … 158 381 λC,p,m,n,stack_cost,max_allowed_stack. ∃s0,prefix,s1,interesting,s2. 159 382 let g ≝ make_global … (pcs_exec … C) p in 160 let C' ≝ pcs_to_cs C g stack_costin383 let C' ≝ pcs_to_cs C g in 161 384 make_initial_state … p = OK ? s0 ∧ 162 385 exec_steps m ?? (cs_exec … C') g s0 = OK ? 〈prefix,s1〉 ∧ … … 164 387 trace_is_label_to_return C' interesting ∧ 165 388 bool_to_Prop (will_return' C' interesting) ∧ 166 le (max_stack C' 0 (prefix@interesting)) max_allowed_stack.389 le (max_stack stack_cost 0 (\snd (intensional_trace_of_trace C' [ ] (prefix@interesting)))) max_allowed_stack. 167 390 168 391 (* TODO: probably ought to be elsewhere; use exec_steps instead
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