Ignore:
Timestamp:
Feb 21, 2013, 11:38:36 AM (8 years ago)
Author:
campbell
Message:

Progress on measurable trace preservation: prefix preserves observable
intensional events (costs and calls).

File:
1 edited

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  • src/common/Measurable.ma

    r2670 r2685  
    1111  cs_labelled : state … cs_exec cs_global → bool;
    1212  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 ] → nat
     13  cs_callee : ∀s. match cs_classify … s with [ cl_call ⇒ True | _ ⇒ False ] → ident
    1414}.
    1515
     
    2525λC,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.
    2626
    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 *)
     28include "common/StatusSimulation.ma".
     29
     30definition intensional_event_of_event : event → list intensional_event ≝
     31λev. match ev with
     32[ EVcost l ⇒ [ IEVcost l ]
     33| EVextcall _ _ _ ⇒ [ ]  (* No equivalent, but there shouldn't be any for now *)
     34].
     35
     36definition intensional_events_of_events : trace → list intensional_event ≝
     37λtr. flatten ? (map ?? intensional_event_of_event tr).
     38
     39let rec intensional_trace_of_trace C (callstack:list ident) (trace:list (cs_state … C × trace)) on trace : list ident × (list intensional_event) ≝
     40match 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
     55definition normal_state : ∀C:classified_system. cs_state … C → bool ≝
     56λC,s. match cs_classify C s with [ cl_other ⇒ true | cl_jump ⇒ true | _ ⇒ false ].
     57
     58lemma 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/ *
     62qed.
     63
     64lemma 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
     70whd in ⊢ (??%?);
     71generalize in match (cs_callee C s);
     72cases (normal_state_inv … NORMAL)
     73#CL >CL normalize nodelta #_
     74cases (intensional_trace_of_trace C callstack trace)
     75#callstack' #tl normalize nodelta
     76%
     77qed.
     78
     79lemma flatten_append : ∀A,l1,l2.
     80  flatten A (l1@l2) = (flatten A l1)@(flatten A l2).
     81#A #l1 #l2
     82elim 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
     91lemma 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
     94change with (flatten ??) in ⊢ (??%(??%%));
     95<map_append >flatten_append %
     96qed.
     97
     98
     99lemma 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)
     103cases (intensional_trace_of_trace ???) //
     104qed.
     105
     106lemma 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)
     113cases (intensional_trace_of_trace ???) #callstack' #trace'
     114normalize nodelta
     115>int_events_append
     116>associative_append %
     117qed.
     118
     119lemma 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)
     126cases (intensional_trace_of_trace C callstack rem) #callstack' #trace'
     127#REM whd in ⊢ (??%?);
     128elim 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
     139lemma 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
     162lemma 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
     173lemma 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)
     180lapply (int_trace_normal_cs C callstack t1 NORMAL)
     181cases (intensional_trace_of_trace ?? t1) #cs #tl #E destruct //
     182qed.
     183
     184lemma 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
     192whd
     193whd in ⊢ (??%%); normalize nodelta
     194generalize in match (cs_callee C s); generalize in match (cs_callee C' s');
     195>CL >CL' normalize nodelta #_ #_
     196cases callstack in E ⊢ %; [2: #stkhd #stktl]
     197normalize nodelta
     198cases (intensional_trace_of_trace ?? rem) #cs_rem #ev_rem normalize nodelta
     199>(int_trace_append_normal … NORMAL) normalize nodelta
     200cases (intensional_trace_of_trace ?? rem') #cs_rem' #ev_rem' normalize nodelta #E
     201destruct @eq_f @eq_f
     202whd in match (gather_trace ??); >int_events_append
     203>associative_append @eq_f
     204elim 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
     216lemma 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
     220qed.
     221
     222lemma 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
     231whd in ⊢ (? → ? → %);
     232whd in ⊢ (? → ? → ??%%); normalize nodelta
     233@generalize_callee
     234@generalize_callee
     235>CL in H ⊢ %; * >CL' in H' ⊢ %; * normalize nodelta #calleef #calleef' #E #CALLEE <CALLEE
     236cases (intensional_trace_of_trace ?? rem) in E ⊢ %; #cs_rem #ev_rem normalize nodelta
     237>(int_trace_append_normal … NORMAL) normalize nodelta
     238cases (intensional_trace_of_trace ?? rem') #cs_rem' #ev_rem' normalize nodelta #E
     239destruct @eq_f @eq_f
     240whd in match (gather_trace ??); >int_events_append
     241>associative_append @eq_f
     242elim 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
     255definition measure_stack : (ident → nat) → nat → list intensional_event → nat × nat ≝
     256λcosts,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
     269definition max_stack : (ident → nat) → nat → list intensional_event → nat ≝
     270λcosts, start, trace. \snd (measure_stack costs start trace).
    48271
    49272lemma foldl_inv : ∀A,B. ∀P:A → Prop. ∀f.
     
    55278| #h #t #IH' #acc #H @IH' @IH @H
    56279] qed.
    57 
     280(*
    58281lemma max_stack_step : ∀C,a,m,a',m',tr,s.
    59282  measure_stack_aux C 〈a,m〉 〈s,tr〉 = 〈a',m'〉 →
     
    107330  >IH %
    108331] qed.
    109  
     332*) 
    110333
    111334(* Check that the trace ends with the return from the starting function and one
     
    140363  pcs_labelled : ∀g. state … pcs_exec g → bool;
    141364  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 ] → nat
     365  pcs_callee : ∀g,s. match pcs_classify g s with [ cl_call ⇒ True | _ ⇒ False ] → ident
    143366}.
    144367
    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 ?).
     368definition pcs_to_cs : ∀C:preclassified_system. global … C → classified_system ≝
     369λC,g.
     370  mk_classified_system (pcs_exec C) g (pcs_labelled C ?) (pcs_classify C ?) (pcs_callee C ?).
    148371
    149372(* FIXME: this definition is unreasonable because it presumes we can easily
     
    158381λC,p,m,n,stack_cost,max_allowed_stack.  ∃s0,prefix,s1,interesting,s2.
    159382  let g ≝ make_global … (pcs_exec … C) p in
    160   let C' ≝ pcs_to_cs C g stack_cost in
     383  let C' ≝ pcs_to_cs C g in
    161384  make_initial_state … p = OK ? s0 ∧
    162385  exec_steps m ?? (cs_exec … C') g s0 = OK ? 〈prefix,s1〉 ∧
     
    164387  trace_is_label_to_return C' interesting ∧
    165388  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.
    167390
    168391(* TODO: probably ought to be elsewhere; use exec_steps instead
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