source: src/correctness.ma @ 3021

Last change on this file since 3021 was 3021, checked in by campbell, 7 years ago

Replace clight_clock_after with a more sensible definition that uses
an intensional trace. Also drop the proof argument for the clight cost
map to avoid proving that costs come from the clight program - it doesn't
seem to give us much of a benefit.

File size: 7.2 KB
Line 
1
2include "compiler.ma".
3
4include "ASM/Interpret2.ma".
5
6include "Clight/labelSimulation.ma".
7
8theorem correct :
9  ∀observe,input_program,output.
10(*  ∀lobject_code,labelled,cost_map. *)
11  compile observe input_program = return output →
12
13  not_wrong … (exec_inf … clight_fullexec input_program) →
14 
15  sim_with_labels
16   (exec_inf … clight_fullexec input_program)
17   (exec_inf … clight_fullexec (c_labelled_clight … output))
18  ∧
19  True (* TODO *).
20
21#observe #input_program #output
22#COMPILE
23#NOT_WRONG
24cases (bind_inversion ????? COMPILE) -COMPILE * * #init_cost #labelled' #rtlabs_program * #FRONTEND #COMPILE
25cases (bind_inversion ????? COMPILE) -COMPILE #lobject_code' * #ASSEMBLER #COMPILE
26whd in COMPILE:(??%%); destruct
27cases (bind_inversion ????? FRONTEND) -FRONTEND #cminor_program * #CMINOR #FRONTEND
28whd in FRONTEND:(??%%); destruct
29
30%
31[ (* Needs switch removal too, now
32     @labelling_sim @NOT_WRONG
33   *) cases daemon
34| @I
35] qed.
36
37
38include "Clight/Clight_classified_system.ma".
39
40(* From measurable on Clight, we will end up with an RTLabs flat trace where
41   we know that there are some m' and n' such that the prefix in Clight matches
42   the prefix in RTLabs given by m', the next n steps in Clight are equivalent
43   to the n' steps in RTLabs, and we have a suitable "will_return" for RTLabs
44   for those n' steps so that we can build a corresponding structured trace.
45   
46   "Equivalent" here means, in particular, that the observables will be the same,
47   and those observables will include the stack space costs.
48 *)
49
50definition in_execution_prefix : execution_prefix Clight_state → costlabel → Prop ≝
51λx,l. Exists … (λtrs. Exists … (λev. ev = EVcost l) (\fst trs)) x.
52
53let 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 ≝
54match l' return λl'. (∀b. Exists … (λx.x=b) l' → Exists … (λx.x=b) l) → A with
55[ nil ⇒ λ_. a
56| cons h t ⇒ λH. foldl_exists_aux A B l t f (f a h (H …)) ?
57].
58[ %1 %
59| #b #H' @H %2 @H'
60] qed.
61
62definition foldl_exists : ∀A,B:Type[0]. ∀l:list B. (A → ∀b:B. Exists … (λx. x = b ) l → A) → A → A ≝
63λA,B,l,f,a.  foldl_exists_aux A B l l f a (λb,H. H).
64
65lemma Exists_lift : ∀A,P,Q,l.
66  (∀x. P x → Q x) →
67  Exists A P l →
68  Exists A Q l.
69#A #P #Q #l elim l
70[ //
71| #h #t #IH #H * [ #H' %1 @H @H' | #H' %2 @IH /2/ ]
72] qed.
73
74include "RTLabs/MeasurableToStructured.ma".
75
76definition clight_clock_after : ∀p:clight_program. nat → (costlabel→ℕ) → res nat ≝
77λp,n,costmap.
78  ! 〈s,itrace〉 ← exec_steps_with_obs Clight_pcs p n;
79  let ctrace ≝ filter_map … (λi. match i with [IEVcost cl ⇒ Some … cl | _ ⇒ None ? ]) itrace in
80  return Σ_{l ∈ ctrace}(costmap l).
81
82include "common/AssocList.ma".
83
84definition lookup_stack_cost : stack_cost_model → ident → option nat ≝
85 λstack_cost,id.
86  assoc_list_lookup ?? id (eq_identifier …) stack_cost.
87
88definition simulates ≝
89  λp: compiler_output.
90  let initial_status ≝ initialise_status … (cm (c_labelled_object_code … p)) in
91  ∀m1,m2.
92   measurable Clight_pcs (c_labelled_clight … p) m1 m2
93    (lookup_stack_cost (c_stack_cost … p)) (c_max_stack … p) →
94  ∀c1,c2.
95   clight_clock_after (c_labelled_clight … p) m1 (c_clight_cost_map … p) = OK … c1 →
96   clight_clock_after (c_labelled_clight … p) m2 (c_clight_cost_map … p) = OK … c2 →
97  ∃n1,n2.
98   observables Clight_pcs (c_labelled_clight … p) m1 m2 =
99   observables (OC_preclassified_system (c_labelled_object_code … p))
100    (c_labelled_object_code … p) n1 n2
101  ∧
102   minus c2 c1 = clock … (execute n2 ? initial_status) - clock … (execute n1 ? initial_status).
103
104include "common/ExtraMonads.ma".
105
106theorem correct' :
107  ∀observe.
108  ∀input_program,output.
109  compile observe input_program = return output →
110  not_wrong … (exec_inf … clight_fullexec input_program) →
111  sim_with_labels
112   (exec_inf … clight_fullexec input_program)
113   (exec_inf … clight_fullexec (c_labelled_clight … output))
114  ∧
115  simulates output.
116#observe #p_in #out
117#H @('bind_inversion H) -H
118** #init_cost #labelled #p_rtlabs #EQ_front_end
119#H @('bind_inversion H) -H
120** #p_asm #stack_costs #globals_size
121#H @('bind_inversion H) -H
122#p_asm'
123#H lapply (opt_eq_from_res ???? H) #EQ_lin_to_asm
124whd in ⊢ (??%%→?); #EQ lapply (sym_eq ??? EQ) -EQ #EQ destruct(EQ)
125#H @('bind_inversion H) -H
126#p_loc #EQ_assembler
127whd in ⊢ (??%%→?); #EQ lapply (sym_eq ??? EQ) -EQ #EQ destruct(EQ)
128#NOT_WRONG %
129[ cases daemon (* TODO *)
130| #m1 #m2 #m1_m2_meas #c1 #c2 #c1_prf #c2_prf
131
132cut (∀n,s1,s2,obs1,obs2.
133          exec_with_observables_n n (RTLabs_init_state p_rtlabs) = return 〈obs1, s1〉 →
134            ∀tlr : trace_label_return (RTLabs_status p_rtlabs) s1 s2.
135            tlr_observables … tlr (function_of … s1) = obs2 →
136            (* maybe instead of function_of the called id can rather be obtained from execution? *)
137     ∃m,p,s_fin.
138   observables (OC_preclassified_system (c_labelled_object_code … p))
139    (c_labelled_object_code … p) m p = return 〈obs1, obs2〉)
140   
141     
142             
143             
144 
145(* start of old simulates 
146
147(* [nth_state_of_with_stack state stack_cost stack_bound exec n] returns [Some s] iff after
148   [n] steps of [exec] we have reached [s] without exceeding the [stack_bound]
149   according to the [stack_cost] function. *)
150axiom nth_state_of_with_stack : ∀state. (state → nat) → nat → execution state io_out io_in → nat → option state.
151axiom nth_state_of : ∀state. execution state io_out io_in → nat → option state.
152
153
154  let cl_trace ≝ exec_inf … clight_fullexec labelled in
155  let asm_trace ≝ exec_inf … ASM_fullexec object_code in
156  not_wrong ? cl_trace →
157  ∀n,s. nth_state_of_with_stack ? stack_cost stack_bound cl_trace n = Some ? s →
158  𝚺m,s'. nth_state_of ? asm_trace m = Some ? s' ∧ s ≃ s'
159
160*)
161
162(* TODO
163
164
165∀input_program.
166! 〈object_code,costlabel_map,labelled,cost_map〉 ← compile input_program
167
168exec_inf … clight_fullexec input_program ≃l exec_inf … clight_fullexec labelled
169
170
171
172exec_inf … clight_fullexec labelled ≈ exec_inf … ASM_fullexec object_code
173(* Should we be lifting labels in some way here? *)
174
175
176
177∀i,f : clight_status.
178  Clight_labelled i →
179  Clight_labelled f →
180∀mx,time.
181  let trace ≝ exec_inf_aux … clight_fullexec labelled i in
182  will_return O O mx time f trace →
183  mx < max_allowed_stack →
184∃!i',f'. i ≃ i' ∧ f ≃ f' ∧ i' 8051~> f' ∧
185  time = clock f' - clock i'.
186
187
188∀s,flat.
189let ge ≝ (globalenvs … labelled) in
190subtrace_of (exec_inf … RTLabs_fullexec labelled) flat →
191RTLabs_cost s = true →
192∀WR : will_return ge 0 s flat.
193let structured_trace_rtlabs ≝ make_label_return' ge 0 s flat ??? WR in
194let labels_rtlabs ≝ flat_label_trace … flat WR in
195∃!initial,final,structured_trace_asm.
196  structured_trace_rtlabs ≈ structured_trace_asm ∧ 
197  clock … code_memory … final = clock … code_memory … initial +
198     (Σ_{i < |labels_rtlabs|} (cost_map (match nth i labels_rtlabs with [ Some k ⇒ k | None ⇒ 0 ])).
199
200
201
202What is ≃l?  Must show that "labelled" does everything that
203"input_program" does, without getting lost in some
204non-terminating loop part way.
205
206*)
207
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