source: src/correctness.ma @ 2832

Last change on this file since 2832 was 2802, checked in by sacerdot, 7 years ago

New file Clight_classified_system with the classified system for Clight.
The code comes from correctness.ma and it is used in the untrusted code
for debugging too.

File size: 6.5 KB
Line 
1
2include "compiler.ma".
3
4include "ASM/Interpret2.ma".
5
6include "Clight/labelSimulation.ma".
7
8theorem correct :
9  ∀input_program,output.
10(*  ∀lobject_code,labelled,cost_map. *)
11  compile input_program = OK ? 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#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
74definition measure_clock : ∀x:execution_prefix Clight_state. ((Σl:costlabel.in_execution_prefix x l)→ℕ) → nat ≝
75λx,costmap. foldl_exists … x
76 (λclock,trs,H.
77    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)
78 0.
79whd @(Exists_lift … H) * #tr1 #s1 #E destruct @(Exists_lift … H') #ev1 #E @E
80qed.
81
82definition clight_clock_after : ∀p:clight_program. nat → ((Σl:costlabel.in_clight_program p l)→ℕ) → option nat ≝
83λp,n,costmap.
84  let x ≝ exec_inf … clight_fullexec p in
85  match split_trace … x n with
86  [ Some traces ⇒
87    Some ? (measure_clock (\fst traces) (λl.costmap «l,?»))
88  | None ⇒ None ?
89  ].
90cases daemon
91qed.
92
93include "common/AssocList.ma".
94
95definition lookup_stack_cost : stack_cost_model → ident → nat ≝
96 λstack_cost,id.
97  match assoc_list_lookup ?? id (eq_identifier …) stack_cost with
98  [ None ⇒ 0 | Some n ⇒ n ].
99
100definition simulates ≝
101  λp: compiler_output.
102  let initial_status ≝ initialise_status … (load_code_memory (oc (c_labelled_object_code … p))) in
103  ∀m1,m2.
104   measurable Clight_pcs (c_labelled_clight … p) m1 m2
105    (lookup_stack_cost (c_stack_cost … p)) (c_max_stack … p) →
106  ∀c1,c2.
107   clight_clock_after (c_labelled_clight … p) m1 (c_clight_cost_map … p) = Some ? c1 →
108   clight_clock_after (c_labelled_clight … p) m2 (c_clight_cost_map … p) = Some ? c2 →
109  ∃n1,n2.
110   observables Clight_pcs (c_labelled_clight … p) m1 m2 =
111   observables (OC_preclassified_system (c_labelled_object_code … p)) it n1 n2
112  ∧
113   minus c2 c1 = clock … (execute n2 ? initial_status) - clock … (execute n1 ? initial_status).
114
115theorem correct' :
116  ∀input_program,output.
117  ∀initial_status.
118  compile input_program = OK ? output →
119  not_wrong … (exec_inf … clight_fullexec input_program) →
120  sim_with_labels
121   (exec_inf … clight_fullexec input_program)
122   (exec_inf … clight_fullexec (c_labelled_clight … output))
123  ∧
124  simulates output.
125 
126(* start of old simulates 
127
128(* [nth_state_of_with_stack state stack_cost stack_bound exec n] returns [Some s] iff after
129   [n] steps of [exec] we have reached [s] without exceeding the [stack_bound]
130   according to the [stack_cost] function. *)
131axiom nth_state_of_with_stack : ∀state. (state → nat) → nat → execution state io_out io_in → nat → option state.
132axiom nth_state_of : ∀state. execution state io_out io_in → nat → option state.
133
134
135  let cl_trace ≝ exec_inf … clight_fullexec labelled in
136  let asm_trace ≝ exec_inf … ASM_fullexec object_code in
137  not_wrong ? cl_trace →
138  ∀n,s. nth_state_of_with_stack ? stack_cost stack_bound cl_trace n = Some ? s →
139  𝚺m,s'. nth_state_of ? asm_trace m = Some ? s' ∧ s ≃ s'
140
141*)
142
143(* TODO
144
145
146∀input_program.
147! 〈object_code,costlabel_map,labelled,cost_map〉 ← compile input_program
148
149exec_inf … clight_fullexec input_program ≃l exec_inf … clight_fullexec labelled
150
151
152
153exec_inf … clight_fullexec labelled ≈ exec_inf … ASM_fullexec object_code
154(* Should we be lifting labels in some way here? *)
155
156
157
158∀i,f : clight_status.
159  Clight_labelled i →
160  Clight_labelled f →
161∀mx,time.
162  let trace ≝ exec_inf_aux … clight_fullexec labelled i in
163  will_return O O mx time f trace →
164  mx < max_allowed_stack →
165∃!i',f'. i ≃ i' ∧ f ≃ f' ∧ i' 8051~> f' ∧
166  time = clock f' - clock i'.
167
168
169∀s,flat.
170let ge ≝ (globalenvs … labelled) in
171subtrace_of (exec_inf … RTLabs_fullexec labelled) flat →
172RTLabs_cost s = true →
173∀WR : will_return ge 0 s flat.
174let structured_trace_rtlabs ≝ make_label_return' ge 0 s flat ??? WR in
175let labels_rtlabs ≝ flat_label_trace … flat WR in
176∃!initial,final,structured_trace_asm.
177  structured_trace_rtlabs ≈ structured_trace_asm ∧ 
178  clock … code_memory … final = clock … code_memory … initial +
179     (Σ_{i < |labels_rtlabs|} (cost_map (match nth i labels_rtlabs with [ Some k ⇒ k | None ⇒ 0 ])).
180
181
182
183What is ≃l?  Must show that "labelled" does everything that
184"input_program" does, without getting lost in some
185non-terminating loop part way.
186
187*)
188
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