1 | |
---|
2 | include "compiler.ma". |
---|
3 | |
---|
4 | include "ASM/Interpret2.ma". |
---|
5 | |
---|
6 | include "Clight/Cexec.ma". |
---|
7 | |
---|
8 | include "Clight/labelSimulation.ma". |
---|
9 | |
---|
10 | theorem correct : |
---|
11 | ∀input_program. |
---|
12 | |
---|
13 | ∀object_code,costlabel_map,labelled,cost_map. |
---|
14 | compile input_program = OK ? 〈〈object_code,costlabel_map〉,❬labelled,cost_map❭〉 → |
---|
15 | |
---|
16 | not_wrong … (exec_inf … clight_fullexec input_program) → |
---|
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 |
---|
25 | #NOT_WRONG |
---|
26 | cases (bind_inversion ????? COMPILE) -COMPILE * * #init_cost #labelled' #rtlabs_program * #FRONTEND #COMPILE |
---|
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 | % |
---|
33 | [ (* Needs switch removal too, now |
---|
34 | @labelling_sim @NOT_WRONG |
---|
35 | *) cases daemon |
---|
36 | | @I |
---|
37 | ] qed. |
---|
38 | |
---|
39 | |
---|
40 | include "Clight/Clight_abstract.ma". |
---|
41 | include "common/Measurable.ma". |
---|
42 | |
---|
43 | (* We could restrict this function to identifiers that are function names in the |
---|
44 | program and lift it (like the lift_cost_map_back_to_front function), but |
---|
45 | let's go with the easier notion of having a total map and ignore all the |
---|
46 | extra stuff. *) |
---|
47 | definition stack_cost_T ≝ ident → nat. |
---|
48 | |
---|
49 | (* Again, in principle we can use the fact that the state must be a call and |
---|
50 | also show some relevant invariants about call states, but we go with the |
---|
51 | easier option of returning zero elsewhere. *) |
---|
52 | definition Clight_stack_cost : |
---|
53 | stack_cost_T → |
---|
54 | cl_genv → |
---|
55 | ∀s:Clight_state. |
---|
56 | match Clight_classify s with [ cl_call ⇒ True | _ ⇒ False ] → |
---|
57 | nat ≝ |
---|
58 | λcost,ge,s,H. |
---|
59 | match s with |
---|
60 | [ Callstate v _ _ _ _ ⇒ |
---|
61 | match v with [ Vptr p ⇒ match symbol_for_block … ge (pblock p) with [ Some id ⇒ cost id | _ ⇒ 0 ] | _ ⇒ 0 ] |
---|
62 | | _ ⇒ 0 |
---|
63 | ]. |
---|
64 | |
---|
65 | definition Clight_pcs : preclassified_system ≝ |
---|
66 | mk_preclassified_system |
---|
67 | clight_fullexec |
---|
68 | (λ_.Clight_labelled) |
---|
69 | (λ_.Clight_classify) |
---|
70 | Clight_stack_cost. |
---|
71 | |
---|
72 | (* From measurable on Clight, we will end up with an RTLabs flat trace where |
---|
73 | we know that there are some m' and n' such that the prefix in Clight matches |
---|
74 | the prefix in RTLabs given by m', the next n steps in Clight are equivalent |
---|
75 | to the n' steps in RTLabs, and we have a suitable "will_return" for RTLabs |
---|
76 | for those n' steps so that we can build a corresponding structured trace. |
---|
77 | |
---|
78 | "Equivalent" here means, in particular, that the observables will be the same, |
---|
79 | and those observables will include the stack space costs. |
---|
80 | *) |
---|
81 | |
---|
82 | axiom observables_8051 : object_code → nat → nat → option ((list trace) × (list trace)). |
---|
83 | |
---|
84 | definition in_execution_prefix : execution_prefix Clight_state → costlabel → Prop ≝ |
---|
85 | λx,l. Exists … (λtrs. Exists … (λev. ev = EVcost l) (\fst trs)) x. |
---|
86 | |
---|
87 | 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 ≝ |
---|
88 | match l' return λl'. (∀b. Exists … (λx.x=b) l' → Exists … (λx.x=b) l) → A with |
---|
89 | [ nil ⇒ λ_. a |
---|
90 | | cons h t ⇒ λH. foldl_exists_aux A B l t f (f a h (H …)) ? |
---|
91 | ]. |
---|
92 | [ %1 % |
---|
93 | | #b #H' @H %2 @H' |
---|
94 | ] qed. |
---|
95 | |
---|
96 | definition foldl_exists : ∀A,B:Type[0]. ∀l:list B. (A → ∀b:B. Exists … (λx. x = b ) l → A) → A → A ≝ |
---|
97 | λA,B,l,f,a. foldl_exists_aux A B l l f a (λb,H. H). |
---|
98 | |
---|
99 | lemma Exists_lift : ∀A,P,Q,l. |
---|
100 | (∀x. P x → Q x) → |
---|
101 | Exists A P l → |
---|
102 | Exists A Q l. |
---|
103 | #A #P #Q #l elim l |
---|
104 | [ // |
---|
105 | | #h #t #IH #H * [ #H' %1 @H @H' | #H' %2 @IH /2/ ] |
---|
106 | ] qed. |
---|
107 | |
---|
108 | definition measure_clock : ∀x:execution_prefix Clight_state. ((Σl:costlabel.in_execution_prefix x l)→ℕ) → nat ≝ |
---|
109 | λx,costmap. foldl_exists … x |
---|
110 | (λclock,trs,H. |
---|
111 | 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) |
---|
112 | 0. |
---|
113 | whd @(Exists_lift … H) * #tr1 #s1 #E destruct @(Exists_lift … H') #ev1 #E @E |
---|
114 | qed. |
---|
115 | |
---|
116 | definition clight_clock_after : ∀p:clight_program. nat → ((Σl:costlabel.in_clight_program p l)→ℕ) → option nat ≝ |
---|
117 | λp,n,costmap. |
---|
118 | let x ≝ exec_inf … clight_fullexec p in |
---|
119 | match split_trace … x n with |
---|
120 | [ Some traces ⇒ |
---|
121 | Some ? (measure_clock (\fst traces) (λl.costmap «l,?»)) |
---|
122 | | None ⇒ None ? |
---|
123 | ]. |
---|
124 | cases daemon |
---|
125 | qed. |
---|
126 | |
---|
127 | axiom initial_8051_status : ∀oc. Status oc. |
---|
128 | |
---|
129 | definition simulates ≝ |
---|
130 | λstack_cost : stack_cost_T. λ stack_bound, labelled, object_code, cost_map. |
---|
131 | let initial_status ≝ initial_8051_status (load_code_memory object_code) in |
---|
132 | ∀m1,m2. measurable Clight_pcs labelled m1 m2 stack_cost stack_bound → |
---|
133 | ∀c1,c2. clight_clock_after labelled m1 cost_map = Some ? c1 → clight_clock_after labelled m2 cost_map = Some ? c2 → |
---|
134 | ∃n1,n2. observables clight_fullexec labelled m1 m2 = observables_8051 object_code n1 n2 ∧ |
---|
135 | c2 - c1 = clock … (execute n2 ? initial_status) - clock … (execute n1 ? initial_status). |
---|
136 | |
---|
137 | axiom compile' : clight_program → res (object_code × costlabel_map × |
---|
138 | (𝚺labelled:clight_program. ((Σl:costlabel.in_clight_program labelled l)→ℕ)) × stack_cost_T × nat). |
---|
139 | |
---|
140 | theorem correct' : |
---|
141 | ∀input_program. |
---|
142 | |
---|
143 | not_wrong … (exec_inf … clight_fullexec input_program) → |
---|
144 | |
---|
145 | ∀object_code,costlabel_map,labelled,cost_map,stack_cost,stack_bound. |
---|
146 | compile' input_program = OK ? 〈〈〈object_code,costlabel_map〉,❬labelled,cost_map❭〉,stack_cost,stack_bound〉 → |
---|
147 | |
---|
148 | sim_with_labels (exec_inf … clight_fullexec input_program) (exec_inf … clight_fullexec labelled) |
---|
149 | ∧ |
---|
150 | |
---|
151 | simulates stack_cost stack_bound labelled object_code cost_map. |
---|
152 | |
---|
153 | |
---|
154 | |
---|
155 | (* start of old simulates |
---|
156 | |
---|
157 | (* [nth_state_of_with_stack state stack_cost stack_bound exec n] returns [Some s] iff after |
---|
158 | [n] steps of [exec] we have reached [s] without exceeding the [stack_bound] |
---|
159 | according to the [stack_cost] function. *) |
---|
160 | axiom nth_state_of_with_stack : ∀state. (state → nat) → nat → execution state io_out io_in → nat → option state. |
---|
161 | axiom nth_state_of : ∀state. execution state io_out io_in → nat → option state. |
---|
162 | |
---|
163 | |
---|
164 | let cl_trace ≝ exec_inf … clight_fullexec labelled in |
---|
165 | let asm_trace ≝ exec_inf … ASM_fullexec object_code in |
---|
166 | not_wrong ? cl_trace → |
---|
167 | ∀n,s. nth_state_of_with_stack ? stack_cost stack_bound cl_trace n = Some ? s → |
---|
168 | 𝚺m,s'. nth_state_of ? asm_trace m = Some ? s' ∧ s ≃ s' |
---|
169 | |
---|
170 | *) |
---|
171 | |
---|
172 | (* TODO |
---|
173 | |
---|
174 | |
---|
175 | ∀input_program. |
---|
176 | ! 〈object_code,costlabel_map,labelled,cost_map〉 ← compile input_program |
---|
177 | |
---|
178 | exec_inf … clight_fullexec input_program ≃l exec_inf … clight_fullexec labelled |
---|
179 | |
---|
180 | ∧ |
---|
181 | |
---|
182 | exec_inf … clight_fullexec labelled ≈ exec_inf … ASM_fullexec object_code |
---|
183 | (* Should we be lifting labels in some way here? *) |
---|
184 | |
---|
185 | ∧ |
---|
186 | |
---|
187 | ∀i,f : clight_status. |
---|
188 | Clight_labelled i → |
---|
189 | Clight_labelled f → |
---|
190 | ∀mx,time. |
---|
191 | let trace ≝ exec_inf_aux … clight_fullexec labelled i in |
---|
192 | will_return O O mx time f trace → |
---|
193 | mx < max_allowed_stack → |
---|
194 | ∃!i',f'. i ≃ i' ∧ f ≃ f' ∧ i' 8051~> f' ∧ |
---|
195 | time = clock f' - clock i'. |
---|
196 | |
---|
197 | |
---|
198 | ∀s,flat. |
---|
199 | let ge ≝ (globalenvs … labelled) in |
---|
200 | subtrace_of (exec_inf … RTLabs_fullexec labelled) flat → |
---|
201 | RTLabs_cost s = true → |
---|
202 | ∀WR : will_return ge 0 s flat. |
---|
203 | let structured_trace_rtlabs ≝ make_label_return' ge 0 s flat ??? WR in |
---|
204 | let labels_rtlabs ≝ flat_label_trace … flat WR in |
---|
205 | ∃!initial,final,structured_trace_asm. |
---|
206 | structured_trace_rtlabs ≈ structured_trace_asm ∧ |
---|
207 | clock … code_memory … final = clock … code_memory … initial + |
---|
208 | (Σ_{i < |labels_rtlabs|} (cost_map (match nth i labels_rtlabs with [ Some k ⇒ k | None ⇒ 0 ])). |
---|
209 | |
---|
210 | |
---|
211 | |
---|
212 | What is ≃l? Must show that "labelled" does everything that |
---|
213 | "input_program" does, without getting lost in some |
---|
214 | non-terminating loop part way. |
---|
215 | |
---|
216 | *) |
---|
217 | |
---|