source: src/common/StructuredTraces.ma @ 1658

Last change on this file since 1658 was 1658, checked in by mulligan, 8 years ago

asm costs changes from today

File size: 6.8 KB
Line 
1include "basics/types.ma".
2include "basics/bool.ma".
3include "basics/jmeq.ma".
4
5inductive status_class : Type[0] ≝
6| cl_return : status_class
7| cl_jump   : status_class
8| cl_call   : status_class
9| cl_other : status_class
10.
11
12record abstract_status : Type[1] ≝ {
13  as_status :> Type[0];
14  as_execute : as_status → as_status → Prop;
15  as_classifier : as_status → status_class → Prop;
16  as_costed : as_status → Prop;
17  as_after_return : (Σs:as_status. as_classifier s cl_call) → as_status → Prop
18}.
19
20inductive trace_ends_with_ret: Type[0] ≝
21  | ends_with_ret: trace_ends_with_ret
22  | doesnt_end_with_ret: trace_ends_with_ret.
23
24inductive trace_label_return (S:abstract_status) : S → S → Type[0] ≝
25  | tlr_base:
26      ∀status_before: S.
27      ∀status_after: S.
28        trace_label_label S ends_with_ret status_before status_after →
29        trace_label_return S status_before status_after
30  | tlr_step:
31      ∀status_initial: S.
32      ∀status_labelled: S.
33      ∀status_final: S.
34        trace_label_label S doesnt_end_with_ret status_initial status_labelled →
35        trace_label_return S status_labelled status_final →
36          trace_label_return S status_initial status_final
37with trace_label_label: trace_ends_with_ret → S → S → Type[0] ≝
38  | tll_base:
39      ∀ends_flag: trace_ends_with_ret.
40      ∀start_status: S.
41      ∀end_status: S.
42        trace_any_label S ends_flag start_status end_status →
43        as_costed S start_status →
44          trace_label_label S ends_flag start_status end_status
45with trace_any_label: trace_ends_with_ret → S → S → Type[0] ≝
46  (* Single steps within a function which reach a label.
47     Note that this is the only case applicable for a jump. *)
48  | tal_base_not_return:
49      ∀start_status: S.
50      ∀final_status: S.
51        as_execute S start_status final_status →
52        (as_classifier S start_status cl_jump ∨
53         as_classifier S start_status cl_other) →
54        as_costed S final_status →
55          trace_any_label S doesnt_end_with_ret start_status final_status
56  | tal_base_return:
57      ∀start_status: S.
58      ∀final_status: S.
59        as_execute S start_status final_status →
60        as_classifier S start_status cl_return →
61          trace_any_label S ends_with_ret start_status final_status
62  (* A call followed by a label on return. *)
63  | tal_base_call:
64      ∀status_pre_fun_call: S.
65      ∀status_start_fun_call: S.
66      ∀status_final: S.
67        as_execute S status_pre_fun_call status_start_fun_call →
68        ∀H:as_classifier S status_pre_fun_call cl_call.
69          as_after_return S (mk_Sig ?? status_pre_fun_call H) status_final →
70          trace_label_return S status_start_fun_call status_final →
71          as_costed S status_final →
72            trace_any_label S doesnt_end_with_ret status_pre_fun_call status_final
73  (* A call followed by a non-empty trace. *)
74  | tal_step_call:
75      ∀end_flag: trace_ends_with_ret.
76      ∀status_pre_fun_call: S.
77      ∀status_start_fun_call: S.
78      ∀status_after_fun_call: S.
79      ∀status_final: S.
80        as_execute S status_pre_fun_call status_start_fun_call →
81        ∀H:as_classifier S status_pre_fun_call cl_call.
82          as_after_return S (mk_Sig ?? status_pre_fun_call H) status_after_fun_call →
83          trace_label_return S status_start_fun_call status_after_fun_call →
84          ¬ as_costed S status_after_fun_call →
85          trace_any_label S end_flag status_after_fun_call status_final →
86            trace_any_label S end_flag status_pre_fun_call status_final
87  | tal_step_default:
88      ∀end_flag: trace_ends_with_ret.
89      ∀status_pre: S.
90      ∀status_init: S.
91      ∀status_end: S.
92        as_execute S status_pre status_init →
93        trace_any_label S end_flag status_init status_end →
94        as_classifier S status_pre cl_other →
95        ¬ (as_costed S status_init) →
96          trace_any_label S end_flag status_pre status_end.
97
98coinductive trace_label_diverges (S:abstract_status) : S → Type[0] ≝
99  | tld_step:
100      ∀status_initial: S.
101      ∀status_labelled: S.
102          trace_label_label S doesnt_end_with_ret status_initial status_labelled →
103          trace_label_diverges S status_labelled →
104            trace_label_diverges S status_initial
105  | tld_base:
106      ∀status_initial: S.
107      ∀status_pre_fun_call: S.
108      ∀status_after_fun_call: S.
109      ∀status_start_fun_call: S.
110        trace_label_call S status_initial status_pre_fun_call →
111        as_execute S status_pre_fun_call status_start_fun_call →
112        ∀H:as_classifier S status_pre_fun_call cl_call.
113          trace_label_diverges S status_start_fun_call →
114            trace_label_diverges S status_initial
115             
116with trace_label_call: S → S → Type[0] ≝
117  | tlc_base:
118      ∀start_status: S.
119      ∀end_status: S.
120        trace_any_call S start_status end_status →
121        as_costed S start_status →
122        trace_label_call S start_status end_status
123       
124with trace_any_call: S → S → Type[0] ≝
125  | tac_base:
126      ∀status: S.
127        as_classifier S status cl_call →
128          trace_any_call S status status
129  | tac_step_call:
130      ∀status_pre_fun_call: S.
131      ∀status_after_fun_call: S.
132      ∀status_final: S.
133      ∀status_start_fun_call: S.
134        as_execute S status_pre_fun_call status_start_fun_call →
135        ∀H:as_classifier S status_pre_fun_call cl_call.
136          as_after_return S (mk_Sig ?? status_pre_fun_call H) status_after_fun_call →
137          trace_label_return S status_start_fun_call status_after_fun_call →
138          ¬ as_costed S status_after_fun_call →
139          trace_any_call S status_after_fun_call status_final →
140            trace_any_call S status_pre_fun_call status_final
141  | tac_step_default:
142      ∀status_pre: S.
143      ∀status_end: S.
144      ∀status_init: S.
145        as_execute S status_pre status_init →
146        trace_any_call S status_init status_end →
147        as_classifier S status_pre cl_other →
148        ¬ (as_costed S status_init) →
149          trace_any_call S status_pre status_end.
150
151let rec trace_any_label_label S s s' f
152  (tr:trace_any_label S f s s') on tr : match f with [ doesnt_end_with_ret ⇒ as_costed S s' | ends_with_ret ⇒ True ] ≝
153match tr return λf,s,s',tr. match f with [ doesnt_end_with_ret ⇒ as_costed S s' | ends_with_ret ⇒ True ] with
154[ tal_base_not_return start final _ _ C ⇒ C
155| tal_base_return _ _  _ _ ⇒ I
156| tal_base_call _ _ _ _ _ _ _ C ⇒ C
157| tal_step_call f pre start after final X C RET LR C' tr' ⇒ trace_any_label_label … tr'
158| tal_step_default f pre init end X tr' C C' ⇒ trace_any_label_label … tr'
159].
160
161lemma trace_label_label_label : ∀S,s,s',f.
162  ∀tr:trace_label_label S f s s'. match f with [ doesnt_end_with_ret ⇒ as_costed S s' | ends_with_ret ⇒ True ].
163#S #s #s' #f #tr
164cases tr
165#f #start #end #tr' #C @(trace_any_label_label … tr')
166qed.
167
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