-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathDEX_BigStep.v
196 lines (179 loc) · 6.91 KB
/
DEX_BigStep.v
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
(** * Bicolano: Big step (interface implementation) *)
(* <Insert License Here>
$Id: BigStep.v 68 2006-02-02 15:06:27Z davidpichardie $ *)
(** Big step semantics.
@author David Pichardie *)
(* Hendra : - Modified to suit DEX program.
- Also trim the system to contain only Arithmetic *)
Require Export DEX_BigStepType.
Require Export DEX_ImplemDomain.
Open Scope type_scope.
Module DEX_BigStep <: DEX_BIGSTEP.
Module DEX_Dom := DEX_ImplemDomain.DEX_Dom.
(* Inductive definition are put in BigStepLoad.v.
They are shared with BigStepType.v *)
Load "DEX_BigStepLoad.v".
Lemma IntraStep_ind_ :
forall (p:DEX_Program) (P:DEX_Method->DEX_IntraNormalState->DEX_IntraNormalState+DEX_ReturnState->Prop),
(forall m s, P m s (inl _ s)) ->
(forall m s r, DEX_exec_return p m s r -> P m s (inr _ r)) ->
(forall m s s' , DEX_exec_intra p m s s' ->
forall r, DEX_IntraStepStar p m s' r -> P m s' r ->
P m s r) ->
(* DEX Method
(forall m s s' ret m' r,
DEX_exec_call p m s ret m' s' (inr _ r) ->
DEX_IntraStepStar p m' s' (inr _ ret) ->
P m' s' (inr _ ret) ->
P m s (inr _ r)) ->
(forall m s s' ret m' s'' r,
DEX_exec_call p m s ret m' s' (inl _ s'') ->
DEX_IntraStepStar p m' s' (inr _ ret) -> P m' s' (inr _ ret) ->
DEX_IntraStepStar p m s'' r -> P m s'' r ->
P m s r) ->
*)
forall m s r, DEX_IntraStep p m s r ->
match r with
| inr r' => P m s (inr _ r')
| inl s' => forall r', DEX_IntraStepStar p m s' r' -> P m s' r' -> P m s r'
end.
Proof.
intros p P H0 Hr Hi Hcr Hc.
fix intra (*4*) 2;intros (*m s*) r Hs;case Hs;clear (*m s*) r Hs;intros.
apply Hr;trivial.
apply Hi with s2;trivial.
Qed.
(* DEX Method
assert (P m' s' (inr DEX_IntraNormalState ret')).
generalize s' (inr DEX_IntraNormalState ret') H1;clear H1 H m s1 s' ret' r.
fix fixp 3;intros s' s Ht;case Ht;clear Ht s' s;intros.
apply H0.
generalize (intra _ _ _ H);clear H;case r;intros.
apply H;trivial. constructor. trivial.
assert (HH:= intra _ _ _ H);simpl in HH.
apply HH;trivial. apply fixp;trivial.
generalize H;clear H;case r.
intros s'' Hcall r' Hint HP.
eapply Hc;eauto.
intros r' Hcall;eapply Hcr;eauto.
Qed.
*)
Lemma IntraStepStar_ind :
forall (p:DEX_Program)
(P : DEX_Method -> DEX_IntraNormalState -> DEX_IntraNormalState + DEX_ReturnState -> Prop),
(forall m s, P m s (inl _ s)) ->
(forall m s r, DEX_exec_return p m s r -> P m s (inr _ r)) ->
(forall m s s' , DEX_exec_intra p m s s' ->
forall r, DEX_IntraStepStar p m s' r -> P m s' r ->
P m s r) ->
(* DEX Method
(forall m s s' ret m' r,
DEX_exec_call p m s ret m' s' (inr _ r) ->
DEX_IntraStepStar p m' s' (inr _ ret) ->
P m' s' (inr _ ret) ->
P m s (inr _ r)) ->
(forall m s s' ret m' s'' r,
DEX_exec_call p m s ret m' s' (inl _ s'') ->
DEX_IntraStepStar p m' s' (inr _ ret) -> P m' s' (inr _ ret) ->
DEX_IntraStepStar p m s'' r -> P m s'' r ->
P m s r) ->
*)
forall m s r, DEX_IntraStepStar p m s r -> P m s r.
Proof.
intros p P H0 Hr Hi (*Hcr Hc*).
fix fixp 4; intros m s' s Ht;case Ht;clear Ht s' s;intros.
apply H0.
generalize (IntraStep_ind_ p P H0 Hr Hi (*Hcr Hc*) _ _ _ H).
case r;intros;trivial.
apply H1;trivial. constructor.
assert (HH:=IntraStep_ind_ p P H0 Hr Hi (*Hcr Hc*) _ _ _ H);simpl in HH.
apply HH;trivial.
apply fixp;trivial.
Qed.
Lemma IntraStepStar_intra_ind :
forall (p:DEX_Program)
(P : DEX_Method -> DEX_IntraNormalState -> DEX_IntraNormalState -> Prop),
(forall m s, P m s s) ->
(forall m s s', DEX_exec_intra p m s s' ->
forall s'', DEX_IntraStepStar_intra p m s' s'' -> P m s' s'' ->
P m s s'') ->
(* DEX Method
(forall m s s1 ret m' s2 s3,
DEX_exec_call p m s ret m' s1 (inl _ s2) ->
DEX_BigStep p m' s1 ret ->
DEX_IntraStepStar_intra p m s2 s3 -> P m s2 s3 ->
P m s s3) ->
*)
forall m s s', DEX_IntraStepStar_intra p m s s' -> P m s s'.
Proof.
intros p P H0 Hi Hc.
assert (forall m s r, DEX_IntraStepStar p m s r ->
forall s', r = inl _ s' -> P m s s').
induction 1 using IntraStepStar_ind;intros;try discriminate;subst;eauto.
inversion H;auto.
intros; eapply H; eauto.
Qed.
Lemma BigStep_ind :
forall (p:DEX_Program)
(P : DEX_Method -> DEX_IntraNormalState -> DEX_ReturnState -> Prop),
(forall m s r, DEX_exec_return p m s r -> P m s r) ->
(forall m s s' , DEX_exec_intra p m s s' ->
forall r, DEX_BigStep p m s' r -> P m s' r ->
P m s r) ->
(* DEX Method
(forall m s s' ret m' r,
DEX_exec_call p m s ret m' s' (inr _ r) ->
DEX_BigStep p m' s' ret ->
P m' s' ret ->
P m s r) ->
(forall m s s' ret m' s'' r,
DEX_exec_call p m s ret m' s' (inl _ s'') ->
DEX_BigStep p m' s' ret -> P m' s' ret ->
DEX_BigStep p m s'' r -> P m s'' r ->
P m s r) ->
*)
forall m s r, DEX_BigStep p m s r -> P m s r.
Proof.
intros p P Hr Hi Hcr Hc.
assert (forall m s R, DEX_IntraStepStar p m s R -> forall r, R = inr _ r -> P m s r).
induction 1 using IntraStepStar_ind;intros r0 Heq;try inversion Heq;subst;
try (eauto;fail).
intros;eapply H;eauto.
Qed.
Lemma ReachableStar_ind :
forall (p:DEX_Program)
(P : (DEX_Method * DEX_IntraNormalState) -> (DEX_Method * DEX_IntraNormalState) -> Prop),
(forall m s, P (m,s) (m,s)) ->
(forall m s s', DEX_exec_intra p m s s' ->
forall m' s'', ClosReflTrans (DEX_ReachableStep p) (m,s') (m',s'') ->
P (m,s') (m',s'') ->
P (m,s) (m',s'')) ->
(* DEX Method
(forall m s s1 ret m' s2,
DEX_exec_call p m s ret m' s1 (inl _ s2) ->
DEX_BigStep p m' s1 ret ->
forall m' s3, ClosReflTrans (DEX_ReachableStep p) (m,s2) (m',s3) ->
P (m,s2) (m',s3) ->
P (m,s) (m',s3)) ->
(forall m pc h l m' l' bm',
DEX_CallStep p m (pc,(h,l)) (m',l') ->
DEX_METHOD.body m' = Some bm' ->
forall m'' s'',
ClosReflTrans (DEX_ReachableStep p)
(m', (DEX_BYTECODEMETHOD.firstAddress bm',(h, l')))
(m'',s'') ->
P (m', (DEX_BYTECODEMETHOD.firstAddress bm',(h, l')))
(m'',s'') ->
P (m, (pc,(h, l))) (m'',s'')) ->
*)
forall ms ms',
ClosReflTrans (DEX_ReachableStep p) ms ms' -> P ms ms'.
Proof.
intros p P H0 Hi Hc Hsc.
induction 1;intros.
destruct a;eauto.
destruct a'' as (m'',s'').
inversion H;subst;eauto.
inversion H2;clear H2;subst;eauto.
Qed.
End DEX_BigStep.