-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathFrame_Validation.v
162 lines (141 loc) · 3.07 KB
/
Frame_Validation.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
Require Import Modal_Library Classical.
Theorem validation_frame_reflexivo_ida:
forall M Ψ,
reflexivity_frame (F M) ->
(M |= .[] Ψ .-> Ψ).
Proof.
intros;
unfold validate_model in *;
simpl in *; intuition.
Qed.
Theorem validation_frame_reflexivo_volta:
forall M Ψ,
(M |= .[] Ψ .-> Ψ) ->
reflexivity_frame (F M).
Proof.
Admitted.
Theorem validation_frame_transitivo_ida:
forall M φ,
transitivity_frame (F M) ->
(M |= .[]φ .-> .[].[]φ).
Proof.
unfold validate_model, transitivity_frame.
simpl; intros.
apply H0.
eapply H; split.
- apply H1.
- assumption.
Qed.
Theorem validation_frame_transitivo_volta:
forall M φ,
(M |= .[]φ .-> .[].[]φ) ->
transitivity_frame (F M).
Proof.
Admitted.
Theorem validation_frame_simetria_ida:
forall M φ,
simmetry_frame (F M) ->
(M |= φ .-> .[] .<> φ).
Proof.
unfold validate_model, simmetry_frame.
simpl in *; intros; exists w.
apply and_comm; split.
- apply H0.
- eauto.
Qed.
Theorem validation_frame_simetria_volta:
forall M φ,
(M |= φ .-> .[] .<> φ) ->
simmetry_frame (F M).
Proof.
Admitted.
Theorem validation_frame_eucliadiana_ida:
forall M φ,
euclidian_frame (F M) ->
(M |= .<> φ .-> .[] .<> φ).
Proof.
unfold euclidian_frame, validate_model.
simpl in *; intros.
edestruct H0.
exists x; split.
- eapply H; split.
* apply H1.
* intuition.
- intuition.
Qed.
Theorem validation_frame_eucliadiana_volta:
forall M φ,
(M |= .<> φ .-> .[] .<> φ) ->
euclidian_frame (F M).
Proof.
Admitted.
Theorem validation_frame_serial_ida:
forall M φ,
serial_frame (F M) ->
(M |= .[] φ .-> .<> φ).
Proof.
unfold validate_model, serial_frame.
simpl in *; intros.
edestruct H.
exists x; split.
- apply H1.
- eauto.
Qed.
Theorem validation_frame_serial_volta:
forall M φ,
(M |= .[] φ .-> .<> φ) ->
serial_frame (F M).
Proof.
Admitted.
Theorem validation_frame_funcional_ida:
forall M φ,
functional_frame (F M) ->
(M |= .<> φ .-> .[] φ).
Proof.
intros; unfold validate_model;
unfold functional_frame in *.
intros w H0 w1 H1.
edestruct H0.
edestruct H with (w:=w) (w':=w1) (w'':=x).
- split.
+ apply H1.
+ intuition.
- intuition.
Qed.
Theorem validation_frame_funcional_volta:
forall M φ,
(M |= .<> φ .-> .[] φ) ->
functional_frame (F M).
Proof.
Admitted.
Theorem validation_frame_densa_ida:
forall M φ,
dense_frame (F M) ->
(M |= .[] .[] φ .-> .[] φ).
Proof.
Admitted.
Theorem validation_frame_densa_volta:
forall M φ,
(M |= .[] .[] φ .-> .[] φ) ->
dense_frame (F M).
Proof.
Admitted.
Theorem validation_frame_convergente_ida:
forall M φ,
convergente_frame (F M) ->
(M |= .<> .[] φ .-> .[] .<> φ).
Proof.
unfold convergente_frame, validate_model.
simpl in *; intros.
edestruct H0.
destruct H with (w:=w) (x:=x) (y:=w').
destruct H0; auto.
exists x0.
split; intuition.
Qed.
Theorem validation_frame_convergente_volta:
forall (M: Model) (φ: modalFormula),
(M |= .<> .[] φ .-> .[] .<> φ) ->
convergente_frame (F M).
Proof.
Admitted.