-
Notifications
You must be signed in to change notification settings - Fork 3
/
Omega.v
190 lines (149 loc) · 4.57 KB
/
Omega.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
(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(* Contribution to the Coq Library V6.3 (July 1999) *)
(* The set of natural numbers *)
Require Import Sets.
Require Import Axioms.
Definition Class_succ (E : Ens) := Union (Paire E (Sing E)).
(*
Inductive Ord : Ens -> Prop :=
Oo : (Ord Vide)
| So : (E:Ens)(Ord E)->(Ord (Class_succ E))
| Lo : (E:Ens)((e:Ens)(IN e E)->(Ord e))->(Ord (Union E))
| Eo : (E,E':Ens)(Ord E)->(EQ E E')->(Ord E').
Hints Resolve Oo So Lo : zfc.
*)
Definition Nat : nat -> Ens.
simple induction 1; intros.
exact Vide.
exact (Class_succ X).
Defined.
(*
Theorem Nat_Ord : (n:nat)(Ord (Nat n)).
Induction n; Simpl; Auto with zfc.
Save.
*)
Definition Omega : Ens := sup nat Nat.
Theorem IN_Class_succ : forall E : Ens, IN E (Class_succ E).
intros E; unfold Class_succ in |- *; unfold Sing in |- *;
apply IN_Union with (Paire E E); auto with zfc.
Qed.
Theorem INC_Class_succ : forall E : Ens, INC E (Class_succ E).
unfold INC in |- *; unfold Class_succ in |- *.
intros.
apply IN_Union with E; auto with zfc.
Qed.
Hint Resolve IN_Class_succ INC_Class_succ: zfc.
Theorem IN_Class_succ_or :
forall E E' : Ens, IN E' (Class_succ E) -> EQ E E' \/ IN E' E.
intros E E' i.
unfold Class_succ in i.
elim (Union_IN (Paire E (Sing E)) E' i).
intros E1; simple induction 1; intros i1 i2.
elim (Paire_IN E (Sing E) E1 i1).
intros; right; apply IN_sound_right with E1; auto with zfc.
intros; left; cut (IN E' (Sing E)).
auto with zfc.
apply IN_sound_right with E1; auto with zfc.
Qed.
Theorem E_not_IN_E : forall E : Ens, IN E E -> F.
simple induction E; intros A f r i.
cut False.
simple induction 1.
elim (IN_EXType (sup A f) (sup A f) i); intros a e.
simpl in a.
change (EQ (sup A f) (f a)) in e.
elim (r a).
apply IN_sound_right with (sup A f); auto with zfc.
exists a; auto with zfc.
Qed.
Theorem Nat_IN_Omega : forall n : nat, IN (Nat n) Omega.
intros; simpl in |- *; exists n; auto with zfc.
Qed.
Hint Resolve Nat_IN_Omega: zfc.
Theorem IN_Omega_EXType :
forall E : Ens, IN E Omega -> EXType _ (fun n : nat => EQ (Nat n) E).
simpl in |- *; simple induction 1.
intros n e.
exists n; auto with zfc.
Qed.
Theorem IN_Nat_EXType :
forall (n : nat) (E : Ens),
IN E (Nat n) -> EXType _ (fun p : nat => EQ E (Nat p)).
simple induction n.
simpl in |- *.
simple induction 1.
simple induction x.
intros.
change (IN E (Class_succ (Nat n0))) in H0.
elim (IN_Class_succ_or (Nat n0) E H0).
intros; exists n0.
auto with zfc.
intros.
elim (H E); auto with zfc.
Qed.
Theorem Omega_EQ_Union : EQ Omega (Union Omega).
apply INC_EQ; unfold INC in |- *.
intros.
elim (IN_Omega_EXType E H); intros n e.
apply IN_Union with (Nat (S n)).
auto with zfc.
apply IN_sound_left with (Nat n).
auto with zfc; try auto with zfc.
change (IN (Nat n) (Class_succ (Nat n))) in |- *; auto with zfc.
intros.
elim (Union_IN Omega E H).
intros e h.
elim h.
intros i1 i2.
elim (IN_Omega_EXType e i1).
intros n e1.
cut (IN E (Nat n)).
intros.
elim (IN_Nat_EXType n E H0); intros.
apply IN_sound_left with (Nat x); auto with zfc.
apply IN_sound_right with e; auto with zfc.
Qed.
(*
Theorem Omega_Ord : (Ord Omega).
apply Eo with (Union Omega).
apply Lo.
intros.
elim (IN_Omega_EXType e H).
intros n ee.
apply Eo with (Nat n); Auto with zfc.
elim n.
auto with zfc.
auto with zfc.
intros.
change (Ord (Class_succ (Nat n0))); Auto with zfc.
apply EQ_sym; Auto with zfc.
apply Omega_EQ_Union.
Save.
*)
Fixpoint Vee (E : Ens) : Ens :=
match E with
| sup A f => Union (sup A (fun a : A => Power (Vee (f a))))
end.
(*
Definition Alpha : (E:Ens)Ens.
(Induction E; Intros A f r).
apply Union.
apply (sup A).
intros a.
exact (Power (r a)).
Save.
Transparent Alpha.
*)