-
Notifications
You must be signed in to change notification settings - Fork 51
/
Copy pathIZF_pair.v
227 lines (182 loc) · 7.7 KB
/
IZF_pair.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
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
(* 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 *)
Require Import IZF_logic.
Require Import IZF_base.
(** Let (X, A, a) and (Y, B, b) be two pointed graphs (with X,Y : Typ1).
The unordered pair formed by the sets represented by these pointed
graphs is itself represented by the pointed graph
((sum X Y), (PAIR X A a Y B b), (out X Y))
whose edge relation (PAIR X A a Y B b) : (Rel (sum X Y)) is
defined by the following four clauses :
1. Delocate A in the new graph via (inl X Y):
if (A x' x), then (PAIR X A a Y B b (inl X Y x') (inl X Y x))
2. Delocate B in the new graph via (inr X Y):
if (B y' y), then (PAIR X A a Y B b (inr X Y y') (inr X Y y))
3. Connect the (image of the) root a to the new root (out X Y):
(PAIR X A a Y B b (inl X Y a) (out X Y))
4. Connect the (image of the) root b to the new root (out X Y):
(PAIR X A a Y B b (inr X Y b) (out X Y))
As usual, we define this relation by a direct impredicative encoding: *)
Definition PAIR (X : Typ1) (A : Rel X) (a : X) (Y : Typ1)
(B : Rel Y) (b : Y) (z' z : sum X Y) :=
forall E : Prop,
(forall x x' : X,
eq (sum X Y) z (inl X Y x) -> eq (sum X Y) z' (inl X Y x') -> A x' x -> E) ->
(forall y y' : Y,
eq (sum X Y) z (inr X Y y) -> eq (sum X Y) z' (inr X Y y') -> B y' y -> E) ->
(eq (sum X Y) z' (inl X Y a) -> eq (sum X Y) z (out X Y) -> E) ->
(eq (sum X Y) z' (inr X Y b) -> eq (sum X Y) z (out X Y) -> E) -> E.
(** The introduction rules corresponding to the 4 clauses of the
definition of (PAIR X A a Y B b) are the following: *)
Lemma PAIR_in1 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y)
(b : Y) (x x' : X), A x' x -> PAIR X A a Y B b (inl X Y x') (inl X Y x).
Proof
fun X A a Y B b x x' H E H1 H2 H3 H4 =>
H1 x x' (eq_refl (sum X Y) (inl X Y x)) (eq_refl (sum X Y) (inl X Y x')) H.
Lemma PAIR_in2 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b y y' : Y),
B y' y -> PAIR X A a Y B b (inr X Y y') (inr X Y y).
Proof
fun X A a Y B b y y' H E H1 H2 H3 H4 =>
H2 y y' (eq_refl (sum X Y) (inr X Y y)) (eq_refl (sum X Y) (inr X Y y')) H.
Lemma PAIR_rt1 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
PAIR X A a Y B b (inl X Y a) (out X Y).
Proof
fun X A a Y B b E H1 H2 H3 H4 =>
H3 (eq_refl (sum X Y) (inl X Y a)) (eq_refl (sum X Y) (out X Y)).
Lemma PAIR_rt2 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
PAIR X A a Y B b (inr X Y b) (out X Y).
Proof
fun X A a Y B b E H1 H2 H3 H4 =>
H4 (eq_refl (sum X Y) (inr X Y b)) (eq_refl (sum X Y) (out X Y)).
(** We first check that the left injection (inl X Y) : X -> (sum X Y)
is a delocation, and deduce that the pointed graphs (X, A, a) and
((sum X Y), (PAIR X A a Y B b), (inl X Y a)) are bisimilar. *)
Lemma PAIR_deloc1 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
deloc X A (sum X Y) (PAIR X A a Y B b) (inl X Y).
Proof.
intros X A a Y B b; unfold deloc in |- *; apply and_intro.
(* Deloc 1 *)
exact (PAIR_in1 X A a Y B b).
(* Deloc 2 (case distinction) *)
intros x z' H; apply H; clear H.
(* Deloc 2, case 1 *)
intros x0 x' H1 H2 H3; apply ex2_intro with x'.
apply (eq_sym _ _ _ (eq_inl_inl X Y x x0 H1)); assumption.
assumption.
(* Deloc 2, case 2 (absurd) *)
intros y y' H1 H2 H3.
apply (eq_inl_inr X Y x y H1).
(* Deloc 2, case 3 (absurd) *)
intros H1 H2; apply (eq_inl_out X Y x H2).
(* Deloc 2, case 4 (absurd) *)
intros H1 H2; apply (eq_inl_out X Y x H2).
Qed.
Lemma PAIR_eqv1 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
EQV X A a (sum X Y) (PAIR X A a Y B b) (inl X Y a).
Proof.
intros; apply EQV_deloc; apply PAIR_deloc1.
Qed.
(** The same for the right injection (inr X Y) : Y -> (sum X Y). *)
Lemma PAIR_deloc2 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
deloc Y B (sum X Y) (PAIR X A a Y B b) (inr X Y).
Proof.
intros X A a Y B b; unfold deloc in |- *; apply and_intro.
(* Deloc 1 *)
exact (PAIR_in2 X A a Y B b).
(* Deloc 2 (case distinction) *)
intros y z' H; apply H; clear H.
(* Deloc 2, case 1 (absurd) *)
intros x x' H1 H2 H3; apply (eq_inr_inl X Y x y H1).
(* Deloc 2, case 2 *)
intros y0 y' H1 H2 H3; apply ex2_intro with y'.
apply (eq_sym _ _ _ (eq_inr_inr X Y y y0 H1)); assumption.
assumption.
(* Deloc 2, case 3 (absurd) *)
intros H1 H2; apply (eq_inr_out X Y y H2).
(* Deloc 2, case 4 (absurd) *)
intros H1 H2; apply (eq_inr_out X Y y H2).
Qed.
Lemma PAIR_eqv2 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
EQV Y B b (sum X Y) (PAIR X A a Y B b) (inr X Y b).
Proof.
intros; apply EQV_deloc; apply PAIR_deloc2.
Qed.
(** From PAIR_eqv1 and PAIR_eqv2, we easily get that the pointed graphs
(X, A, a) and (Y, B, b) are elements of the unordered pair. *)
Lemma pairing_intro1 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
ELT X A a (sum X Y) (PAIR X A a Y B b) (out X Y).
Proof.
intros X A a Y B b; apply ELT_intro with (inl X Y a).
apply PAIR_rt1. apply PAIR_eqv1.
Qed.
Lemma pairing_intro2 :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
ELT Y B b (sum X Y) (PAIR X A a Y B b) (out X Y).
Proof.
intros X A a Y B b; apply ELT_intro with (inr X Y b).
apply PAIR_rt2. apply PAIR_eqv2.
Qed.
(** And conversely, (X, A, a) and (Y, B, b) are the only elements of
the pair, up to bisimulation. *)
Lemma pairing_elim :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y)
(b : Y) (Z : Typ1) (C : Rel Z) (c : Z),
ELT Z C c (sum X Y) (PAIR X A a Y B b) (out X Y) ->
or (EQV Z C c X A a) (EQV Z C c Y B b).
Proof.
intros X A a Y B b Z C c H.
apply H; clear H; intros c' H H1.
apply H; clear H.
(* Case 1 (absurd) *)
intros x x' H2 H3 H4; apply (eq_out_inl X Y x H2).
(* Case 2 (absurd) *)
intros y y' H2 H3 H4; apply (eq_out_inr X Y y H2).
(* Case 3 *)
intros H2 H3; apply or_inl.
apply EQV_trans with (sum X Y) (PAIR X A a Y B b) (inl X Y a).
apply H2; assumption. apply EQV_sym; apply PAIR_eqv1.
(* Case 4 *)
intros H2 H3; apply or_inr.
apply EQV_trans with (sum X Y) (PAIR X A a Y B b) (inr X Y b).
apply H2; assumption. apply EQV_sym; apply PAIR_eqv2.
Qed.
(** By collecting the last three lemmas, we obtain the desired result: *)
Theorem pairing :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y)
(b : Y) (Z : Typ1) (C : Rel Z) (c : Z),
iff (ELT Z C c (sum X Y) (PAIR X A a Y B b) (out X Y))
(or (EQV Z C c X A a) (EQV Z C c Y B b)).
Proof.
intros; unfold iff in |- *; apply and_intro.
(* Forward implication *)
intro; apply pairing_elim; assumption.
(* Backward implication: case distinction *)
intro H; apply H; clear H; intro H.
(* First case *)
apply ELT_compat_l with X A a.
assumption. apply pairing_intro1.
(* Second case *)
apply ELT_compat_l with Y B b.
assumption. apply pairing_intro2.
Qed.