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Pierre Letouzey
natded
Commits
debb9b68
Commit
debb9b68
authored
Jul 22, 2020
by
Samuel Ben Hamou
Browse files
Fin preuve union.
parent
50553df9
Changes
1
Hide whitespace changes
Inline
Side-by-side
ZF.v
View file @
debb9b68
...
...
@@ -236,12 +236,14 @@ Qed.
Lemma
union
:
IsTheorem
Intuiti
ZF
(
∀∀∃∀
(#
0
∈
#
1
<->
#
0
∈
#
3
\
/
#
0
∈
#
2
)).
Proof
.
thm
.
exists
[
pairing
;
union
].
exists
[
pairing
;
union
;
compat_right
;
eq_refl
].
split
;
auto
.
-
simpl
.
constructor
.
+
left
.
calc
.
+
constructor
;
[
left
;
calc
|
auto
].
-
set
(
Γ
:=
[
_
;
_
]).
+
constructor
.
left
.
calc
.
constructor
.
left
.
calc
.
constructor
;
[
left
;
calc
|
auto
].
-
set
(
Γ
:=
[
_
;
_
;
_
]).
app_R_All_i
"A"
A
.
app_R_All_i
"B"
B
.
inst_axiom
pairing
[
A
;
B
];
cbn
in
*
.
fold
A
.
fold
B
.
fold
A
in
H
.
fold
B
in
H
.
reIff
.
...
...
@@ -272,11 +274,39 @@ Proof.
set
(
Ax
:=
∀
_
<->
_
\
/
_
).
inst_axiom
Ax
[
y
].
exact
H
.
--
apply
R_Or_i1
.
(
*
todo
avec
compat_left
*
)
admit
.
apply
R_Imp_e
with
(
A
:=
x
∈
y
/
\
y
=
A
).
++
inst_axiom
compat_right
[
A
;
y
;
x
].
++
apply
R_And_i
;
apply
R_Ax
;
calc
.
--
apply
R_Or_i2
.
(
*
todo
*
)
admit
.
apply
R_Imp_e
with
(
A
:=
x
∈
y
/
\
y
=
B
).
++
inst_axiom
compat_right
[
B
;
y
;
x
].
++
apply
R_And_i
;
apply
R_Ax
;
calc
.
+
apply
R_Imp_i
.
apply
R
'_
Or_e
.
*
admit
.
\ No newline at end of file
*
set
(
Ax
:=
∀
_
∈
U
<->
_
).
inst_axiom
Ax
[
x
].
cbn
in
H
.
fold
U
in
H
.
fold
C
in
H
.
fold
x
in
H
.
fold
Ax
in
H
.
apply
R_And_e2
in
H
.
apply
R_Imp_e
with
(
A
:=
(
∃
#
0
∈
C
/
\
x
∈
#
0
));
[
assumption
|
].
apply
R_Ex_i
with
(
t
:=
A
).
cbn
.
fold
C
.
fold
A
.
fold
x
.
apply
R_And_i
.
--
set
(
Ax2
:=
∀
_
<->
_
).
inst_axiom
Ax2
[
A
].
cbn
in
H0
.
fold
B
in
H0
.
fold
C
in
H0
.
fold
A
in
H0
.
fold
Ax
in
H0
.
apply
R_And_e2
in
H0
.
apply
R_Imp_e
with
(
A
:=
A
=
A
\
/
A
=
B
);
[
assumption
|
].
apply
R_Or_i1
.
inst_axiom
eq_refl
[
A
].
--
apply
R
'_
Ax
.
*
set
(
Ax
:=
∀
_
∈
U
<->
_
).
inst_axiom
Ax
[
x
].
cbn
in
H
.
fold
U
in
H
.
fold
C
in
H
.
fold
x
in
H
.
fold
Ax
in
H
.
apply
R_And_e2
in
H
.
apply
R_Imp_e
with
(
A
:=
(
∃
#
0
∈
C
/
\
x
∈
#
0
));
[
assumption
|
].
apply
R_Ex_i
with
(
t
:=
B
).
cbn
.
fold
C
.
fold
B
.
fold
x
.
apply
R_And_i
.
--
set
(
Ax2
:=
∀
_
<->
_
).
inst_axiom
Ax2
[
B
].
cbn
in
H0
.
fold
B
in
H0
.
fold
C
in
H0
.
fold
A
in
H0
.
fold
Ax
in
H0
.
apply
R_And_e2
in
H0
.
apply
R_Imp_e
with
(
A
:=
B
=
A
\
/
B
=
B
);
[
assumption
|
].
apply
R_Or_i2
.
inst_axiom
eq_refl
[
B
].
--
apply
R
'_
Ax
.
Qed
.
\ No newline at end of file
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