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Micaela Mayero
Numerical Analysis in Coq
Commits
44897cb9
Commit
44897cb9
authored
2 years ago
by
François Clément
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Renaming + cleaning (of doubles and useless results).
Mv def of continuity (to Set_system_def_any).
parent
115b9ac9
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Lebesgue/Set_theory/Set_system/Topology.v
+26
-69
26 additions, 69 deletions
Lebesgue/Set_theory/Set_system/Topology.v
with
26 additions
and
69 deletions
Lebesgue/Set_theory/Set_system/Topology.v
+
26
−
69
View file @
44897cb9
...
...
@@ -16,12 +16,12 @@ COPYING file for more details.
(
**
General
topology
-
bases
and
continuity
(
definitions
and
properties
).
Generators
of
topologies
are
also
called
subbase
,
or
prebase
.
*
)
Generators
of
topologies
are
also
called
subbase
s
,
or
prebase
s
.
*
)
Require
Import
Set_system
.
Section
Basis_Facts
3
.
Section
Basis_Facts
1
.
Context
{
U
:
Type
}
.
Variable
T
:
set_system
U
.
...
...
@@ -29,77 +29,36 @@ Variable PB : set_system U.
Hypothesis
HT
:
is_Open
T
.
Hypothesis
HPB
:
is_Basisp
T
PB
.
Lemma
Basisp_full_unionp_any
:
full
(
unionp_any
PB
).
Proof
.
apply
is_Basisp_wFull
with
T
;
try
apply
is_Open_equiv
;
easy
.
Qed
.
Lemma
Basisp_Union_any_inter
:
Union_any_inter
PB
.
Proof
.
apply
is_Basisp_Inter
with
T
;
try
apply
is_Open_equiv
;
easy
.
Qed
.
Lemma
Basisp_is_gen
:
T
=
Open
PB
.
Proof
.
destruct
HPB
as
[
HPB1
HPB2
];
rewrite
<-
HT
;
apply
Open_ext
.
(
*
*
)
intros
A
HA
;
rewrite
(
HPB2
_
HA
).
pose
(
PB_A
:=
fun
B
=>
PB
B
/
\
incl
B
A
);
fold
PB_A
.
destruct
(
empty_dec
PB_A
)
as
[
H
|
H
].
destruct
(
empty_dec
(
Subset
A
PB
))
as
[
H
|
H
].
(
*
.
*
)
rewrite
empty_equiv
in
H
;
rewrite
H
,
unionp_any_nullary
.
apply
Open_wEmpty
.
(
*
.
*
)
apply
Open_Union_any
;
try
easy
.
unfold
PB_A
;
intros
B
HB
;
apply
Open_Gen
;
easy
.
intros
B
HB
;
induction
HB
;
apply
Open_Gen
;
easy
.
(
*
*
)
intros
B
HB
;
apply
Open_Gen
;
auto
.
Qed
.
Lemma
Basisp_fullset
:
unionp_any
PB
=
fullset
.
Proof
.
rewrite
<-
full_equiv
;
apply
is_Basisp_wFull
with
T
;
try
easy
.
apply
is_Open_equiv
;
easy
.
Qed
.
End
Basis_Facts1
.
Lemma
Basisp_Union_any_inter
:
Union_any_inter
PB
.
Proof
.
apply
is_Basisp_Inter
with
T
;
try
easy
.
apply
is_Open_equiv
;
easy
.
Qed
.
End
Basis_Facts3
.
Section
Basis_Facts4
.
Context
{
U
Idx
:
Type
}
.
Variable
T
:
set_system
U
.
(
*
Note
that
the
following
results
do
not
need
T
to
be
a
topology
!
*
)
Lemma
all_is_Basisf
:
is_Basisf
T
(
skolem
T
).
Proof
.
split
.
intros
[
x
Hx
];
easy
.
intros
A
HA
;
apply
set_ext_equiv
;
split
;
intros
x
.
intros
;
exists
(
exist
_
_
HA
);
easy
.
intros
[[
B
HB1
]
[
HB2
HB3
]];
auto
.
Qed
.
Lemma
all_is_Basisp
:
is_Basisp
T
T
.
Proof
.
split
;
try
easy
.
intros
A
HA
;
apply
set_ext_equiv
;
split
;
intros
x
Hx
.
exists
A
;
easy
.
destruct
Hx
as
[
B
[[
HB1
HB2
]
HB3
]];
auto
.
Qed
.
Variable
fB
:
Idx
->
set
U
.
(
*
Useful
?
*
)
Lemma
Basisf_to_Basisp
:
is_Basisf
T
fB
->
is_Basisp
T
(
unskolem
fB
).
Proof
.
apply
(
proj1
(
is_Basisf_is_Basisp_equiv
_
_
)).
Qed
.
Variable
PB
:
set_system
U
.
(
*
Useful
?
*
)
Lemma
Basisf_of_Basisp
:
is_Basisp
T
PB
->
is_Basisf
T
(
skolem
PB
).
Proof
.
apply
(
proj1
(
is_Basisp_is_Basisf_equiv
_
_
)).
Qed
.
Section
Basis_Facts2
.
Context
{
U1
U2
:
Type
}
.
Variable
T1
:
set_system
U1
.
...
...
@@ -115,10 +74,10 @@ apply Incl_trans with (Preimage f T2); try easy.
apply
Preimage_monot
;
easy
.
Qed
.
End
Basis_Facts
4
.
End
Basis_Facts
2
.
Section
Basis_Facts
5
.
Section
Basis_Facts
3
.
Context
{
U1
U2
Idx1
Idx2
:
Type
}
.
Variable
T1
:
set_system
U1
.
...
...
@@ -162,27 +121,25 @@ intros [i [Hx1 Hx2]]; auto.
*
)
Admitted
.
End
Basis_Facts
5
.
End
Basis_Facts
3
.
Section
Continuous_
Def
.
Section
Continuous_
fun_Facts1
.
Context
{
U1
U2
:
Type
}
.
(
*
Universes
.
*
)
Variable
genU1
:
set_system
U1
.
(
*
Generator
,
or
subbase
.
*
)
Variable
genU2
:
set_system
U2
.
(
*
Generator
,
or
subbase
.
*
)
Definition
continuous_fun
:
set
(
U1
->
U2
)
:=
fun
f
=>
Incl
(
Preimage
f
(
Open
genU2
))
(
Open
genU1
).
Lemma
continuous_fun_ext
:
forall
f
g
,
same_fun
f
g
->
continuous_fun
f
->
continuous_fun
g
.
forall
f
g
,
same_fun
f
g
->
continuous_fun
genU1
genU2
f
->
continuous_fun
genU1
genU2
g
.
Proof
.
intros
f
g
H
Hf
_
[
A2
HA2
].
rewrite
<-
(
preimage_ext_fun
f
);
try
easy
;
apply
Hf
;
easy
.
Qed
.
Lemma
continuous_fun_equiv
:
forall
f
,
continuous_fun
f
<->
Incl
(
Preimage
f
genU2
)
(
Open
genU1
).
forall
f
,
continuous_fun
genU1
genU2
f
<->
Incl
(
Preimage
f
genU2
)
(
Open
genU1
).
Proof
.
intros
f
;
split
;
intros
Hf
.
intros
_
[
A2
HA2
];
apply
Hf
,
Im
,
Open_Gen
;
easy
.
...
...
@@ -194,10 +151,10 @@ apply Open_Inter_finite; easy.
apply
Open_Unionf_any
;
easy
.
Qed
.
End
Continuous_
Def
.
End
Continuous_
fun_Facts1
.
Section
Continuous_Facts
1
.
Section
Continuous_
fun_
Facts
2
.
Context
{
U1
U2
:
Type
}
.
Variable
genU1
:
set_system
U1
.
...
...
@@ -223,7 +180,7 @@ split; intros Hf A1 HA1; induction HA1 as [A2 HA2].
Admitted
.
End
Continuous_Facts
1
.
End
Continuous_
fun_
Facts
2
.
(
*
TODO
:
add
section
(
s
)
on
UniformSpace
.
...
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