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Pierre Letouzey
natded
Commits
f0abfe1e
Commit
f0abfe1e
authored
Jul 07, 2020
by
Samuel Ben Hamou
Browse files
Création de ZF.v et ajout d'axiomes.
parent
e9cefa54
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ZF.v
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f0abfe1e
(
**
*
The
Zermelo
Fraenkel
set
theory
and
its
Coq
model
*
)
(
**
The
NatDed
development
,
Pierre
Letouzey
,
2019.
This
file
is
released
under
the
CC0
License
,
see
the
LICENSE
file
*
)
Require
Import
Defs
NameProofs
Mix
Meta
Theories
PreModels
Models
.
Import
ListNotations
.
Local
Open
Scope
bool_scope
.
Local
Open
Scope
eqb_scope
.
(
**
The
Peano
axioms
*
)
Definition
ZFSign
:=
Finite
.
to_infinite
zf_sign
.
Notation
"x = y"
:=
(
Pred
"="
[
x
;
y
])
:
formula_scope
.
Notation
"x ∈ y"
:=
(
Pred
"∈"
[
x
;
y
])
:
formula_scope
.
Module
ZFAx
.
Local
Open
Scope
formula_scope
.
Definition
eq_refl
:=
∀
(#
0
=
#
0
).
Definition
eq_sym
:=
∀∀
(#
1
=
#
0
>
#
0
=
#
1
).
Definition
eq_trans
:=
∀∀∀
(#
2
=
#
1
/
\
#
1
=
#
0
>
#
2
=
#
0
).
Definition
compat_left
:=
∀∀∀
(#
0
=
#
1
/
\
#
0
∈
#
2
>
#
1
∈
#
2
).
Definition
compat_right
:=
∀∀∀
(#
0
∈
#
1
/
\
#
1
=
#
2
>
#
0
∈
#
2
).
Definition
ext
:=
∀∀
(
∀
(#
2
∈
#
0
<>
#
2
∈
#
1
)
>
#
0
=
#
1
).
Definition
pairing
:=
∀∀∃∀
(#
3
∈
#
2
<>
#
3
=
#
0
\
/
#
3
=
#
1
).
Definition
union
:=
∀∃∀
(#
2
∈
#
1
<>
∃
(#
3
∈
#
0
/
\
#
2
∈
#
3
).
Definition
powerset
:=
∀∃∀
(#
2
∈
#
1
<>
∀
(#
3
∈
#
2
>
#
3
∈
#
0
)).
Definition
infinity
:=
∃
(
∃
(#
1
∈
#
0
/
\
∀
not
#
2
∈
#
1
)
/
\
∀
(#
3
∈
#
0
>
(
∃
(#
4
∈
#
0
/
\
(
∀
(#
5
∈
#
4
<>
#
5
=
#
3
\
/
#
5
∈
#
3
)))))).
Definition
axioms_list
:=
[
eq_refl
;
eq_symp
;
eq_trans
;
compat_left
;
compat_right
;
ext
;
pairing
;
union
;
powerset
;
infinity
].
(
**
Beware
,
[
bsubst
]
is
ok
below
for
turning
[#
0
]
into
[
Succ
#
0
],
but
only
since
it
contains
now
a
[
lift
]
of
substituted
terms
inside
quantifiers
.
And
the
unconventional
[
∀
]
before
[
A
[
0
]]
is
to
get
the
right
bounded
vars
(
Hack
!
).
*
)
Definition
comprehension_schema
A
:=
nForall
(
Nat
.
pred
(
level
A
))
(
Definition
induction_schema
A_x
:=
let
A_0
:=
bsubst
0
Zero
A_x
in
let
A_Sx
:=
bsubst
0
(
Succ
(#
0
))
A_x
in
nForall
(
Nat
.
pred
(
level
A_x
))
(((
∀
A_0
)
/
\
(
∀
(
A_x
>
A_Sx
)))
>
∀
A_x
).
Local
Close
Scope
formula_scope
.
Definition
IsAx
A
:=
List
.
In
A
axioms_list
\
/
exists
B
,
A
=
induction_schema
B
/
\
check
PeanoSign
B
=
true
/
\
FClosed
B
.
Lemma
WfAx
A
:
IsAx
A
>
Wf
PeanoSign
A
.
Proof
.
intros
[
IN

(
B
&
>
&
HB
&
HB
'
)].

apply
Wf_iff
.
unfold
axioms_list
in
IN
.
simpl
in
IN
.
intuition
;
subst
;
reflexivity
.

repeat
split
;
unfold
induction_schema
;
cbn
.
+
rewrite
nForall_check
.
cbn
.
rewrite
!
check_bsubst
,
HB
;
auto
.
+
red
.
rewrite
nForall_level
.
cbn
.
assert
(
level
(
bsubst
0
Zero
B
)
<=
level
B
).
{
apply
level_bsubst
'
.
auto
.
}
assert
(
level
(
bsubst
0
(
Succ
(
BVar
0
))
B
)
<=
level
B
).
{
apply
level_bsubst
'
.
auto
.
}
omega
with
*
.
+
apply
nForall_fclosed
.
red
.
cbn
.
assert
(
FClosed
(
bsubst
0
Zero
B
)).
{
red
.
rewrite
bsubst_fvars
.
intro
x
.
rewrite
Names
.
union_spec
.
cbn
.
red
in
HB
'
.
intuition
.
}
assert
(
FClosed
(
bsubst
0
(
Succ
(
BVar
0
))
B
)).
{
red
.
rewrite
bsubst_fvars
.
intro
x
.
rewrite
Names
.
union_spec
.
cbn

[
Names
.
union
].
rewrite
Names
.
union_spec
.
generalize
(
HB
'
x
)
(
@
Names
.
empty_spec
x
).
intuition
.
}
unfold
FClosed
in
*
.
intuition
.
Qed
.
End
PeanoAx
.
Local
Open
Scope
string
.
Local
Open
Scope
formula_scope
.
Definition
PeanoTheory
:=
{
sign
:=
PeanoSign
;
IsAxiom
:=
PeanoAx
.
IsAx
;
WfAxiom
:=
PeanoAx
.
WfAx
}
.
Import
PeanoAx
.
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