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Pierre Letouzey
natded
Commits
6e23647c
Commit
6e23647c
authored
Aug 04, 2020
by
Pierre Letouzey
Browse files
Extra : recip de CurryHoward
parent
d20e7429
Changes
1
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Lambda.v
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6e23647c
...
...
@@ 4,14 +4,16 @@
(
**
The
NatDed
development
,
Pierre
Letouzey
,
Samuel
Ben
Hamou
,
2019

2020.
This
file
is
released
under
the
CC0
License
,
see
the
LICENSE
file
*
)
Require
Import
Defs
Mix
List
.
Require
Import
Utils
Defs
Mix
Meta
.
Import
ListNotations
.
Open
Scope
list
.
Local
Open
Scope
list
.
Local
Open
Scope
eqb_scope
.
Inductive
term
:=

Var
:
nat
>
term

App
:
term
>
term
>
term

Abs
:
term
>
term

One
:
term

Nabla
:
term
>
term

Couple
:
term
>
term
>
term

Pi1
:
term
>
term
...
...
@@ 24,6 +26,7 @@ Inductive type :=

Arr
:
type
>
type
>
type

Atom
:
name
>
type

Bot
:
type

Unit
:
type

And
:
type
>
type
>
type

Or
:
type
>
type
>
type
.
...
...
@@ 33,6 +36,7 @@ Inductive typed : context > term > type > Prop :=

Type_Var
:
forall
Γ
τ
n
,
nth_error
Γ
n
=
Some
τ
>
typed
Γ
(
Var
n
)
τ

Type_App
:
forall
Γ
u
v
σ
τ
,
typed
Γ
u
(
Arr
σ
τ
)
>
typed
Γ
v
σ
>
typed
Γ
(
App
u
v
)
τ

Type_Abs
:
forall
Γ
u
σ
τ
,
typed
(
σ
::
Γ
)
u
τ
>
typed
Γ
(
Abs
u
)
(
Arr
σ
τ
)

Type_One
:
forall
Γ
,
typed
Γ
One
Unit

Type_Nabla
:
forall
Γ
u
τ
,
typed
Γ
u
Bot
>
typed
Γ
(
Nabla
u
)
τ

Type_Couple
:
forall
Γ
u
v
σ
τ
,
typed
Γ
u
σ
>
typed
Γ
v
τ
>
typed
Γ
(
Couple
u
v
)
(
And
σ
τ
)
...
...
@@ 43,8 +47,11 @@ Inductive typed : context > term > type > Prop :=

Type_I1
:
forall
Γ
u
σ
τ
,
typed
Γ
u
σ
>
typed
Γ
(
I1
u
)
(
Or
σ
τ
)

Type_I2
:
forall
Γ
u
σ
τ
,
typed
Γ
u
τ
>
typed
Γ
(
I2
u
)
(
Or
σ
τ
).
Hint
Constructors
typed
.
Notation
"u @ v"
:=
(
App
u
v
)
(
at
level
20
)
:
lambda_scope
.
Notation
"∇ u"
:=
(
Nabla
u
)
(
at
level
20
)
:
lambda_scope
.
(
*
Notation
"u , v"
:=
(
Couple
u
v
)
(
at
level
20
)
:
lambda_scope
.
*
)
Notation
"σ > τ"
:=
(
Arr
σ
τ
)
:
lambda_scope
.
Notation
"σ /\ τ"
:=
(
And
σ
τ
)
:
lambda_scope
.
Notation
"σ \/ τ"
:=
(
Or
σ
τ
)
:
lambda_scope
.
...
...
@@ 62,6 +69,7 @@ Fixpoint to_form (t : type) : formula :=
match
t
with

Arr
u
v
=>
((
to_form
u
)
>
(
to_form
v
))
%
form

Atom
a
=>
Pred
a
[]

Unit
=>
True

Bot
=>
False

And
u
v
=>
((
to_form
u
)
/
\
(
to_form
v
))
%
form

Or
u
v
=>
((
to_form
u
)
\
/
(
to_form
v
))
%
form
...
...
@@ 72,36 +80,126 @@ Definition to_ctxt (Γ : context) := List.map to_form Γ.
Theorem
CurryHoward
Γ
u
τ
:
typed
Γ
u
τ
>
Pr
Intuiti
(
to_ctxt
Γ
⊢
to_form
τ
).
Proof
.
induction
1.
induction
1
;
cbn
.

apply
R_Ax
.
apply
in_map
.
apply
nth_error_In
with
(
n
:=
n
);
auto
.

apply
R_Imp_e
with
(
A
:=
to_form
σ
);
intuition
.

apply
R_Imp_i
.
intuition
.

apply
R_Tr_i
.

apply
R_Fa_e
.
intuition
.

apply
R_And_i
;
intuition
.

apply
R_And_e1
with
(
B
:=
to_form
τ
).
intuition
.

apply
R_And_e2
with
(
A
:=
to_form
σ
).
intuition
.

apply
R_Or_e
with
(
A
:=
to_form
τ
1
)
(
B
:=
to_form
τ
2
);
intuition
.

apply
R_Or_i1
with
(
B
:=
to_form
τ
).
intuition
.

apply
R_Or_i2
with
(
A
:=
to_form
σ
).
intuition
.

e
apply
R_Or_e
;
eauto
.

now
apply
R_Or_i1
.

now
apply
R_Or_i2
.
Qed
.
Lemma
ex
:
Pr
Intuiti
([]
⊢
(
Pred
"A"
[]
/
\
Pred
"B"
[])
>
(
Pred
"A"
[]
\
/
Pred
"B"
[])).
Proof
.
set
(
A
:=
Pred
"A"
[]).
set
(
B
:=
Pred
"B"
[]).
set
(
form
:=
(
_
>
_
)
%
form
).
set
(
typ
:=
Atom
"A"
/
\
Atom
"B"
>
Atom
"A"
\
/
Atom
"B"
).
assert
(
to_form
typ
=
form
).
{
intuition
.
}
rewrite
<
H
.
assert
(
to_ctxt
[]
=
[]).
{
intuition
.
}
rewrite
<
H0
.
set
(
typ
:=
Atom
"A"
/
\
Atom
"B"
>
Atom
"A"
\
/
Atom
"B"
).
change
(
Pr
J
(
to_ctxt
[]
⊢
to_form
typ
)).
apply
CurryHoward
with
(
u
:=
(
Abs
(
I1
(
Pi1
(#
0
))))).
unfold
typ
.
apply
Type_Abs
.
apply
Type_I1
.
apply
Type_Pi1
with
(
τ
:=
Atom
"B"
).
apply
Type_Var
.
intuition
.
easy
.
Qed
.
Instance
eqb_type
:
Eqb
type
:=
fix
eqb_type
(
u
v
:
type
)
:=
match
u
,
v
with

Unit
,
Unit

Bot
,
Bot
=>
true

Atom
a
,
Atom
b
=>
a
=?
b

Arr
u1
u2
,
Arr
v1
v2

And
u1
u2
,
And
v1
v2

Or
u1
u2
,
Or
v1
v2
=>
eqb_type
u1
v1
&&
eqb_type
u2
v2

_
,
_
=>
false
end
.
Instance
eqbspec_type
:
EqbSpec
type
.
Proof
.
red
.
induction
x
;
destruct
y
;
cbn
;
try
constructor
;
try
easy
.

destruct
(
IHx1
y1
),
(
IHx2
y2
);
cbn
;
constructor
;
congruence
.

case
eqbspec
;
constructor
;
congruence
.

destruct
(
IHx1
y1
),
(
IHx2
y2
);
cbn
;
constructor
;
congruence
.

destruct
(
IHx1
y1
),
(
IHx2
y2
);
cbn
;
constructor
;
congruence
.
Qed
.
Lemma
list_index_nth_error
{
A
}
`
(
EqbSpec
A
)
n
x
l
:
list_index
x
l
=
Some
n
>
nth_error
l
n
=
Some
x
.
Proof
.
revert
n
.
induction
l
;
intros
n
;
cbn
;
try
easy
.
case
eqbspec
.
now
intros
>
[
=
<
].
intros
N
.
destruct
(
list_index
x
l
);
try
easy
.
cbn
.
intros
[
=
<
].
cbn
;
auto
.
Qed
.
Fixpoint
of_form
(
f
:
formula
)
:
type
:=
match
f
with

True
=>
Unit

False
=>
Bot

Pred
a
_
=>
Atom
a

Not
A
=>
Arr
(
of_form
A
)
Bot

(
A
>
B
)
=>
Arr
(
of_form
A
)
(
of_form
B
)

(
A
/
\
B
)
=>
And
(
of_form
A
)
(
of_form
B
)

(
A
\
/
B
)
=>
Or
(
of_form
A
)
(
of_form
B
)

Quant
_
A
=>
of_form
A
end
%
form
.
Definition
of_ctxt
(
Γ
:
Mix
.
context
)
:=
List
.
map
of_form
Γ
.
Lemma
of_to_form
T
:
of_form
(
to_form
T
)
=
T
.
Proof
.
induction
T
;
cbn
;
now
f_equal
.
Qed
.
Lemma
of_form_bsubst
n
t
A
:
of_form
(
bsubst
n
t
A
)
=
of_form
A
.
Proof
.
revert
n
t
;
induction
A
;
cbn
;
intros
;
auto
.

now
f_equal
.

destruct
o
;
f_equal
;
auto
.
Qed
.
Theorem
CurryHoward_recip
seq
:
Pr
Intuiti
seq
>
let
'
(
Γ
⊢
A
)
:=
seq
in
exists
u
,
typed
(
of_ctxt
Γ
)
u
(
of_form
A
).
Proof
.
induction
1
;
cbn
in
*
.

unfold
of_ctxt
.
apply
(
in_map
of_form
)
in
H
.
rewrite
<
list_index_in
in
H
.
destruct
(
list_index
(
of_form
A
)
(
map
of_form
Γ
))
eqn
:
E
;
try
easy
.
exists
(
Var
n
).
constructor
.
apply
list_index_nth_error
;
eauto
with
*
.

exists
One
;
auto
.

destruct
IHPr
as
(
u
,
P
).
exists
(
Nabla
u
);
auto
.

destruct
IHPr
as
(
u
,
P
).
exists
(
Abs
u
);
auto
.

destruct
IHPr1
as
(
u
,
P
),
IHPr2
as
(
v
,
Q
).
exists
(
v
@
u
);
eauto
.

destruct
IHPr1
as
(
u
,
P
),
IHPr2
as
(
v
,
Q
).
exists
(
Couple
u
v
);
eauto
.

destruct
IHPr
as
(
u
,
P
).
exists
(
Pi1
u
);
eauto
.

destruct
IHPr
as
(
u
,
P
).
exists
(
Pi2
u
);
eauto
.

destruct
IHPr
as
(
u
,
P
).
exists
(
I1
u
);
auto
.

destruct
IHPr
as
(
u
,
P
).
exists
(
I2
u
);
auto
.

destruct
IHPr1
as
(
u
,
P
),
IHPr2
as
(
v
,
Q
),
IHPr3
as
(
w
,
R
).
exists
(
Case
u
v
w
);
eauto
.

destruct
IHPr
as
(
u
,
P
).
exists
(
Abs
u
);
auto
.

destruct
IHPr1
as
(
u
,
P
),
IHPr2
as
(
v
,
Q
).
exists
(
u
@
v
);
eauto
.

destruct
IHPr
as
(
u
,
P
).
rewrite
of_form_bsubst
in
P
.
now
exists
u
.

destruct
IHPr
as
(
u
,
P
).
exists
u
.
now
rewrite
of_form_bsubst
.

destruct
IHPr
as
(
u
,
P
).
exists
u
.
now
rewrite
of_form_bsubst
in
P
.

destruct
IHPr1
as
(
u
,
P
),
IHPr2
as
(
v
,
Q
).
rewrite
of_form_bsubst
in
Q
.
exists
(
Abs
v
@
u
);
eauto
.

easy
.
Qed
.
Theorem
CurryHoward_recip
'
Γ
A
:
Pr
Intuiti
(
Γ
⊢
A
)
>
exists
u
,
typed
(
of_ctxt
Γ
)
u
(
of_form
A
).
Proof
.
apply
CurryHoward_recip
.
Qed
.
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