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stdpp
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76d280b9
Commit
76d280b9
authored
6 years ago
by
Ralf Jung
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add telescopic versions of the Coq quantifiers
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add telescopic versions of the Coq quantifiers
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theories/telescopes.v
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76d280b9
...
...
@@ -141,3 +141,45 @@ Notation "'λ..' x .. y , e" :=
(
tele_app
$
tele_bind
(
λ
x
,
.
.
(
tele_app
$
tele_bind
(
λ
y
,
e
))
.
.
))
(
at
level
200
,
x
binder
,
y
binder
,
right
associativity
,
format
"'[ ' 'λ..' x .. y ']' , e"
)
.
(** Telescopic quantifiers *)
Definition
tforall
{
TT
:
tele
}
(
Ψ
:
TT
→
Prop
)
:
Prop
:=
tele_fold
(
λ
(
T
:
Type
)
(
b
:
T
→
Prop
),
∀
x
:
T
,
b
x
)
(
λ
x
,
x
)
(
tele_bind
Ψ
)
.
Arguments
tforall
{
!
_}
_
/.
Definition
texist
{
TT
:
tele
}
(
Ψ
:
TT
→
Prop
)
:
Prop
:=
tele_fold
ex
(
λ
x
,
x
)
(
tele_bind
Ψ
)
.
Arguments
texist
{
!
_}
_
/.
Notation
"'∀..' x .. y , P"
:=
(
tforall
(
λ
x
,
.
.
(
tforall
(
λ
y
,
P
))
.
.
))
(
at
level
200
,
x
binder
,
y
binder
,
right
associativity
,
format
"∀.. x .. y , P"
)
:
stdpp_scope
.
Notation
"'∃..' x .. y , P"
:=
(
texist
(
λ
x
,
.
.
(
texist
(
λ
y
,
P
))
.
.
))
(
at
level
200
,
x
binder
,
y
binder
,
right
associativity
,
format
"∃.. x .. y , P"
)
:
stdpp_scope
.
Lemma
tforall_forall
{
TT
:
tele
}
(
Ψ
:
TT
→
Prop
)
:
(
tforall
Ψ
)
↔
(
∀
x
,
Ψ
x
)
.
Proof
.
symmetry
.
unfold
tforall
.
induction
TT
as
[|
X
ft
IH
]
.
-
simpl
.
split
.
+
done
.
+
intros
?
p
.
rewrite
(
tele_arg_O_inv
p
)
.
done
.
-
simpl
.
split
;
intros
Hx
a
.
+
rewrite
<-
IH
.
done
.
+
destruct
(
tele_arg_S_inv
a
)
as
[
x
[
pf
->
]]
.
revert
pf
.
setoid_rewrite
IH
.
done
.
Qed
.
Lemma
texist_exist
{
TT
:
tele
}
(
Ψ
:
TT
→
Prop
)
:
(
texist
Ψ
)
↔
(
ex
Ψ
)
.
Proof
.
symmetry
.
induction
TT
as
[|
X
ft
IH
]
.
-
simpl
.
split
.
+
intros
[
p
Hp
]
.
rewrite
(
tele_arg_O_inv
p
)
in
Hp
.
done
.
+
intros
.
by
exists
TargO
.
-
simpl
.
split
;
intros
[
p
Hp
];
revert
Hp
.
+
destruct
(
tele_arg_S_inv
p
)
as
[
x
[
pf
->
]]
.
intros
?
.
exists
x
.
rewrite
<-
(
IH
x
(
λ
a
,
Ψ
(
TargS
x
a
)))
.
eauto
.
+
rewrite
<-
(
IH
p
(
λ
a
,
Ψ
(
TargS
p
a
)))
.
intros
[??]
.
eauto
.
Qed
.
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