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stdpp
Commits
e4e6c36d
Commit
e4e6c36d
authored
Feb 23, 2019
by
Robbert Krebbers
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Some properties of `list_to_set_disj`.
parent
37762bb8
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#14974
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in 7 minutes and 44 seconds
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theories/gmultiset.v
theories/gmultiset.v
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theories/gmultiset.v
View file @
e4e6c36d
...
...
@@ -12,7 +12,7 @@ Proof. solve_decision. Defined.
Program
Instance
gmultiset_countable
`
{
Countable
A
}
:
Countable
(
gmultiset
A
)
:
=
{|
encode
X
:
=
encode
(
gmultiset_car
X
)
;
decode
p
:
=
GMultiSet
<$>
decode
p
encode
X
:
=
encode
(
gmultiset_car
X
)
;
decode
p
:
=
GMultiSet
<$>
decode
p
|}.
Next
Obligation
.
intros
A
??
[
X
]
;
simpl
.
by
rewrite
decode_encode
.
Qed
.
...
...
@@ -250,7 +250,29 @@ Proof.
by
rewrite
multiplicity_singleton
,
multiplicity_empty
.
Qed
.
(* Properties of the elements operation *)
(** Conversion from lists *)
Lemma
list_to_set_disj_nil
:
list_to_set_disj
[]
=@{
gmultiset
A
}
∅
.
Proof
.
done
.
Qed
.
Lemma
list_to_set_disj_cons
x
l
:
list_to_set_disj
(
x
::
l
)
=@{
gmultiset
A
}
{[
x
]}
⊎
list_to_set_disj
l
.
Proof
.
done
.
Qed
.
Lemma
list_to_set_disj_app
l1
l2
:
list_to_set_disj
(
l1
++
l2
)
=@{
gmultiset
A
}
list_to_set_disj
l1
⊎
list_to_set_disj
l2
.
Proof
.
induction
l1
as
[|
x
l1
IH
]
;
simpl
.
-
by
rewrite
(
left_id_L
_
_
).
-
by
rewrite
IH
,
(
assoc_L
_
).
Qed
.
Global
Instance
list_to_set_disj_perm
:
Proper
((
≡
ₚ
)
==>
(=))
(
list_to_set_disj
(
C
:
=
gmultiset
A
)).
Proof
.
induction
1
as
[|
x
l
l'
_
IH
|
x
y
l
|
l
l'
l''
_
IH1
_
IH2
]
;
simpl
;
auto
.
-
by
rewrite
IH
.
-
by
rewrite
!(
assoc_L
_
),
(
comm_L
_
{[
x
]}).
-
by
rewrite
IH1
.
Qed
.
(** Properties of the elements operation *)
Lemma
gmultiset_elements_empty
:
elements
(
∅
:
gmultiset
A
)
=
[].
Proof
.
unfold
elements
,
gmultiset_elements
;
simpl
.
by
rewrite
map_to_list_empty
.
...
...
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