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stdpp
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37762bb8
Commit
37762bb8
authored
Feb 23, 2019
by
Robbert Krebbers
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Bit of refactoring for multiset singleton.
parent
34a3138e
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theories/gmultiset.v
theories/gmultiset.v
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theories/gmultiset.v
View file @
37762bb8
...
...
@@ 83,6 +83,13 @@ Lemma multiplicity_singleton x : multiplicity x {[ x ]} = 1.
Proof
.
unfold
multiplicity
;
simpl
.
by
rewrite
lookup_singleton
.
Qed
.
Lemma
multiplicity_singleton_ne
x
y
:
x
≠
y
→
multiplicity
x
{[
y
]}
=
0
.
Proof
.
intros
.
unfold
multiplicity
;
simpl
.
by
rewrite
lookup_singleton_ne
.
Qed
.
Lemma
multiplicity_singleton'
x
y
:
multiplicity
x
{[
y
]}
=
if
decide
(
x
=
y
)
then
1
else
0
.
Proof
.
destruct
(
decide
_
)
as
[>].

by
rewrite
multiplicity_singleton
.

by
rewrite
multiplicity_singleton_ne
.
Qed
.
Lemma
multiplicity_union
X
Y
x
:
multiplicity
x
(
X
∪
Y
)
=
multiplicity
x
X
`
max
`
multiplicity
x
Y
.
Proof
.
...
...
@@ 117,10 +124,9 @@ Global Instance gmultiset_simple_set : SemiSet A (gmultiset A).
Proof
.
split
.

intros
x
.
rewrite
elem_of_multiplicity
,
multiplicity_empty
.
lia
.

intros
x
y
.
destruct
(
decide
(
x
=
y
))
as
[>].
+
rewrite
elem_of_multiplicity
,
multiplicity_singleton
.
split
;
auto
with
lia
.
+
rewrite
elem_of_multiplicity
,
multiplicity_singleton_ne
by
done
.
by
split
;
auto
with
lia
.

intros
x
y
.
rewrite
elem_of_multiplicity
,
multiplicity_singleton'
.
destruct
(
decide
(
x
=
y
))
;
intuition
lia
.

intros
X
Y
x
.
rewrite
!
elem_of_multiplicity
,
multiplicity_union
.
lia
.
Qed
.
Global
Instance
gmultiset_elem_of_dec
:
RelDecision
(
∈
@{
gmultiset
A
}).
...
...
@@ 411,9 +417,8 @@ Proof. rewrite (comm_L (⊎)). apply gmultiset_disj_union_subset_l. Qed.
Lemma
gmultiset_elem_of_singleton_subseteq
x
X
:
x
∈
X
↔
{[
x
]}
⊆
X
.
Proof
.
rewrite
elem_of_multiplicity
.
split
.

intros
Hx
y
;
destruct
(
decide
(
x
=
y
))
as
[>].
+
rewrite
multiplicity_singleton
;
lia
.
+
rewrite
multiplicity_singleton_ne
by
done
;
lia
.

intros
Hx
y
.
rewrite
multiplicity_singleton'
.
destruct
(
decide
(
y
=
x
))
;
naive_solver
lia
.

intros
Hx
.
generalize
(
Hx
x
).
rewrite
multiplicity_singleton
.
lia
.
Qed
.
...
...
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