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Tej Chajed
stdpp
Commits
953a4d67
Commit
953a4d67
authored
8 years ago
by
Robbert Krebbers
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Some properties about set disjointness.
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theories/collections.v
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953a4d67
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@@ -41,6 +41,25 @@ Section simple_collection.
Lemma
elem_of_disjoint
X
Y
:
X
⊥
Y
↔
∀
x
,
x
∈
X
→
x
∈
Y
→
False
.
Proof
.
done
.
Qed
.
Global
Instance
disjoint_sym
:
Symmetric
(
@
disjoint
C
_)
.
Proof
.
intros
??
.
rewrite
!
elem_of_disjoint
;
naive_solver
.
Qed
.
Lemma
disjoint_empty_l
Y
:
∅
⊥
Y
.
Proof
.
rewrite
elem_of_disjoint
;
intros
x
;
by
rewrite
elem_of_empty
.
Qed
.
Lemma
disjoint_empty_r
X
:
X
⊥
∅.
Proof
.
rewrite
(
symmetry_iff
_);
apply
disjoint_empty_l
.
Qed
.
Lemma
disjoint_singleton_l
x
Y
:
{[
x
]}
⊥
Y
↔
x
∉
Y
.
Proof
.
rewrite
elem_of_disjoint
;
setoid_rewrite
elem_of_singleton
;
naive_solver
.
Qed
.
Lemma
disjoint_singleton_r
y
X
:
X
⊥
{[
y
]}
↔
y
∉
X
.
Proof
.
rewrite
(
symmetry_iff
(
⊥
))
.
apply
disjoint_singleton_l
.
Qed
.
Lemma
disjoint_union_l
X1
X2
Y
:
X1
∪
X2
⊥
Y
↔
X1
⊥
Y
∧
X2
⊥
Y
.
Proof
.
rewrite
!
elem_of_disjoint
;
setoid_rewrite
elem_of_union
;
naive_solver
.
Qed
.
Lemma
disjoint_union_r
X
Y1
Y2
:
X
⊥
Y1
∪
Y2
↔
X
⊥
Y1
∧
X
⊥
Y2
.
Proof
.
rewrite
!
(
symmetry_iff
(
⊥
)
X
)
.
apply
disjoint_union_l
.
Qed
.
Lemma
collection_positive_l
X
Y
:
X
∪
Y
≡
∅
→
X
≡
∅.
Proof
.
rewrite
!
elem_of_equiv_empty
.
setoid_rewrite
elem_of_union
.
naive_solver
.
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