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Glen Mével
stdpp
Commits
b7460a79
Commit
b7460a79
authored
8 years ago
by
Robbert Krebbers
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Some tweaks to minimal.
parent
7a5e6ea4
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theories/collections.v
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-12
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theories/collections.v
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b7460a79
...
...
@@ -999,6 +999,7 @@ End seq_set.
Definition
minimal
`{
ElemOf
A
C
}
(
R
:
relation
A
)
(
x
:
A
)
(
X
:
C
)
:
Prop
:=
∀
y
,
y
∈
X
→
R
y
x
→
R
x
y
.
Instance
:
Params
(
@
minimal
)
5
.
Typeclasses
Opaque
minimal
.
Section
minimal
.
Context
`{
SimpleCollection
A
C
}
{
R
:
relation
A
}
.
...
...
@@ -1006,18 +1007,19 @@ Section minimal.
Global
Instance
minimal_proper
x
:
Proper
(
@
equiv
C
_
==>
iff
)
(
minimal
R
x
)
.
Proof
.
intros
X
X'
y
;
unfold
minimal
;
set_solver
.
Qed
.
Lemma
minimal_anti_symm
`{
!
AntiSymm
(
=
)
R
}
x
X
:
Lemma
minimal_anti_symm_1
`{
!
AntiSymm
(
=
)
R
}
X
x
y
:
minimal
R
x
X
→
y
∈
X
→
R
y
x
→
x
=
y
.
Proof
.
intros
Hmin
??
.
apply
(
anti_symm
_);
auto
.
Qed
.
Lemma
minimal_anti_symm
`{
!
AntiSymm
(
=
)
R
}
X
x
:
minimal
R
x
X
↔
∀
y
,
y
∈
X
→
R
y
x
→
x
=
y
.
Proof
.
unfold
minimal
;
split
;
[|
naive_solver
]
.
intros
Hmin
y
??
.
apply
(
anti_symm
_);
auto
.
Qed
.
Lemma
minimal_strict
`{
!
StrictOrder
R
}
x
X
:
Proof
.
unfold
minimal
;
naive_solver
eauto
using
minimal_anti_symm_1
.
Qed
.
Lemma
minimal_strict_1
`{
!
StrictOrder
R
}
X
x
y
:
minimal
R
x
X
→
y
∈
X
→
¬
R
y
x
.
Proof
.
intros
Hmin
??
.
destruct
(
irreflexivity
R
x
);
trans
y
;
auto
.
Qed
.
Lemma
minimal_strict
`{
!
StrictOrder
R
}
X
x
:
minimal
R
x
X
↔
∀
y
,
y
∈
X
→
¬
R
y
x
.
Proof
.
unfold
minimal
;
split
;
[|
naive_solver
]
.
intros
Hmin
y
??
.
destruct
(
irreflexivity
R
x
);
trans
y
;
auto
.
Qed
.
Proof
.
unfold
minimal
;
split
;
[
eauto
using
minimal_strict_1
|
naive_solver
]
.
Qed
.
Lemma
empty_minimal
x
:
minimal
R
x
∅.
Proof
.
unfold
minimal
;
set_solver
.
Qed
.
...
...
@@ -1034,7 +1036,6 @@ Section minimal.
Lemma
minimal_weaken
`{
!
Transitive
R
}
X
x
x'
:
minimal
R
x
X
→
R
x'
x
→
minimal
R
x'
X
.
Proof
.
intros
Hmin
?
y
??
.
trans
x
;
[
done
|]
.
by
eapply
(
Hmin
y
),
transitivity
.
intros
Hmin
?
y
??
.
trans
x
;
[
done
|]
.
by
eapply
(
Hmin
y
),
transitivity
.
Qed
.
End
minimal
.
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