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stdpp
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f5abe554
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f5abe554
authored
5 years ago
by
David Swasey
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Add `coGset`.
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cogset
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f5abe554
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@@ -34,6 +34,7 @@ theories/numbers.v
theories/nmap.v
theories/zmap.v
theories/coPset.v
theories/coGset.v
theories/lexico.v
theories/propset.v
theories/decidable.v
...
...
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theories/coGset.v
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f5abe554
(* Copyright (c) 2020, Coq-std++ developers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file implements the type [coGset A] of finite/cofinite sets
of elements of any countable type [A].
Note that [coGset positive] cannot represent all elements of [coPset]
(e.g., [coPset_suffixes], [coPset_l], and [coPset_r] construct
infinite sets that cannot be represented). *)
From
stdpp
Require
Export
sets
countable
.
From
stdpp
Require
Import
decidable
finite
gmap
coPset
.
(* Set Default Proof Using "Type". *)
Inductive
coGset
`{
Countable
A
}
:=
|
FinGSet
(
X
:
gset
A
)
|
CoFinGset
(
X
:
gset
A
)
.
Arguments
coGset
_
{_
_}
:
assert
.
Instance
coGset_eq_dec
`{
Countable
A
}
:
EqDecision
(
coGset
A
)
.
Proof
.
solve_decision
.
Defined
.
Instance
coGset_countable
`{
Countable
A
}
:
Countable
(
coGset
A
)
.
Proof
.
apply
(
inj_countable'
(
λ
X
,
match
X
with
FinGSet
X
=>
inl
X
|
CoFinGset
X
=>
inr
X
end
)
(
λ
s
,
match
s
with
inl
X
=>
FinGSet
X
|
inr
X
=>
CoFinGset
X
end
))
.
by
intros
[]
.
Qed
.
Section
coGset
.
Context
`{
Countable
A
}
.
Global
Instance
coGset_elem_of
:
ElemOf
A
(
coGset
A
)
:=
λ
x
X
,
match
X
with
FinGSet
X
=>
x
∈
X
|
CoFinGset
X
=>
x
∉
X
end
.
Global
Instance
coGset_empty
:
Empty
(
coGset
A
)
:=
FinGSet
∅.
Global
Instance
coGset_top
:
Top
(
coGset
A
)
:=
CoFinGset
∅.
Global
Instance
coGset_singleton
:
Singleton
A
(
coGset
A
)
:=
λ
x
,
FinGSet
{[
x
]}
.
Global
Instance
coGset_union
:
Union
(
coGset
A
)
:=
λ
X
Y
,
match
X
,
Y
with
|
FinGSet
X
,
FinGSet
Y
=>
FinGSet
(
X
∪
Y
)
|
CoFinGset
X
,
CoFinGset
Y
=>
CoFinGset
(
X
∩
Y
)
|
FinGSet
X
,
CoFinGset
Y
=>
CoFinGset
(
Y
∖
X
)
|
CoFinGset
X
,
FinGSet
Y
=>
CoFinGset
(
X
∖
Y
)
end
.
Global
Instance
coGset_intersection
:
Intersection
(
coGset
A
)
:=
λ
X
Y
,
match
X
,
Y
with
|
FinGSet
X
,
FinGSet
Y
=>
FinGSet
(
X
∩
Y
)
|
CoFinGset
X
,
CoFinGset
Y
=>
CoFinGset
(
X
∪
Y
)
|
FinGSet
X
,
CoFinGset
Y
=>
FinGSet
(
X
∖
Y
)
|
CoFinGset
X
,
FinGSet
Y
=>
FinGSet
(
Y
∖
X
)
end
.
Global
Instance
coGset_difference
:
Difference
(
coGset
A
)
:=
λ
X
Y
,
match
X
,
Y
with
|
FinGSet
X
,
FinGSet
Y
=>
FinGSet
(
X
∖
Y
)
|
CoFinGset
X
,
CoFinGset
Y
=>
FinGSet
(
Y
∖
X
)
|
FinGSet
X
,
CoFinGset
Y
=>
FinGSet
(
X
∩
Y
)
|
CoFinGset
X
,
FinGSet
Y
=>
CoFinGset
(
X
∪
Y
)
end
.
Global
Instance
coGset_set
:
TopSet
A
(
coGset
A
)
.
Proof
.
split
;
[
split
;
[
split
|
|]|]
.
-
by
intros
??
.
-
intros
x
y
.
unfold
elem_of
,
coGset_elem_of
;
simpl
.
by
rewrite
elem_of_singleton
.
-
intros
[
X
|
X
]
[
Y
|
Y
]
x
;
unfold
elem_of
,
coGset_elem_of
,
coGset_union
;
simpl
.
+
set_solver
.
+
by
rewrite
not_elem_of_difference
,
(
comm
(
∨
))
.
+
by
rewrite
not_elem_of_difference
.
+
by
rewrite
not_elem_of_intersection
.
-
intros
[]
[];
unfold
elem_of
,
coGset_elem_of
,
coGset_intersection
;
set_solver
.
-
intros
[
X
|
X
]
[
Y
|
Y
]
x
;
unfold
elem_of
,
coGset_elem_of
,
coGset_difference
;
simpl
.
+
set_solver
.
+
rewrite
elem_of_intersection
.
destruct
(
decide
(
x
∈
Y
));
tauto
.
+
set_solver
.
+
rewrite
elem_of_difference
.
destruct
(
decide
(
x
∈
Y
));
tauto
.
-
done
.
Qed
.
End
coGset
.
Instance
coGset_elem_of_dec
`{
Countable
A
}
:
RelDecision
(
∈@
{
coGset
A
})
:=
λ
x
X
,
match
X
with
|
FinGSet
X
=>
decide_rel
elem_of
x
X
|
CoFinGset
X
=>
not_dec
(
decide_rel
elem_of
x
X
)
end
.
Section
infinite
.
Context
`{
Countable
A
,
Infinite
A
}
.
Global
Instance
coGset_leibniz
:
LeibnizEquiv
(
coGset
A
)
.
Proof
.
intros
[
X
|
X
]
[
Y
|
Y
];
rewrite
elem_of_equiv
;
unfold
elem_of
,
coGset_elem_of
;
simpl
;
intros
HXY
.
-
f_equal
.
by
apply
leibniz_equiv
.
-
by
destruct
(
exist_fresh
(
X
∪
Y
))
as
[?
[?
?
%
HXY
]
%
not_elem_of_union
]
.
-
by
destruct
(
exist_fresh
(
X
∪
Y
))
as
[?
[?
%
HXY
?]
%
not_elem_of_union
]
.
-
f_equal
.
apply
leibniz_equiv
;
intros
x
.
by
apply
not_elem_of_iff
.
Qed
.
Global
Instance
coGset_equiv_dec
:
RelDecision
(
≡@
{
coGset
A
})
.
Proof
.
refine
(
λ
X
Y
,
cast_if
(
decide
(
X
=
Y
)));
abstract
(
by
fold_leibniz
)
.
Defined
.
Global
Instance
coGset_disjoint_dec
:
RelDecision
(
##@
{
coGset
A
})
.
Proof
.
refine
(
λ
X
Y
,
cast_if
(
decide
(
X
∩
Y
=
∅
)));
abstract
(
by
rewrite
disjoint_intersection_L
)
.
Defined
.
Global
Instance
coGset_subseteq_dec
:
RelDecision
(
⊆@
{
coGset
A
})
.
Proof
.
refine
(
λ
X
Y
,
cast_if
(
decide
(
X
∪
Y
=
Y
)));
abstract
(
by
rewrite
subseteq_union_L
)
.
Defined
.
Definition
coGset_finite
(
X
:
coGset
A
)
:
bool
:=
match
X
with
FinGSet
_
=>
true
|
CoFinGset
_
=>
false
end
.
Lemma
coGset_finite_spec
X
:
set_finite
X
↔
coGset_finite
X
.
Proof
.
destruct
X
as
[
X
|
X
];
unfold
set_finite
,
elem_of
at
1
,
coGset_elem_of
;
simpl
.
-
split
;
[
done
|
intros
_]
.
exists
(
elements
X
)
.
set_solver
.
-
split
;
[
intros
[
Y
HXY
]
%
(
pred_finite_set
(
C
:=
gset
A
))|
done
]
.
by
destruct
(
exist_fresh
(
X
∪
Y
))
as
[?
[?
%
HXY
?]
%
not_elem_of_union
]
.
Qed
.
Global
Instance
coGset_finite_dec
(
X
:
coGset
A
)
:
Decision
(
set_finite
X
)
.
Proof
.
refine
(
cast_if
(
decide
(
coGset_finite
X
)));
abstract
(
by
rewrite
coGset_finite_spec
)
.
Defined
.
End
infinite
.
(** * Pick elements from infinite sets *)
Definition
cogpick
`{
Countable
A
,
Infinite
A
}
(
X
:
coGset
A
)
:
A
:=
fresh
(
match
X
with
FinGSet
_
=>
∅
|
CoFinGset
X
=>
X
end
)
.
Lemma
cogpick_elem_of
`{
Countable
A
,
Infinite
A
}
X
:
¬
set_finite
X
→
cogpick
X
∈
X
.
Proof
.
unfold
cogpick
.
destruct
X
as
[
X
|
X
];
rewrite
coGset_finite_spec
;
simpl
.
done
.
by
intros
_;
apply
is_fresh
.
Qed
.
(** * Conversion to and from gset *)
Definition
coGset_to_gset
`{
Countable
A
}
(
X
:
coGset
A
)
:
gset
A
:=
match
X
with
FinGSet
X
=>
X
|
CoFinGset
_
=>
∅
end
.
Definition
gset_to_coGset
`{
Countable
A
}
:
gset
A
→
coGset
A
:=
FinGSet
.
Section
to_gset
.
Context
`{
Countable
A
,
Infinite
A
}
.
Lemma
elem_of_coGset_to_gset
(
X
:
coGset
A
)
x
:
set_finite
X
→
x
∈
coGset_to_gset
X
↔
x
∈
X
.
Proof
.
rewrite
coGset_finite_spec
.
by
destruct
X
.
Qed
.
Lemma
elem_of_gset_to_coGset
(
X
:
gset
A
)
x
:
x
∈
gset_to_coGset
X
↔
x
∈
X
.
Proof
.
done
.
Qed
.
Lemma
gset_to_coGset_finite
(
X
:
gset
A
)
:
set_finite
(
gset_to_coGset
X
)
.
Proof
.
by
rewrite
coGset_finite_spec
.
Qed
.
End
to_gset
.
(** * Conversion to coPset *)
Definition
coGset_to_coPset
(
X
:
coGset
positive
)
:
coPset
:=
match
X
with
|
FinGSet
X
=>
gset_to_coPset
X
|
CoFinGset
X
=>
⊤
∖
gset_to_coPset
X
end
.
Lemma
elem_of_coGset_to_coPset
X
x
:
x
∈
coGset_to_coPset
X
↔
x
∈
X
.
Proof
.
destruct
X
as
[
X
|
X
];
simpl
.
by
rewrite
elem_of_gset_to_coPset
.
by
rewrite
elem_of_difference
,
elem_of_gset_to_coPset
,
(
left_id
True
(
∧
))
.
Qed
.
(** * Inefficient conversion to arbitrary sets with a top element *)
(** This shows that, when [A] is countable, [coGset A] is initial
among sets with [∪], [∩], [∖], [∅], [{[_]}], and [⊤]. *)
Definition
coGset_to_top_set
`{
Countable
A
,
Empty
C
,
Singleton
A
C
,
Union
C
,
Top
C
,
Difference
C
}
(
X
:
coGset
A
)
:
C
:=
match
X
with
|
FinGSet
X
=>
list_to_set
(
elements
X
)
|
CoFinGset
X
=>
⊤
∖
list_to_set
(
elements
X
)
end
.
Lemma
elem_of_coGset_to_top_set
`{
Countable
A
,
TopSet
A
C
}
X
x
:
x
∈@
{
C
}
coGset_to_top_set
X
↔
x
∈
X
.
Proof
.
destruct
X
;
set_solver
.
Qed
.
(** * Domain of finite maps *)
Instance
coGset_dom
`{
Countable
K
}
{
A
}
:
Dom
(
gmap
K
A
)
(
coGset
K
)
:=
λ
m
,
gset_to_coGset
(
dom
_
m
)
.
Instance
coGset_dom_spec
`{
Countable
K
}
:
FinMapDom
K
(
gmap
K
)
(
coGset
K
)
.
Proof
.
split
;
try
apply
_
.
intros
B
m
i
.
unfold
dom
,
coGset_dom
.
by
rewrite
elem_of_gset_to_coGset
,
elem_of_dom
.
Qed
.
Typeclasses
Opaque
coGset_elem_of
coGset_empty
coGset_top
coGset_singleton
.
Typeclasses
Opaque
coGset_union
coGset_intersection
coGset_difference
.
Typeclasses
Opaque
coGset_dom
.
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