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stdpp
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8a43c1bb
Commit
8a43c1bb
authored
Mar 02, 2015
by
Robbert Krebbers
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Remove duplicate mem_allocable notion.
parent
8734dd1c
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2
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2 changed files
with
41 additions
and
12 deletions
+41
12
theories/collections.v
theories/collections.v
+35
12
theories/fin_map_dom.v
theories/fin_map_dom.v
+6
0
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theories/collections.v
View file @
8a43c1bb
...
...
@@ 430,16 +430,19 @@ End more_quantifiers.
(** * Fresh elements *)
(** We collect some properties on the [fresh] operation. In particular we
generalize [fresh] to generate lists of fresh elements. *)
Section
fresh
.
Context
`
{
FreshSpec
A
C
}
.
Fixpoint
fresh_list
`
{
Fresh
A
C
,
Union
C
,
Singleton
A
C
}
(
n
:
nat
)
(
X
:
C
)
:
list
A
:
=
match
n
with

0
=>
[]

S
n
=>
let
x
:
=
fresh
X
in
x
::
fresh_list
n
({[
x
]}
∪
X
)
end
.
Inductive
Forall_fresh
`
{
ElemOf
A
C
}
(
X
:
C
)
:
list
A
→
Prop
:
=

Forall_fresh_nil
:
Forall_fresh
X
[]

Forall_fresh_cons
x
xs
:
x
∉
xs
→
x
∉
X
→
Forall_fresh
X
xs
→
Forall_fresh
X
(
x
::
xs
).
Definition
fresh_sig
(
X
:
C
)
:
{
x
:
A

x
∉
X
}
:
=
exist
(
∉
X
)
(
fresh
X
)
(
is_fresh
X
).
Fixpoint
fresh_list
(
n
:
nat
)
(
X
:
C
)
:
list
A
:
=
match
n
with

0
=>
[]

S
n
=>
let
x
:
=
fresh
X
in
x
::
fresh_list
n
({[
x
]}
∪
X
)
end
.
Section
fresh
.
Context
`
{
FreshSpec
A
C
}.
Global
Instance
fresh_proper
:
Proper
((
≡
)
==>
(=))
fresh
.
Proof
.
intros
???.
by
apply
fresh_proper_alt
,
elem_of_equiv
.
Qed
.
...
...
@@ 448,18 +451,38 @@ Section fresh.
intros
?
n
>.
induction
n
as
[
n
IH
]
;
intros
??
E
;
f_equal'
;
[
by
rewrite
E
].
apply
IH
.
by
rewrite
E
.
Qed
.
Lemma
Forall_fresh_NoDup
X
xs
:
Forall_fresh
X
xs
→
NoDup
xs
.
Proof
.
induction
1
;
by
constructor
.
Qed
.
Lemma
Forall_fresh_elem_of
X
xs
x
:
Forall_fresh
X
xs
→
x
∈
xs
→
x
∉
X
.
Proof
.
intros
HX
;
revert
x
;
rewrite
<
Forall_forall
.
by
induction
HX
;
constructor
.
Qed
.
Lemma
Forall_fresh_alt
X
xs
:
Forall_fresh
X
xs
↔
NoDup
xs
∧
∀
x
,
x
∈
xs
→
x
∉
X
.
Proof
.
split
;
eauto
using
Forall_fresh_NoDup
,
Forall_fresh_elem_of
.
rewrite
<
Forall_forall
.
intros
[
Hxs
Hxs'
].
induction
Hxs
;
decompose_Forall_hyps
;
constructor
;
auto
.
Qed
.
Lemma
fresh_list_length
n
X
:
length
(
fresh_list
n
X
)
=
n
.
Proof
.
revert
X
.
induction
n
;
simpl
;
auto
.
Qed
.
Lemma
fresh_list_is_fresh
n
X
x
:
x
∈
fresh_list
n
X
→
x
∉
X
.
Proof
.
revert
X
.
induction
n
as
[
n
IH
]
;
intros
X
;
simpl
;
[
by
rewrite
elem_of_nil
].
revert
X
.
induction
n
as
[
n
IH
]
;
intros
X
;
simpl
;
[
by
rewrite
elem_of_nil
].
rewrite
elem_of_cons
;
intros
[>
Hin
]
;
[
apply
is_fresh
].
apply
IH
in
Hin
;
solve_elem_of
.
Qed
.
Lemma
fresh_list_nodup
n
X
:
NoDup
(
fresh_list
n
X
).
Lemma
NoDup_fresh_list
n
X
:
NoDup
(
fresh_list
n
X
).
Proof
.
revert
X
.
induction
n
;
simpl
;
constructor
;
auto
.
intros
Hin
.
apply
fresh_list_is_fresh
in
Hin
.
solve_elem_of
.
intros
Hin
;
apply
fresh_list_is_fresh
in
Hin
;
solve_elem_of
.
Qed
.
Lemma
Forall_fresh_list
X
n
:
Forall_fresh
X
(
fresh_list
n
X
).
Proof
.
rewrite
Forall_fresh_alt
;
eauto
using
NoDup_fresh_list
,
fresh_list_is_fresh
.
Qed
.
End
fresh
.
...
...
theories/fin_map_dom.v
View file @
8a43c1bb
...
...
@@ 44,6 +44,12 @@ Proof.
intros
E
.
apply
map_empty
.
intros
.
apply
not_elem_of_dom
.
rewrite
E
.
solve_elem_of
.
Qed
.
Lemma
dom_alter
{
A
}
f
(
m
:
M
A
)
i
:
dom
D
(
alter
f
i
m
)
≡
dom
D
m
.
Proof
.
apply
elem_of_equiv
;
intros
j
;
rewrite
!
elem_of_dom
;
unfold
is_Some
.
destruct
(
decide
(
i
=
j
))
;
simplify_map_equality'
;
eauto
.
destruct
(
m
!!
j
)
;
naive_solver
.
Qed
.
Lemma
dom_insert
{
A
}
(
m
:
M
A
)
i
x
:
dom
D
(<[
i
:
=
x
]>
m
)
≡
{[
i
]}
∪
dom
D
m
.
Proof
.
apply
elem_of_equiv
.
intros
j
.
rewrite
elem_of_union
,
!
elem_of_dom
.
...
...
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