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Simon Spies
stdpp
Commits
be4eb648
Commit
be4eb648
authored
Nov 26, 2018
by
Robbert Krebbers
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Versions of `elem_of_list_split` that give first or last element.
parent
f8f6d0a9
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theories/list.v
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theories/list.v
View file @
be4eb648
...
...
@@ 647,6 +647,28 @@ Proof.
induction
1
as
[
x
l

x
y
l
?
[
l1
[
l2
>]]]
;
[
by
eexists
[],
l
].
by
exists
(
y
::
l1
),
l2
.
Qed
.
Lemma
elem_of_list_split_l
`
{
EqDecision
A
}
l
x
:
x
∈
l
→
∃
l1
l2
,
l
=
l1
++
x
::
l2
∧
x
∉
l1
.
Proof
.
induction
1
as
[
x
l

x
y
l
?
IH
].
{
exists
[],
l
.
rewrite
elem_of_nil
.
naive_solver
.
}
destruct
(
decide
(
x
=
y
))
as
[>?].

exists
[],
l
.
rewrite
elem_of_nil
.
naive_solver
.

destruct
IH
as
(
l1
&
l2
&
>
&
?).
exists
(
y
::
l1
),
l2
.
rewrite
elem_of_cons
.
naive_solver
.
Qed
.
Lemma
elem_of_list_split_r
`
{
EqDecision
A
}
l
x
:
x
∈
l
→
∃
l1
l2
,
l
=
l1
++
x
::
l2
∧
x
∉
l2
.
Proof
.
induction
l
as
[
y
l
IH
]
using
rev_ind
.
{
by
rewrite
elem_of_nil
.
}
destruct
(
decide
(
x
=
y
))
as
[>].

exists
l
,
[].
rewrite
elem_of_nil
.
naive_solver
.

rewrite
elem_of_app
,
elem_of_list_singleton
.
intros
[?
>]
;
try
done
.
destruct
IH
as
(
l1
&
l2
&
>
&
?)
;
auto
.
exists
l1
,
(
l2
++
[
y
]).
rewrite
elem_of_app
,
elem_of_list_singleton
,
<(
assoc_L
(++)).
naive_solver
.
Qed
.
Lemma
elem_of_list_lookup_1
l
x
:
x
∈
l
→
∃
i
,
l
!!
i
=
Some
x
.
Proof
.
induction
1
as
[????
IH
]
;
[
by
exists
0
].
...
...
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