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David Swasey
coqstdpp
Commits
fcc1f0de
Commit
fcc1f0de
authored
Aug 14, 2013
by
Robbert Krebbers
Browse files
More efficient conversion of pmap to association lists.
parent
bc659ba4
Changes
1
Hide whitespace changes
Inline
Sidebyside
theories/pmap.v
View file @
fcc1f0de
...
...
@@ 257,76 +257,84 @@ Lemma Plookup_fmap {A B} (f : A → B) (t : Pmap_raw A) i :
fmap
f
t
!!
i
=
fmap
f
(
t
!!
i
).
Proof
.
revert
i
.
induction
t
.
done
.
by
intros
[??]
;
simpl
.
Qed
.
Fixpoint
Pto_list_raw
{
A
}
(
j
:
positive
)
(
t
:
Pmap_raw
A
)
:
list
(
positive
*
A
)
:
=
Fixpoint
Pto_list_raw
{
A
}
(
j
:
positive
)
(
t
:
Pmap_raw
A
)
(
acc
:
list
(
positive
*
A
))
:
list
(
positive
*
A
)
:
=
match
t
with

PLeaf
=>
[]

PNode
l
o
r
=>
default
[]
o
(
λ
x
,
[(
Preverse
j
,
x
)])
++
Pto_list_raw
(
j
~
0
)
l
++
Pto_list_raw
(
j
~
1
)
r

PLeaf
=>
acc

PNode
l
o
r
=>
default
[]
o
(
λ
x
,
[(
Preverse
j
,
x
)])
++
Pto_list_raw
(
j
~
0
)
l
(
Pto_list_raw
(
j
~
1
)
r
acc
)
end
%
list
.
Lemma
Pelem_of_to_list_aux
{
A
}
(
t
:
Pmap_raw
A
)
j
i
x
:
(
i
,
x
)
∈
Pto_list_raw
j
t
↔
∃
i'
,
i
=
i'
++
Preverse
j
∧
t
!!
i'
=
Some
x
.
Lemma
Pelem_of_to_list
{
A
}
(
t
:
Pmap_raw
A
)
j
i
acc
x
:
(
i
,
x
)
∈
Pto_list_raw
j
t
acc
↔
(
∃
i'
,
i
=
i'
++
Preverse
j
∧
t
!!
i'
=
Some
x
)
∨
(
i
,
x
)
∈
acc
.
Proof
.
split
.
*
revert
j
.
induction
t
as
[?
IHl
[?]
?
IHr
]
;
intros
j
;
simpl
.
+
by
rewrite
?elem_of_nil
.
+
rewrite
elem_of_cons
,
!
elem_of_app
.
intros
[?[??]].

simplify_equality
.
exists
1
.
by
rewrite
(
left_id_L
1
(++))%
positive
.

destruct
(
IHl
(
j
~
0
))
as
(
i'
&?&?)
;
trivial
;
subst
.
exists
(
i'
~
0
).
by
rewrite
Preverse_xO
,
(
associative_L
_
).

destruct
(
IHr
(
j
~
1
))
as
(
i'
&?&?)
;
trivial
;
subst
.
exists
(
i'
~
1
).
by
rewrite
Preverse_xI
,
(
associative_L
_
).
+
rewrite
!
elem_of_app
.
intros
[??].

destruct
(
IHl
(
j
~
0
))
as
(
i'
&?&?)
;
trivial
;
subst
.
exists
(
i'
~
0
).
by
rewrite
Preverse_xO
,
(
associative_L
_
).

destruct
(
IHr
(
j
~
1
))
as
(
i'
&?&?)
;
trivial
;
subst
.
exists
(
i'
~
1
).
by
rewrite
Preverse_xI
,
(
associative_L
_
).
*
intros
(
i'
&
?&
Hi'
)
;
subst
.
revert
i'
j
Hi'
.
induction
t
as
[?
IHl
[?]
?
IHr
]
;
intros
i
j
;
simpl
.
+
done
.
+
rewrite
elem_of_cons
,
elem_of_app
.
destruct
i
as
[
i

i
]
;
simpl
in
*.

right
.
right
.
specialize
(
IHr
i
(
j
~
1
)).
rewrite
Preverse_xI
,
(
associative_L
_
)
in
IHr
.
auto
.

right
.
left
.
specialize
(
IHl
i
(
j
~
0
)).
rewrite
Preverse_xO
,
(
associative_L
_
)
in
IHl
.
auto
.

left
.
simplify_equality
.
by
rewrite
(
left_id_L
1
(++))%
positive
.
+
rewrite
elem_of_app
.
destruct
i
as
[
i

i
]
;
simpl
in
*.

right
.
specialize
(
IHr
i
(
j
~
1
)).
rewrite
Preverse_xI
,
(
associative_L
_
)
in
IHr
.
auto
.

left
.
specialize
(
IHl
i
(
j
~
0
)).
rewrite
Preverse_xO
,
(
associative_L
_
)
in
IHl
.
auto
.

done
.
Qed
.
Lemma
Pelem_of_to_list
{
A
}
(
t
:
Pmap_raw
A
)
i
x
:
(
i
,
x
)
∈
Pto_list_raw
1
t
↔
t
!!
i
=
Some
x
.
Proof
.
rewrite
Pelem_of_to_list_aux
.
split
.
by
intros
(
i'
&>&?).
intros
.
by
exists
i
.
{
revert
j
acc
.
induction
t
as
[
l
IHl
[
y
]
r
IHr
]
;
intros
j
acc
;
simpl
.
*
by
right
.
*
rewrite
elem_of_cons
.
intros
[??]
;
simplify_equality
.
{
left
;
exists
1
.
by
rewrite
(
left_id_L
1
(++))%
positive
.
}
destruct
(
IHl
(
j
~
0
)
(
Pto_list_raw
j
~
1
r
acc
))
as
[(
i'
&>&?)?]
;
auto
.
{
left
;
exists
(
i'
~
0
).
by
rewrite
Preverse_xO
,
(
associative_L
_
).
}
destruct
(
IHr
(
j
~
1
)
acc
)
as
[(
i'
&>&?)?]
;
auto
.
left
;
exists
(
i'
~
1
).
by
rewrite
Preverse_xI
,
(
associative_L
_
).
*
intros
.
destruct
(
IHl
(
j
~
0
)
(
Pto_list_raw
j
~
1
r
acc
))
as
[(
i'
&>&?)?]
;
auto
.
{
left
;
exists
(
i'
~
0
).
by
rewrite
Preverse_xO
,
(
associative_L
_
).
}
destruct
(
IHr
(
j
~
1
)
acc
)
as
[(
i'
&>&?)?]
;
auto
.
left
;
exists
(
i'
~
1
).
by
rewrite
Preverse_xI
,
(
associative_L
_
).
}
revert
t
j
i
acc
.
assert
(
∀
t
j
i
acc
,
(
i
,
x
)
∈
acc
→
(
i
,
x
)
∈
Pto_list_raw
j
t
acc
)
as
help
.
{
intros
t
;
induction
t
as
[
l
IHl
[
y
]
r
IHr
]
;
intros
j
i
acc
;
simpl
;
rewrite
?elem_of_cons
;
auto
.
}
intros
t
j
?
acc
[(
i
&>&
Hi
)?]
;
[
by
auto
].
revert
j
i
acc
Hi
.
induction
t
as
[
l
IHl
[
y
]
r
IHr
]
;
intros
j
i
acc
?
;
simpl
.
*
done
.
*
rewrite
elem_of_cons
.
destruct
i
as
[
i

i
]
;
simplify_equality'
.
+
right
.
apply
help
.
specialize
(
IHr
(
j
~
1
)
i
).
rewrite
Preverse_xI
,
(
associative_L
_
)
in
IHr
.
by
apply
IHr
.
+
right
.
specialize
(
IHl
(
j
~
0
)
i
).
rewrite
Preverse_xO
,
(
associative_L
_
)
in
IHl
.
by
apply
IHl
.
+
left
.
by
rewrite
(
left_id_L
1
(++))%
positive
.
*
destruct
i
as
[
i

i
]
;
simplify_equality'
.
+
apply
help
.
specialize
(
IHr
(
j
~
1
)
i
).
rewrite
Preverse_xI
,
(
associative_L
_
)
in
IHr
.
by
apply
IHr
.
+
specialize
(
IHl
(
j
~
0
)
i
).
rewrite
Preverse_xO
,
(
associative_L
_
)
in
IHl
.
by
apply
IHl
.
Qed
.
Lemma
Pto_list_nodup
{
A
}
j
(
t
:
Pmap_raw
A
)
:
NoDup
(
Pto_list_raw
j
t
).
Lemma
Pto_list_nodup
{
A
}
j
(
t
:
Pmap_raw
A
)
acc
:
(
∀
i
x
,
(
i
++
Preverse
j
,
x
)
∈
acc
→
t
!!
i
=
None
)
→
NoDup
acc
→
NoDup
(
Pto_list_raw
j
t
acc
).
Proof
.
revert
j
.
induction
t
as
[?
IHl
[?]
?
IHr
]
;
simpl
.
*
constructor
.
*
intros
.
rewrite
NoDup_cons
,
NoDup_app
.
split_ands
;
trivial
.
+
rewrite
elem_of_app
,
!
Pelem_of_to_list_aux
.
intros
[(
i
&
Hi
&?)(
i
&
Hi
&?)].

rewrite
Preverse_xO
in
Hi
.
apply
(
f_equal
Plength
)
in
Hi
.
rewrite
!
Papp_length
in
Hi
.
simpl
in
Hi
.
lia
.

rewrite
Preverse_xI
in
Hi
.
apply
(
f_equal
Plength
)
in
Hi
.
rewrite
!
Papp_length
in
Hi
.
simpl
in
Hi
.
lia
.
+
intros
[??].
rewrite
!
Pelem_of_to_list_aux
.
intros
(
i1
&?&?)
(
i2
&
Hi
&?)
;
subst
.
rewrite
Preverse_xO
,
Preverse_xI
,
!(
associative_L
_
)
in
Hi
.
by
apply
(
injective
(++
_
))
in
Hi
.
*
intros
.
rewrite
NoDup_app
.
split_ands
;
trivial
.
intros
[??].
rewrite
!
Pelem_of_to_list_aux
.
intros
(
i1
&?&?)
(
i2
&
Hi
&?)
;
subst
.
rewrite
Preverse_xO
,
Preverse_xI
,
!(
associative_L
_
)
in
Hi
.
by
apply
(
injective
(++
_
))
in
Hi
.
revert
j
acc
.
induction
t
as
[
l
IHl
[
y
]
r
IHr
]
;
simpl
;
intros
j
acc
Hin
?.
*
done
.
*
repeat
constructor
.
{
rewrite
Pelem_of_to_list
.
intros
[(
i
&
Hi
&?)
Hj
].
{
apply
(
f_equal
Plength
)
in
Hi
.
rewrite
Preverse_xO
,
!
Papp_length
in
Hi
;
simpl
in
*
;
lia
.
}
rewrite
Pelem_of_to_list
in
Hj
.
destruct
Hj
as
[(
i
&
Hi
&?)
Hj
].
{
apply
(
f_equal
Plength
)
in
Hi
.
rewrite
Preverse_xI
,
!
Papp_length
in
Hi
;
simpl
in
*
;
lia
.
}
specialize
(
Hin
1
y
).
rewrite
(
left_id_L
1
(++))%
positive
in
Hin
.
discriminate
(
Hin
Hj
).
}
apply
IHl
.
{
intros
i
x
.
rewrite
Pelem_of_to_list
.
intros
[(?&
Hi
&?)
Hi
].
+
rewrite
Preverse_xO
,
Preverse_xI
,
!(
associative_L
_
)
in
Hi
.
by
apply
(
injective
(++
_
))
in
Hi
.
+
apply
(
Hin
(
i
~
0
)
x
).
by
rewrite
Preverse_xO
,
(
associative_L
_
)
in
Hi
.
}
apply
IHr
;
auto
.
intros
i
x
Hi
.
apply
(
Hin
(
i
~
1
)
x
).
by
rewrite
Preverse_xI
,
(
associative_L
_
)
in
Hi
.
*
apply
IHl
.
{
intros
i
x
.
rewrite
Pelem_of_to_list
.
intros
[(?&
Hi
&?)
Hi
].
+
rewrite
Preverse_xO
,
Preverse_xI
,
!(
associative_L
_
)
in
Hi
.
by
apply
(
injective
(++
_
))
in
Hi
.
+
apply
(
Hin
(
i
~
0
)
x
).
by
rewrite
Preverse_xO
,
(
associative_L
_
)
in
Hi
.
}
apply
IHr
;
auto
.
intros
i
x
Hi
.
apply
(
Hin
(
i
~
1
)
x
).
by
rewrite
Preverse_xI
,
(
associative_L
_
)
in
Hi
.
Qed
.
Global
Instance
Pto_list
{
A
}
:
FinMapToList
positive
A
(
Pmap
A
)
:
=
λ
t
,
Pto_list_raw
1
(
`
t
).
λ
t
,
Pto_list_raw
1
(
`
t
)
[]
.
Fixpoint
Pmerge_aux
`
(
f
:
option
A
→
option
B
)
(
t
:
Pmap_raw
A
)
:
Pmap_raw
B
:
=
match
t
with
...
...
@@ 385,7 +393,11 @@ Proof.
*
intros
??
[??]
??.
by
apply
Plookup_alter_ne
.
*
intros
???
[??].
by
apply
Plookup_fmap
.
*
intros
?
[??].
apply
Pto_list_nodup
.
*
intros
?
[??].
apply
Pelem_of_to_list
.
+
intros
??.
by
rewrite
elem_of_nil
.
+
constructor
.
*
intros
?
[??]
i
x
;
unfold
map_to_list
,
Pto_list
.
rewrite
Pelem_of_to_list
,
elem_of_nil
.
split
.
by
intros
[(?&>&?)].
by
left
;
exists
i
.
*
intros
???
??
[??]
[??]
?.
by
apply
Pmerge_spec
.
Qed
.
...
...
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