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Why3
why3
Commits
86c3b64c
Commit
86c3b64c
authored
Feb 12, 2014
by
Andrei Paskevich
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update Coq realization for Map.Permut
parent
c93f9849
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1
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with
41 additions
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48 deletions
+41
-48
lib/coq/map/MapPermut.v
lib/coq/map/MapPermut.v
+41
-48
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lib/coq/map/MapPermut.v
View file @
86c3b64c
...
...
@@ -10,6 +10,15 @@ Definition map_eq_sub {a:Type} {a_WT:WhyType a} (a1:(@map.Map.map Z _
a
a_WT
))
(
a2
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
(
l
:
Z
)
(
u
:
Z
)
:
Prop
:=
forall
(
i
:
Z
),
((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
->
((
map
.
Map
.
get
a1
i
)
=
(
map
.
Map
.
get
a2
i
)).
(
*
Why3
assumption
*
)
Definition
exchange
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
(
a2
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
(
l
:
Z
)
(
u
:
Z
)
(
i
:
Z
)
(
j
:
Z
)
:
Prop
:=
((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
/
\
(((
l
<=
j
)
%
Z
/
\
(
j
<
u
)
%
Z
)
/
\
(((
map
.
Map
.
get
a1
i
)
=
(
map
.
Map
.
get
a2
j
))
/
\
(((
map
.
Map
.
get
a1
j
)
=
(
map
.
Map
.
get
a2
i
))
/
\
forall
(
k
:
Z
),
((
l
<=
k
)
%
Z
/
\
(
k
<
u
)
%
Z
)
->
((
~
(
k
=
i
))
->
((
~
(
k
=
j
))
->
((
map
.
Map
.
get
a1
k
)
=
(
map
.
Map
.
get
a2
k
))))))).
(
*
Why3
assumption
*
)
Inductive
permut
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
)
->
(
@
map
.
Map
.
map
Z
_
a
a_WT
)
->
Z
->
Z
->
Prop
:=
...
...
@@ -21,60 +30,38 @@ Inductive permut {a:Type} {a_WT:WhyType a} : (@map.Map.map Z _ a a_WT) ->
((
@
permut
_
_
)
a1
a2
l
u
)
->
(((
@
permut
_
_
)
a2
a3
l
u
)
->
((
@
permut
_
_
)
a1
a3
l
u
))
|
permut_exchange
:
forall
(
a1
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
(
a2
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
)),
forall
(
l
:
Z
)
(
u
:
Z
)
(
i
:
Z
)
(
j
:
Z
),
((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
->
(((
l
<=
j
)
%
Z
/
\
(
j
<
u
)
%
Z
)
->
((
~
(
i
=
j
))
->
(((
map
.
Map
.
get
a1
i
)
=
(
map
.
Map
.
get
a2
j
))
->
(((
map
.
Map
.
get
a1
j
)
=
(
map
.
Map
.
get
a2
i
))
->
((
forall
(
k
:
Z
),
((
l
<=
k
)
%
Z
/
\
(
k
<
u
)
%
Z
)
->
((
~
(
k
=
i
))
->
((
~
(
k
=
j
))
->
((
map
.
Map
.
get
a1
k
)
=
(
map
.
Map
.
get
a2
k
)))))
->
((
@
permut
_
_
)
a1
a2
l
u
)))))).
Z
_
a
a_WT
)),
forall
(
l
:
Z
)
(
u
:
Z
)
(
i
:
Z
)
(
j
:
Z
),
(
exchange
a1
a2
l
u
i
j
)
->
((
@
permut
_
_
)
a1
a2
l
u
).
(
*
Why3
goal
*
)
Lemma
permut_sym
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
(
a2
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
)),
forall
(
l
:
Z
)
(
u
:
Z
),
(
permut
a1
a2
l
u
)
->
(
permut
a2
a1
l
u
).
intros
a
a_WT
a1
a2
l
u
h1
.
induction
h1
;
intuition
.
apply
permut_refl
;
unfold
map_eq_sub
in
*
;
intuition
.
rewrite
<-
H
;
auto
.
apply
permut_trans
with
a2
;
auto
.
apply
permut_exchange
with
j
i
;
auto
.
intros
;
rewrite
<-
H4
;
auto
.
Qed
.
(
*
Why3
goal
*
)
Lemma
permut_exchange_set
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
)),
forall
(
l
:
Z
)
(
u
:
Z
)
(
i
:
Z
)
(
j
:
Z
),
((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
->
(((
l
<=
j
)
%
Z
/
\
(
j
<
u
)
%
Z
)
->
(
permut
a1
Lemma
exchange_set
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
(
l
:
Z
)
(
u
:
Z
)
(
i
:
Z
)
(
j
:
Z
),
((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
->
(((
l
<=
j
)
%
Z
/
\
(
j
<
u
)
%
Z
)
->
(
exchange
a1
(
map
.
Map
.
set
(
map
.
Map
.
set
a1
i
(
map
.
Map
.
get
a1
j
))
j
(
map
.
Map
.
get
a1
i
))
l
u
)).
u
i
j
)).
intros
a
a_WT
a1
l
u
i
j
(
h1
,
h2
)
(
h3
,
h4
).
assert
(
h
:
i
=
j
\
/
i
<>
j
)
by
omega
.
destruct
h
.
subst
.
apply
permut_refl
.
red
;
intros
.
assert
(
h
:
i
=
j
\
/
i
<>
j
)
by
omega
.
destruct
h
.
subst
.
rewrite
Map
.
Select_eq
;
auto
.
rewrite
Map
.
Select_neq
.
rewrite
Map
.
Select_neq
;
auto
.
auto
.
apply
permut_exchange
with
i
j
;
auto
.
unfold
exchange
.
intuition
.
rewrite
Map
.
Select_eq
;
auto
.
assert
(
H
:
i
=
j
\
/
i
<>
j
)
by
omega
.
destruct
H
.
rewrite
H
;
rewrite
Map
.
Select_eq
;
auto
.
rewrite
Map
.
Select_neq
;
auto
.
rewrite
Map
.
Select_eq
;
auto
.
intros
.
rewrite
Map
.
Select_neq
;
auto
.
rewrite
Map
.
Select_neq
;
auto
.
Qed
.
(
*
Why3
goal
*
)
Lemma
permut_exists
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
(
a2
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
,
forall
(
l
:
Z
)
(
u
:
Z
),
(
permut
a1
a2
l
u
)
->
forall
(
i
:
Z
),
((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
->
exists
j
:
Z
,
((
l
<=
j
)
%
Z
/
\
(
j
<
u
)
%
Z
)
/
\
((
map
.
Map
.
get
a2
i
)
=
(
map
.
Map
.
get
a1
j
)).
forall
(
a1
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
(
a2
:
(
@
map
.
Map
.
map
Z
_
a
a_WT
))
(
l
:
Z
)
(
u
:
Z
)
(
i
:
Z
),
(
permut
a1
a2
l
u
)
->
(((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
->
exists
j
:
Z
,
((
l
<=
j
)
%
Z
/
\
(
j
<
u
)
%
Z
)
/
\
((
map
.
Map
.
get
a1
j
)
=
(
map
.
Map
.
get
a2
i
)
)).
Proof
.
intros
a
a_WT
a1
a2
l
u
h1
i
Hi
.
assert
((
exists
j
,
(
l
<=
j
<
u
)
%
Z
/
\
Map
.
get
a
2
i
=
Map
.
get
a1
j
)
/
\
(
exists
j
,
(
l
<=
j
<
u
)
%
Z
/
\
Map
.
get
a
1
i
=
Map
.
get
a2
j
)).
intros
a
a_WT
a1
a2
l
u
i
h1
Hi
.
assert
((
exists
j
,
(
l
<=
j
<
u
)
%
Z
/
\
Map
.
get
a
1
j
=
Map
.
get
a2
i
)
/
\
(
exists
j
,
(
l
<=
j
<
u
)
%
Z
/
\
Map
.
get
a
2
j
=
Map
.
get
a1
i
)).
2
:
easy
.
revert
i
Hi
.
induction
h1
.
...
...
@@ -97,7 +84,9 @@ exists k.
split
;
try
easy
.
now
transitivity
(
Map
.
get
a2
j
).
(
*
exchange
*
)
intros
k
Hk
.
revert
H
.
unfold
exchange
.
intros
[
Hi
[
Hj
[
Ha1ia2j
[
Ha1ja2i
H
]]]]
k
Hk
.
destruct
(
Z_eq_dec
k
i
)
as
[
Hki
|
Hki
].
split
;
exists
j
;
...
...
@@ -113,13 +102,17 @@ split ;
split
;
exists
k
;
split
;
try
easy
.
now
apply
sym_eq
,
H4
;
intuition
.
now
apply
H4
;
intuition
.
now
apply
H
;
intuition
.
now
apply
sym_eq
,
H
;
intuition
.
Qed
.
(
*
Unused
content
named
permut_sub_concat
intros
a
a_WT
a1
a2
l
m
u
h1
h2
.
induction
h1
.
(
*
Unused
content
named
permut_sym
intros
a
a_WT
a1
a2
l
u
h1
.
induction
h1
;
intuition
.
apply
permut_refl
;
unfold
map_eq_sub
in
*
;
intuition
.
rewrite
<-
H
;
auto
.
apply
permut_trans
with
a2
;
auto
.
apply
permut_exchange
with
j
i
;
auto
.
intros
;
rewrite
<-
H4
;
auto
.
Qed
.
*
)
*
)
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