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Simon Spies
stdpp
Commits
d67bf19d
Commit
d67bf19d
authored
Jun 05, 2015
by
Robbert Krebbers
Browse files
Prove function rules.
parent
60c8d501
Changes
3
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Showing
3 changed files
with
23 additions
and
4 deletions
+23
4
theories/collections.v
theories/collections.v
+8
0
theories/fin_maps.v
theories/fin_maps.v
+6
4
theories/list.v
theories/list.v
+9
0
No files found.
theories/collections.v
View file @
d67bf19d
...
@@ 178,6 +178,7 @@ Tactic Notation "decompose_elem_of" hyp(H) :=
...
@@ 178,6 +178,7 @@ Tactic Notation "decompose_elem_of" hyp(H) :=
let
H1
:
=
fresh
H
in
let
H2
:
=
fresh
H
in
apply
elem_of_guard
in
H
;
let
H1
:
=
fresh
H
in
let
H2
:
=
fresh
H
in
apply
elem_of_guard
in
H
;
destruct
H
as
[
H1
H2
]
;
go
H2
destruct
H
as
[
H1
H2
]
;
go
H2

_
∈
of_option
_
=>
apply
elem_of_of_option
in
H

_
∈
of_option
_
=>
apply
elem_of_of_option
in
H

_
∈
of_list
_
=>
apply
elem_of_of_list
in
H

_
=>
idtac

_
=>
idtac
end
in
go
H
.
end
in
go
H
.
Tactic
Notation
"decompose_elem_of"
:
=
Tactic
Notation
"decompose_elem_of"
:
=
...
@@ 221,6 +222,8 @@ Ltac unfold_elem_of :=
...
@@ 221,6 +222,8 @@ Ltac unfold_elem_of :=

context
[
_
∈
_
≫
=
_
]
=>
setoid_rewrite
elem_of_bind
in
H

context
[
_
∈
_
≫
=
_
]
=>
setoid_rewrite
elem_of_bind
in
H

context
[
_
∈
mjoin
_
]
=>
setoid_rewrite
elem_of_join
in
H

context
[
_
∈
mjoin
_
]
=>
setoid_rewrite
elem_of_join
in
H

context
[
_
∈
guard
_;
_
]
=>
setoid_rewrite
elem_of_guard
in
H

context
[
_
∈
guard
_;
_
]
=>
setoid_rewrite
elem_of_guard
in
H

context
[
_
∈
of_option
_
]
=>
setoid_rewrite
elem_of_of_option
in
H

context
[
_
∈
of_list
_
]
=>
setoid_rewrite
elem_of_of_list
in
H
end
)
;
end
)
;
repeat
match
goal
with
repeat
match
goal
with


context
[
_
⊆
_
]
=>
setoid_rewrite
elem_of_subseteq


context
[
_
⊆
_
]
=>
setoid_rewrite
elem_of_subseteq
...
@@ 239,6 +242,8 @@ Ltac unfold_elem_of :=
...
@@ 239,6 +242,8 @@ Ltac unfold_elem_of :=


context
[
_
∈
_
≫
=
_
]
=>
setoid_rewrite
elem_of_bind


context
[
_
∈
_
≫
=
_
]
=>
setoid_rewrite
elem_of_bind


context
[
_
∈
mjoin
_
]
=>
setoid_rewrite
elem_of_join


context
[
_
∈
mjoin
_
]
=>
setoid_rewrite
elem_of_join


context
[
_
∈
guard
_;
_
]
=>
setoid_rewrite
elem_of_guard


context
[
_
∈
guard
_;
_
]
=>
setoid_rewrite
elem_of_guard


context
[
_
∈
of_option
_
]
=>
setoid_rewrite
elem_of_of_option


context
[
_
∈
of_list
_
]
=>
setoid_rewrite
elem_of_of_list
end
.
end
.
(** The tactic [solve_elem_of tac] composes the above tactic with [intuition].
(** The tactic [solve_elem_of tac] composes the above tactic with [intuition].
...
@@ 485,6 +490,9 @@ Section fresh.
...
@@ 485,6 +490,9 @@ Section fresh.
rewrite
<
Forall_forall
.
rewrite
<
Forall_forall
.
intros
[
Hxs
Hxs'
].
induction
Hxs
;
decompose_Forall_hyps
;
constructor
;
auto
.
intros
[
Hxs
Hxs'
].
induction
Hxs
;
decompose_Forall_hyps
;
constructor
;
auto
.
Qed
.
Qed
.
Lemma
Forall_fresh_subseteq
X
Y
xs
:
Forall_fresh
X
xs
→
Y
⊆
X
→
Forall_fresh
Y
xs
.
Proof
.
rewrite
!
Forall_fresh_alt
;
esolve_elem_of
.
Qed
.
Lemma
fresh_list_length
n
X
:
length
(
fresh_list
n
X
)
=
n
.
Lemma
fresh_list_length
n
X
:
length
(
fresh_list
n
X
)
=
n
.
Proof
.
revert
X
.
induction
n
;
simpl
;
auto
.
Qed
.
Proof
.
revert
X
.
induction
n
;
simpl
;
auto
.
Qed
.
...
...
theories/fin_maps.v
View file @
d67bf19d
...
@@ 240,10 +240,12 @@ Proof.
...
@@ 240,10 +240,12 @@ Proof.
by
destruct
(
decide
(
i
=
j
))
as
[>?]
;
by
destruct
(
decide
(
i
=
j
))
as
[>?]
;
rewrite
?lookup_alter
,
?fmap_None
,
?lookup_alter_ne
.
rewrite
?lookup_alter
,
?fmap_None
,
?lookup_alter_ne
.
Qed
.
Qed
.
Lemma
alter_None
{
A
}
(
f
:
A
→
A
)
m
i
:
m
!!
i
=
None
→
alter
f
i
m
=
m
.
Lemma
alter_id
{
A
}
(
f
:
A
→
A
)
m
i
:
(
∀
x
,
m
!!
i
=
Some
x
→
f
x
=
x
)
→
alter
f
i
m
=
m
.
Proof
.
Proof
.
intros
Hi
.
apply
map_eq
.
intros
j
.
by
destruct
(
decide
(
i
=
j
))
as
[>?]
;
intros
Hi
;
apply
map_eq
;
intros
j
;
destruct
(
decide
(
i
=
j
))
as
[>?].
rewrite
?lookup_alter
,
?Hi
,
?lookup_alter_ne
.
{
rewrite
lookup_alter
;
destruct
(
m
!!
j
)
;
f_equal'
;
auto
.
}
by
rewrite
lookup_alter_ne
by
done
.
Qed
.
Qed
.
(** ** Properties of the [delete] operation *)
(** ** Properties of the [delete] operation *)
...
@@ 340,7 +342,7 @@ Proof.
...
@@ 340,7 +342,7 @@ Proof.
destruct
(
decide
(
i
=
j
))
as
[>]
;
destruct
(
decide
(
i
=
j
))
as
[>]
;
rewrite
?lookup_insert
,
?lookup_insert_ne
;
intuition
congruence
.
rewrite
?lookup_insert
,
?lookup_insert_ne
;
intuition
congruence
.
Qed
.
Qed
.
Lemma
insert_
lookup
{
A
}
(
m
:
M
A
)
i
x
:
m
!!
i
=
Some
x
→
<[
i
:
=
x
]>
m
=
m
.
Lemma
insert_
id
{
A
}
(
m
:
M
A
)
i
x
:
m
!!
i
=
Some
x
→
<[
i
:
=
x
]>
m
=
m
.
Proof
.
Proof
.
intros
;
apply
map_eq
;
intros
j
;
destruct
(
decide
(
i
=
j
))
as
[>]
;
intros
;
apply
map_eq
;
intros
j
;
destruct
(
decide
(
i
=
j
))
as
[>]
;
by
rewrite
?lookup_insert
,
?lookup_insert_ne
by
done
.
by
rewrite
?lookup_insert
,
?lookup_insert_ne
by
done
.
...
...
theories/list.v
View file @
d67bf19d
...
@@ 172,6 +172,15 @@ Definition zipped_map {A B} (f : list A → list A → A → B) :
...
@@ 172,6 +172,15 @@ Definition zipped_map {A B} (f : list A → list A → A → B) :
list
A
→
list
A
→
list
B
:
=
fix
go
l
k
:
=
list
A
→
list
A
→
list
B
:
=
fix
go
l
k
:
=
match
k
with
[]
=>
[]

x
::
k
=>
f
l
k
x
::
go
(
x
::
l
)
k
end
.
match
k
with
[]
=>
[]

x
::
k
=>
f
l
k
x
::
go
(
x
::
l
)
k
end
.
Definition
imap2_go
{
A
B
C
}
(
f
:
nat
→
A
→
B
→
C
)
:
nat
→
list
A
→
list
B
→
list
C
:
=
fix
go
(
n
:
nat
)
(
l
:
list
A
)
(
k
:
list
B
)
:
=
match
l
,
k
with

[],
_

_
,
[]
=>
[]

x
::
l
,
y
::
k
=>
f
n
x
y
::
go
(
S
n
)
l
k
end
.
Definition
imap2
{
A
B
C
}
(
f
:
nat
→
A
→
B
→
C
)
:
list
A
→
list
B
→
list
C
:
=
imap2_go
f
0
.
Inductive
zipped_Forall
{
A
}
(
P
:
list
A
→
list
A
→
A
→
Prop
)
:
Inductive
zipped_Forall
{
A
}
(
P
:
list
A
→
list
A
→
A
→
Prop
)
:
list
A
→
list
A
→
Prop
:
=
list
A
→
list
A
→
Prop
:
=

zipped_Forall_nil
l
:
zipped_Forall
P
l
[]

zipped_Forall_nil
l
:
zipped_Forall
P
l
[]
...
...
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