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Iris
Iris
Commits
6608490c
Commit
6608490c
authored
Mar 21, 2016
by
Robbert Krebbers
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COFE structure on lists.
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6608490c
From
iris
.
algebra
Require
Export
option
.
From
iris
.
prelude
Require
Export
list
.
Section
cofe
.
Context
{
A
:
cofeT
}.
Instance
list_dist
:
Dist
(
list
A
)
:
=
λ
n
,
Forall2
(
dist
n
).
Global
Instance
cons_ne
n
:
Proper
(
dist
n
==>
dist
n
==>
dist
n
)
(@
cons
A
)
:
=
_
.
Global
Instance
app_ne
n
:
Proper
(
dist
n
==>
dist
n
==>
dist
n
)
(@
app
A
)
:
=
_
.
Global
Instance
length_ne
n
:
Proper
(
dist
n
==>
(=))
(@
length
A
)
:
=
_
.
Global
Instance
tail_ne
n
:
Proper
(
dist
n
==>
dist
n
)
(@
tail
A
)
:
=
_
.
Global
Instance
take_ne
n
:
Proper
(
dist
n
==>
dist
n
)
(@
take
A
n
)
:
=
_
.
Global
Instance
drop_ne
n
:
Proper
(
dist
n
==>
dist
n
)
(@
drop
A
n
)
:
=
_
.
Global
Instance
list_lookup_ne
n
i
:
Proper
(
dist
n
==>
dist
n
)
(
lookup
(
M
:
=
list
A
)
i
).
Proof
.
intros
???.
by
apply
dist_option_Forall2
,
Forall2_lookup
.
Qed
.
Global
Instance
list_alter_ne
n
f
i
:
Proper
(
dist
n
==>
dist
n
)
f
→
Proper
(
dist
n
==>
dist
n
)
(
alter
(
M
:
=
list
A
)
f
i
)
:
=
_
.
Global
Instance
list_insert_ne
n
i
:
Proper
(
dist
n
==>
dist
n
==>
dist
n
)
(
insert
(
M
:
=
list
A
)
i
)
:
=
_
.
Global
Instance
list_inserts_ne
n
i
:
Proper
(
dist
n
==>
dist
n
==>
dist
n
)
(@
list_inserts
A
i
)
:
=
_
.
Global
Instance
list_delete_ne
n
i
:
Proper
(
dist
n
==>
dist
n
)
(
delete
(
M
:
=
list
A
)
i
)
:
=
_
.
Global
Instance
option_list_ne
n
:
Proper
(
dist
n
==>
dist
n
)
(@
option_list
A
).
Proof
.
intros
???
;
by
apply
Forall2_option_list
,
dist_option_Forall2
.
Qed
.
Global
Instance
list_filter_ne
n
P
`
{
∀
x
,
Decision
(
P
x
)}
:
Proper
(
dist
n
==>
iff
)
P
→
Proper
(
dist
n
==>
dist
n
)
(
filter
(
B
:
=
list
A
)
P
)
:
=
_
.
Global
Instance
replicate_ne
n
:
Proper
(
dist
n
==>
dist
n
)
(@
replicate
A
n
)
:
=
_
.
Global
Instance
reverse_ne
n
:
Proper
(
dist
n
==>
dist
n
)
(@
reverse
A
)
:
=
_
.
Global
Instance
last_ne
n
:
Proper
(
dist
n
==>
dist
n
)
(@
last
A
).
Proof
.
intros
???
;
by
apply
dist_option_Forall2
,
Forall2_last
.
Qed
.
Global
Instance
resize_ne
n
:
Proper
(
dist
n
==>
dist
n
==>
dist
n
)
(@
resize
A
n
)
:
=
_
.
Program
Definition
list_chain
(
c
:
chain
(
list
A
))
(
x
:
A
)
(
k
:
nat
)
:
chain
A
:
=
{|
chain_car
n
:
=
from_option
x
(
c
n
!!
k
)
|}.
Next
Obligation
.
intros
c
x
k
n
i
?.
by
rewrite
/=
(
chain_cauchy
c
n
i
).
Qed
.
Instance
list_compl
:
Compl
(
list
A
)
:
=
λ
c
,
match
c
0
with
|
[]
=>
[]
|
x
::
_
=>
compl
∘
list_chain
c
x
<$>
seq
0
(
length
(
c
0
))
end
.
Definition
list_cofe_mixin
:
CofeMixin
(
list
A
).
Proof
.
split
.
-
intros
l
k
.
rewrite
equiv_Forall2
-
Forall2_forall
.
split
;
induction
1
;
constructor
;
intros
;
try
apply
equiv_dist
;
auto
.
-
apply
_
.
-
rewrite
/
dist
/
list_dist
.
eauto
using
Forall2_impl
,
dist_S
.
-
intros
n
c
;
rewrite
/
compl
/
list_compl
.
destruct
(
c
0
)
as
[|
x
l
]
eqn
:
Hc0
at
1
.
{
by
destruct
(
chain_cauchy
c
0
n
)
;
auto
with
omega
.
}
rewrite
-(
λ
H
,
length_ne
_
_
_
(
chain_cauchy
c
0
n
H
))
;
last
omega
.
apply
Forall2_lookup
=>
i
;
apply
dist_option_Forall2
.
rewrite
list_lookup_fmap
.
destruct
(
decide
(
i
<
length
(
c
n
)))
;
last
first
.
{
rewrite
lookup_seq_ge
?lookup_ge_None_2
;
auto
with
omega
.
}
rewrite
lookup_seq
//=
(
conv_compl
n
(
list_chain
c
_
_
))
/=.
by
destruct
(
lookup_lt_is_Some_2
(
c
n
)
i
)
as
[?
->].
Qed
.
Canonical
Structure
listC
:
=
CofeT
list_cofe_mixin
.
Global
Instance
list_discrete
:
Discrete
A
→
Discrete
listC
.
Proof
.
induction
2
;
constructor
;
try
apply
(
timeless
_
)
;
auto
.
Qed
.
Global
Instance
nil_timeless
:
Timeless
(@
nil
A
).
Proof
.
inversion_clear
1
;
constructor
.
Qed
.
Global
Instance
cons_timeless
x
l
:
Timeless
x
→
Timeless
l
→
Timeless
(
x
::
l
).
Proof
.
intros
??
;
inversion_clear
1
;
constructor
;
by
apply
timeless
.
Qed
.
End
cofe
.
Arguments
listC
:
clear
implicits
.
(** Functor *)
Instance
list_fmap_ne
{
A
B
:
cofeT
}
(
f
:
A
→
B
)
n
:
Proper
(
dist
n
==>
dist
n
)
f
→
Proper
(
dist
n
==>
dist
n
)
(
fmap
(
M
:
=
list
)
f
).
Proof
.
intros
Hf
l
k
?
;
by
eapply
Forall2_fmap
,
Forall2_impl
;
eauto
.
Qed
.
Definition
listC_map
{
A
B
}
(
f
:
A
-
n
>
B
)
:
listC
A
-
n
>
listC
B
:
=
CofeMor
(
fmap
f
:
listC
A
→
listC
B
).
Instance
listC_map_ne
A
B
n
:
Proper
(
dist
n
==>
dist
n
)
(@
listC_map
A
B
).
Proof
.
intros
f
f'
?
l
;
by
apply
Forall2_fmap
,
Forall_Forall2
,
Forall_true
.
Qed
.
Program
Definition
listCF
(
F
:
cFunctor
)
:
cFunctor
:
=
{|
cFunctor_car
A
B
:
=
listC
(
cFunctor_car
F
A
B
)
;
cFunctor_map
A1
A2
B1
B2
fg
:
=
listC_map
(
cFunctor_map
F
fg
)
|}.
Next
Obligation
.
by
intros
F
A1
A2
B1
B2
n
f
g
Hfg
;
apply
listC_map_ne
,
cFunctor_ne
.
Qed
.
Next
Obligation
.
intros
F
A
B
x
.
rewrite
/=
-{
2
}(
list_fmap_id
x
).
apply
list_fmap_setoid_ext
=>
y
.
apply
cFunctor_id
.
Qed
.
Next
Obligation
.
intros
F
A1
A2
A3
B1
B2
B3
f
g
f'
g'
x
.
rewrite
/=
-
list_fmap_compose
.
apply
list_fmap_setoid_ext
=>
y
;
apply
cFunctor_compose
.
Qed
.
Instance
listCF_contractive
F
:
cFunctorContractive
F
→
cFunctorContractive
(
listCF
F
).
Proof
.
by
intros
?
A1
A2
B1
B2
n
f
g
Hfg
;
apply
listC_map_ne
,
cFunctor_contractive
.
Qed
.
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