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5cc172cf
Commit
5cc172cf
authored
Feb 05, 2016
by
Robbert Krebbers
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Combinators for iFunctors.
parent
30c12c95
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program_logic/functor.v
program_logic/functor.v
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program_logic/functor.v
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5cc172cf
Require
Export
algebra
.
cmra
.
Require
Import
algebra
.
agree
algebra
.
excl
algebra
.
auth
.
Require
Import
algebra
.
option
algebra
.
fin_maps
.
(** * Functors from COFE to CMRA *)
(* The Iris program logic is parametrized by a functor from the category of
COFEs to the category of CMRAs, which is instantiated with [laterC iProp]. The
[laterC iProp] can be used to construct impredicate CMRAs, such as the stored
propositions using the agreement CMRA. *)
Structure
iFunctor
:
=
IFunctor
{
ifunctor_car
:
>
cofeT
→
cmraT
;
ifunctor_map
{
A
B
}
(
f
:
A

n
>
B
)
:
ifunctor_car
A

n
>
ifunctor_car
B
;
...
...
@@ 13,11 +20,97 @@ Existing Instances ifunctor_map_ne ifunctor_map_mono.
Lemma
ifunctor_map_ext
(
Σ
:
iFunctor
)
{
A
B
}
(
f
g
:
A

n
>
B
)
m
:
(
∀
x
,
f
x
≡
g
x
)
→
ifunctor_map
Σ
f
m
≡
ifunctor_map
Σ
g
m
.
Proof
.
by
intros
;
apply
equiv_dist
=>
n
;
apply
ifunctor_map_ne
=>
?
;
apply
equiv_dist
.
Proof
.
by
intros
;
apply
(
ne_proper
(@
ifunctor_map
Σ
A
B
)).
Qed
.
(** * Functor combinators *)
(** We create a functor combinators for all CMRAs in the algebra directory.
These combinators can be used to conveniently construct the global CMRA of
the Iris program logic. Note that we have explicitly built in functor
composition into these combinators, instead of having a notion of a functor
from the category of CMRAs to the category of CMRAs which we can compose. This
way we can convenient deal with (indexed) products in a uniform way. *)
Program
Definition
constF
(
B
:
cmraT
)
:
iFunctor
:
=
{
ifunctor_car
A
:
=
B
;
ifunctor_map
A1
A2
f
:
=
cid
}.
Solve
Obligations
with
done
.
Program
Definition
prodF
(
Σ
1
Σ
2
:
iFunctor
)
:
iFunctor
:
=
{
ifunctor_car
A
:
=
prodRA
(
Σ
1
A
)
(
Σ
2
A
)
;
ifunctor_map
A
B
f
:
=
prodC_map
(
ifunctor_map
Σ
1
f
)
(
ifunctor_map
Σ
2
f
)
}.
Next
Obligation
.
by
intros
Σ
1
Σ
2
A
B
n
f
g
Hfg
;
apply
prodC_map_ne
;
apply
ifunctor_map_ne
.
Qed
.
Next
Obligation
.
by
intros
Σ
1
Σ
2
A
[??]
;
rewrite
/=
!
ifunctor_map_id
.
Qed
.
Next
Obligation
.
by
intros
Σ
1
Σ
2
A
B
C
f
g
[??]
;
rewrite
/=
!
ifunctor_map_compose
.
Qed
.
Program
Definition
iprodF
{
A
}
(
Σ
:
A
→
iFunctor
)
:
iFunctor
:
=
{
ifunctor_car
B
:
=
iprodRA
(
λ
x
,
Σ
x
B
)
;
ifunctor_map
B1
B2
f
:
=
iprodC_map
(
λ
x
,
ifunctor_map
(
Σ
x
)
f
)
;
}.
Next
Obligation
.
by
intros
A
Σ
B1
B2
n
f
f'
?
g
;
apply
iprodC_map_ne
=>
x
;
apply
ifunctor_map_ne
.
Qed
.
Next
Obligation
.
intros
A
Σ
B
g
.
rewrite
/=
{
2
}(
iprod_map_id
g
).
apply
iprod_map_ext
=>
x
;
apply
ifunctor_map_id
.
Qed
.
Next
Obligation
.
intros
A
Σ
B1
B2
B3
f1
f2
g
.
rewrite
/=

iprod_map_compose
.
apply
iprod_map_ext
=>
y
;
apply
ifunctor_map_compose
.
Qed
.
Program
Definition
agreeF
:
iFunctor
:
=
{
ifunctor_car
:
=
agreeRA
;
ifunctor_map
:
=
@
agreeC_map
}.
Solve
Obligations
with
done
.
Program
Definition
exclF
:
iFunctor
:
=
{
ifunctor_car
:
=
exclRA
;
ifunctor_map
:
=
@
exclC_map
}.
Next
Obligation
.
by
intros
A
x
;
rewrite
/=
excl_map_id
.
Qed
.
Next
Obligation
.
by
intros
A
B
C
f
g
x
;
rewrite
/=
excl_map_compose
.
Qed
.
Program
Definition
authF
(
Σ
:
iFunctor
)
:
iFunctor
:
=
{
ifunctor_car
:
=
authRA
∘
Σ
;
ifunctor_map
A
B
:
=
authC_map
∘
ifunctor_map
Σ
}.
Next
Obligation
.
by
intros
Σ
A
B
n
f
g
Hfg
;
apply
authC_map_ne
,
ifunctor_map_ne
.
Qed
.
Next
Obligation
.
intros
Σ
A
x
.
rewrite
/=
{
2
}(
auth_map_id
x
).
apply
auth_map_ext
=>
y
;
apply
ifunctor_map_id
.
Qed
.
Next
Obligation
.
intros
Σ
A
B
C
f
g
x
.
rewrite
/=

auth_map_compose
.
apply
auth_map_ext
=>
y
;
apply
ifunctor_map_compose
.
Qed
.
Program
Definition
optionF
(
Σ
:
iFunctor
)
:
iFunctor
:
=
{
ifunctor_car
:
=
optionRA
∘
Σ
;
ifunctor_map
A
B
:
=
optionC_map
∘
ifunctor_map
Σ
}.
Next
Obligation
.
by
intros
Σ
A
B
n
f
g
Hfg
;
apply
optionC_map_ne
,
ifunctor_map_ne
.
Qed
.
Next
Obligation
.
intros
Σ
A
x
.
rewrite
/=
{
2
}(
option_fmap_id
x
).
apply
option_fmap_setoid_ext
=>
y
;
apply
ifunctor_map_id
.
Qed
.
Next
Obligation
.
intros
Σ
A
B
C
f
g
x
.
rewrite
/=

option_fmap_compose
.
apply
option_fmap_setoid_ext
=>
y
;
apply
ifunctor_map_compose
.
Qed
.
Program
Definition
iFunctor_const
(
icmra
:
cmraT
)
{
icmra_empty
:
Empty
icmra
}
{
icmra_identity
:
CMRAIdentity
icmra
}
:
iFunctor
:
=
{
ifunctor_car
A
:
=
icmra
;
ifunctor_map
A
B
f
:
=
cid
}.
Solve
Obligations
with
done
.
\ No newline at end of file
Program
Definition
mapF
K
`
{
Countable
K
}
(
Σ
:
iFunctor
)
:
iFunctor
:
=
{
ifunctor_car
:
=
mapRA
K
∘
Σ
;
ifunctor_map
A
B
:
=
mapC_map
∘
ifunctor_map
Σ
}.
Next
Obligation
.
by
intros
K
??
Σ
A
B
n
f
g
Hfg
;
apply
mapC_map_ne
,
ifunctor_map_ne
.
Qed
.
Next
Obligation
.
intros
K
??
Σ
A
x
.
rewrite
/=
{
2
}(
map_fmap_id
x
).
apply
map_fmap_setoid_ext
=>
?
y
_;
apply
ifunctor_map_id
.
Qed
.
Next
Obligation
.
intros
K
??
Σ
A
B
C
f
g
x
.
rewrite
/=

map_fmap_compose
.
apply
map_fmap_setoid_ext
=>
?
y
_;
apply
ifunctor_map_compose
.
Qed
.
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