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79d2785b
Commit
79d2785b
authored
Feb 05, 2016
by
Ralf Jung
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experimenting with a global CMRA that is an indexed product
parent
48674eed
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program_logic/auth.v
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79d2785b
Require
Export
algebra
.
auth
.
Require
Import
program_logic
.
functor
program_logic
.
language
program_logic
.
weakestpre
.
(* RJ: This is a workinprogress playground.
FIXME: Finish or remove. *)
Section
auth
.
(* TODO what should be implicit, what explicit? *)
Context
{
Λ
:
language
}.
Context
{
C
:
nat
→
cmraT
}.
Context
(
i
:
nat
).
Context
{
A
:
cmraT
}.
Hypothesis
Ci
:
C
i
=
authRA
A
.
Let
Σ
:
iFunctor
:=
iprodF
(
mapF
positive
∘
constF
∘
C
).
Definition
tr
(
a
:
authRA
A
)
:
C
i
.
rewrite
Ci
.
exact
a
.
Defined
.
Definition
tr'
(
c
:
C
i
)
:
authRA
A
.
rewrite

Ci
.
exact
c
.
Defined
.
Lemma
tr'_tr
a
:
tr'
$
tr
a
=
a
.
Proof
.
rewrite
/
tr'
/
tr
.
by
destruct
Ci
.
Qed
.
Lemma
tr_tr'
c
:
tr
$
tr'
c
=
c
.
Proof
.
rewrite
/
tr'
/
tr
.
by
destruct
Ci
.
Qed
.
Lemma
tr_proper
:
Proper
((
≡
)
==>
(
≡
))
tr
.
Proof
.
move
=>
a1
a2
Heq
.
rewrite
/
tr
.
by
destruct
Ci
.
Qed
.
Lemma
Ci_op
(
c1
c2
:
C
i
)
:
c1
⋅
c2
=
tr
(
tr'
c1
⋅
tr'
c2
).
Proof
.
rewrite
/
tr'
/
tr
.
by
destruct
Ci
.
Qed
.
Definition
Tr
j
{
H
:
i
=
j
}
(
c
:
C
i
)
:
C
j
.
rewrite

H
.
exact
c
.
Defined
.
(* FIXME RJ: I'd rather not have to specify Σ by hand here. *)
Definition
ownNothing
:
iProp
Λ
Σ
:=
ownG
(
Σ
:=
Σ
)
(
fun
j
=>
(
∅
:
mapRA
positive
(
C
j
))).
Definition
ownA
(
p
:
positive
)
(
a
:
authRA
A
)
:
iProp
Λ
Σ
:=
ownG
(
Σ
:=
Σ
)
(
fun
j
=>
match
decide
(
i
=
j
)
with

left
Heq
=>
<[
p
:=
Tr
j
(
H
:=
Heq
)
$
tr
a
]>
∅

right
Hneq
=>
∅
end
).
Lemma
ownA_op
p
a1
a2
:
(
ownA
p
a1
★
ownA
p
a2
)%
I
≡
ownA
p
(
a1
⋅
a2
).
Proof
.
rewrite
/
ownA

ownG_op
.
apply
ownG_proper
=>
j
/=.
rewrite
iprod_lookup_op
.
destruct
(
decide
(
i
=
j
)).

move
=>
q
.
destruct
e
.
rewrite
lookup_op
/=.
destruct
(
decide
(
p
=
q
));
first
subst
q
.
+
rewrite
!
lookup_insert
.
rewrite
/
op
/
cmra_op
/=.
f_equiv
.
rewrite
Ci_op
.
apply
tr_proper
.
rewrite
!
tr'_tr
.
reflexivity
.
+
by
rewrite
!
lookup_insert_ne
//.

by
rewrite
left_id
.
Qed
.
End
auth
.
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