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George Pirlea
Iris
Commits
81ed7343
Commit
81ed7343
authored
Jan 05, 2017
by
Ralf Jung
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COFE for sigma types
parent
6bbc6b49
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theories/algebra/ofe.v
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theories/algebra/ofe.v
View file @
81ed7343
...
...
@@ -955,6 +955,59 @@ Proof.
destruct
n
as
[|
n
]
;
simpl
in
*
;
first
done
.
apply
cFunctor_ne
,
Hfg
.
Qed
.
(** Sigma *)
Section
sigma
.
Context
{
A
:
ofeT
}
{
f
:
A
→
Prop
}.
(* TODO: Find a better place for this Equiv instance. It also
should not depend on A being an OFE. *)
Instance
sig_equiv
:
Equiv
(
sig
f
)
:
=
λ
x1
x2
,
(
proj1_sig
x1
)
≡
(
proj1_sig
x2
).
Instance
sig_dist
:
Dist
(
sig
f
)
:
=
λ
n
x1
x2
,
(
proj1_sig
x1
)
≡
{
n
}
≡
(
proj1_sig
x2
).
Global
Lemma
exist_ne
:
∀
n
x1
x2
,
x1
≡
{
n
}
≡
x2
→
∀
(
H1
:
f
x1
)
(
H2
:
f
x2
),
(
exist
f
x1
H1
)
≡
{
n
}
≡
(
exist
f
x2
H2
).
Proof
.
intros
n
??
Hx
??.
exact
Hx
.
Qed
.
Global
Instance
proj1_sig_ne
:
Proper
(
dist
n
==>
dist
n
)
(@
proj1_sig
_
f
).
Proof
.
intros
n
[]
[]
?.
done
.
Qed
.
Definition
sig_ofe_mixin
:
OfeMixin
(
sig
f
).
Proof
.
split
.
-
intros
x
y
.
unfold
dist
,
sig_dist
,
equiv
,
sig_equiv
.
destruct
x
,
y
.
apply
equiv_dist
.
-
unfold
dist
,
sig_dist
.
intros
n
.
split
;
[
intros
[]
|
intros
[]
[]
|
intros
[]
[]
[]]
;
simpl
;
try
done
.
intros
.
by
etrans
.
-
intros
n
[]
[].
unfold
dist
,
sig_dist
.
apply
dist_S
.
Qed
.
Canonical
Structure
sigC
:
ofeT
:
=
OfeT
(
sig
f
)
sig_ofe_mixin
.
Global
Class
LimitPreserving
`
{
Cofe
A
}
:
Prop
:
=
limit_preserving
:
∀
c
:
chain
A
,
(
∀
n
,
f
(
c
n
))
→
f
(
compl
c
).
Program
Definition
sig_compl
`
{
LimitPreserving
}
:
Compl
sigC
:
=
λ
c
,
exist
f
(
compl
(
chain_map
proj1_sig
c
))
_
.
Next
Obligation
.
intros
?
Hlim
c
.
apply
Hlim
.
move
=>
n
/=.
destruct
(
c
n
).
done
.
Qed
.
Program
Definition
sig_cofe
`
{
LimitPreserving
}
:
Cofe
sigC
:
=
{|
compl
:
=
sig_compl
|}.
Next
Obligation
.
intros
?
Hlim
n
c
.
apply
(
conv_compl
n
(
chain_map
proj1_sig
c
)).
Qed
.
Global
Instance
sig_timeless
(
x
:
sig
f
)
:
Timeless
(
proj1_sig
x
)
→
Timeless
x
.
Proof
.
intros
?
y
.
destruct
x
,
y
.
unfold
dist
,
sig_dist
,
equiv
,
sig_equiv
.
apply
(
timeless
_
).
Qed
.
Global
Instance
sig_discrete_cofe
:
Discrete
A
→
Discrete
sigC
.
Proof
.
intros
?
[??]
[??].
rewrite
/
dist
/
equiv
/
ofe_dist
/
ofe_equiv
/=.
rewrite
/
sig_dist
/
sig_equiv
/=.
apply
discrete_timeless
.
Qed
.
End
sigma
.
Arguments
sigC
{
A
}
f
.
(** Notation for writing functors *)
Notation
"∙"
:
=
idCF
:
cFunctor_scope
.
Notation
"T -c> F"
:
=
(
ofe_funCF
T
%
type
F
%
CF
)
:
cFunctor_scope
.
...
...
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