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Amin Timany
iris-coq
Commits
1c48ea12
Commit
1c48ea12
authored
8 years ago
by
Robbert Krebbers
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Cofe on the function space A → B where A : Type and B : cofeT.
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algebra/cofe.v
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@@ -207,6 +207,39 @@ Section fixpoint.
Proof
.
setoid_rewrite
equiv_dist
;
naive_solver
eauto
using
fixpoint_ne
.
Qed
.
End
fixpoint
.
(** Function space *)
Definition
cofe_fun
(
A
:
Type
)
(
B
:
cofeT
)
:=
A
→
B
.
Section
cofe_fun
.
Context
{
A
:
Type
}
{
B
:
cofeT
}
.
Instance
cofe_fun_equiv
:
Equiv
(
cofe_fun
A
B
)
:=
λ
f
g
,
∀
x
,
f
x
≡
g
x
.
Instance
cofe_fun_dist
:
Dist
(
cofe_fun
A
B
)
:=
λ
n
f
g
,
∀
x
,
f
x
≡
{
n
}
≡
g
x
.
Program
Definition
cofe_fun_chain
`
(
c
:
chain
(
cofe_fun
A
B
))
(
x
:
A
)
:
chain
B
:=
{|
chain_car
n
:=
c
n
x
|}
.
Next
Obligation
.
intros
c
x
n
i
?
.
by
apply
(
chain_cauchy
c
)
.
Qed
.
Program
Instance
cofe_fun_compl
:
Compl
(
cofe_fun
A
B
)
:=
λ
c
x
,
compl
(
cofe_fun_chain
c
x
)
.
Definition
cofe_fun_cofe_mixin
:
CofeMixin
(
cofe_fun
A
B
)
.
Proof
.
split
.
-
intros
f
g
;
split
;
[
intros
Hfg
n
k
;
apply
equiv_dist
,
Hfg
|]
.
intros
Hfg
k
;
apply
equiv_dist
=>
n
;
apply
Hfg
.
-
intros
n
;
split
.
+
by
intros
f
x
.
+
by
intros
f
g
?
x
.
+
by
intros
f
g
h
??
x
;
trans
(
g
x
)
.
-
by
intros
n
f
g
?
x
;
apply
dist_S
.
-
intros
n
c
x
.
apply
(
conv_compl
n
(
cofe_fun_chain
c
x
))
.
Qed
.
Canonical
Structure
cofe_funC
:=
CofeT
(
cofe_fun
A
B
)
cofe_fun_cofe_mixin
.
End
cofe_fun
.
Arguments
cofe_funC
:
clear
implicits
.
Notation
"A -c> B"
:=
(
cofe_funC
A
B
)
(
at
level
99
,
B
at
level
200
,
right
associativity
)
.
Instance
cofe_fun_inhabited
{
A
}
{
B
:
cofeT
}
`{
Inhabited
B
}
:
Inhabited
(
A
-
c
>
B
)
:=
populate
(
λ
_,
inhabitant
)
.
(** Non-expansive function space *)
Record
cofe_mor
(
A
B
:
cofeT
)
:
Type
:=
CofeMor
{
cofe_mor_car
:>
A
→
B
;
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