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Rice Wine
Iris
Commits
edff0fe3
Commit
edff0fe3
authored
Nov 17, 2015
by
Robbert Krebbers
Browse files
Use Inhabited type class for fixpoint and solver.
parent
d9bae949
Changes
3
Hide whitespace changes
Inline
Side-by-side
iris/cofe.v
View file @
edff0fe3
...
...
@@ -93,35 +93,36 @@ Section cofe.
End
cofe
.
(** Fixpoint *)
Program
Definition
fixpoint_chain
`
{
Cofe
A
}
(
f
:
A
→
A
)
`
{!
Contractive
f
}
(
x
:
A
)
:
chain
A
:
=
{|
chain_car
i
:
=
Nat
.
iter
i
f
x
|}.
Program
Definition
fixpoint_chain
`
{
Cofe
A
,
Inhabited
A
}
(
f
:
A
→
A
)
`
{!
Contractive
f
}
:
chain
A
:
=
{|
chain_car
i
:
=
Nat
.
iter
i
f
inhabitant
|}.
Next
Obligation
.
intros
A
????
f
?
x
n
;
induction
n
as
[|
n
IH
]
;
intros
i
?
;
[
done
|].
destruct
i
as
[|
i
]
;
simpl
;
try
lia
;
apply
contractive
,
IH
;
auto
with
lia
.
Qed
.
Program
Definition
fixpoint
`
{
Cofe
A
}
(
f
:
A
→
A
)
`
{!
Contractive
f
}
(
x
:
A
)
:
A
:
=
compl
(
fixpoint_chain
f
x
).
Program
Definition
fixpoint
`
{
Cofe
A
,
Inhabited
A
}
(
f
:
A
→
A
)
`
{!
Contractive
f
}
:
A
:
=
compl
(
fixpoint_chain
f
).
Section
fixpoint
.
Context
`
{
Cofe
A
}
(
f
:
A
→
A
)
`
{!
Contractive
f
}.
Lemma
fixpoint_unfold
x
:
fixpoint
f
x
≡
f
(
fixpoint
f
x
).
Context
`
{
Cofe
A
,
Inhabited
A
}
(
f
:
A
→
A
)
`
{!
Contractive
f
}.
Lemma
fixpoint_unfold
:
fixpoint
f
≡
f
(
fixpoint
f
).
Proof
.
apply
equiv_dist
;
intros
n
;
unfold
fixpoint
.
rewrite
(
conv_compl
(
fixpoint_chain
f
x
)
n
).
by
rewrite
(
chain_cauchy
(
fixpoint_chain
f
x
)
n
(
S
n
))
at
1
by
lia
.
rewrite
(
conv_compl
(
fixpoint_chain
f
)
n
).
by
rewrite
(
chain_cauchy
(
fixpoint_chain
f
)
n
(
S
n
))
at
1
by
lia
.
Qed
.
Lemma
fixpoint_ne
(
g
:
A
→
A
)
`
{!
Contractive
g
}
x
y
n
:
(
∀
z
,
f
z
={
n
}=
g
z
)
→
fixpoint
f
x
={
n
}=
fixpoint
g
y
.
Lemma
fixpoint_ne
(
g
:
A
→
A
)
`
{!
Contractive
g
}
n
:
(
∀
z
,
f
z
={
n
}=
g
z
)
→
fixpoint
f
={
n
}=
fixpoint
g
.
Proof
.
intros
Hfg
;
unfold
fixpoint
;
rewrite
(
conv_compl
(
fixpoint_chain
f
x
)
n
),
(
conv_compl
(
fixpoint_chain
g
y
)
n
).
intros
Hfg
;
unfold
fixpoint
.
rewrite
(
conv_compl
(
fixpoint_chain
f
)
n
),
(
conv_compl
(
fixpoint_chain
g
)
n
).
induction
n
as
[|
n
IH
]
;
simpl
in
*
;
[
done
|].
rewrite
Hfg
;
apply
contractive
,
IH
;
auto
using
dist_S
.
Qed
.
Lemma
fixpoint_proper
(
g
:
A
→
A
)
`
{!
Contractive
g
}
x
:
(
∀
x
,
f
x
≡
g
x
)
→
fixpoint
f
x
≡
fixpoint
g
x
.
Lemma
fixpoint_proper
(
g
:
A
→
A
)
`
{!
Contractive
g
}
:
(
∀
x
,
f
x
≡
g
x
)
→
fixpoint
f
≡
fixpoint
g
.
Proof
.
setoid_rewrite
equiv_dist
;
naive_solver
eauto
using
fixpoint_ne
.
Qed
.
End
fixpoint
.
Global
Opaque
fixpoint
.
(** Function space *)
Structure
cofeMor
(
A
B
:
cofeT
)
:
Type
:
=
CofeMor
{
...
...
@@ -170,6 +171,8 @@ Lemma cofe_mor_ext {A B} (f g : cofeMor A B) : f ≡ g ↔ ∀ x, f x ≡ g x.
Proof
.
done
.
Qed
.
Canonical
Structure
cofe_mor
(
A
B
:
cofeT
)
:
cofeT
:
=
CofeT
(
cofeMor
A
B
).
Infix
"-n>"
:
=
cofe_mor
(
at
level
45
,
right
associativity
).
Instance
cofe_more_inhabited
(
A
B
:
cofeT
)
`
{
Inhabited
B
}
:
Inhabited
(
A
-
n
>
B
)
:
=
populate
(
CofeMor
(
λ
_
,
inhabitant
)).
(** Identity and composition *)
Definition
cid
{
A
}
:
A
-
n
>
A
:
=
CofeMor
id
.
...
...
iris/cofe_solver.v
View file @
edff0fe3
...
...
@@ -2,7 +2,7 @@ Require Export iris.cofe.
Section
solver
.
Context
(
F
:
cofeT
→
cofeT
→
cofeT
).
Context
(
p
:
F
(
CofeT
unit
)
(
CofeT
unit
)).
Context
`
{
Finhab
:
Inhabited
(
F
(
CofeT
unit
)
(
CofeT
unit
))
}
.
Context
(
map
:
∀
{
A1
A2
B1
B2
:
cofeT
},
((
A2
-
n
>
A1
)
*
(
B1
-
n
>
B2
))
→
(
F
A1
B1
-
n
>
F
A2
B2
)).
Arguments
map
{
_
_
_
_
}
_
.
...
...
@@ -22,7 +22,7 @@ Proof. by rewrite <-!cofe_mor_ext; intros Hf Hg Hx; rewrite Hf, Hg, Hx. Qed.
Fixpoint
A
(
k
:
nat
)
:
cofeT
:
=
match
k
with
0
=>
CofeT
unit
|
S
k
=>
F
(
A
k
)
(
A
k
)
end
.
Fixpoint
f
{
k
}
:
A
k
-
n
>
A
(
S
k
)
:
=
match
k
with
0
=>
CofeMor
(
λ
_
,
p
)
|
S
k
=>
map
(
g
,
f
)
end
match
k
with
0
=>
CofeMor
(
λ
_
,
inhabitant
)
|
S
k
=>
map
(
g
,
f
)
end
with
g
{
k
}
:
A
(
S
k
)
-
n
>
A
k
:
=
match
k
with
0
=>
CofeMor
(
λ
_
,
()
:
CofeT
())
|
S
k
=>
map
(
f
,
g
)
end
.
Definition
f_S
k
(
x
:
A
(
S
k
))
:
f
x
=
map
(
g
,
f
)
x
:
=
eq_refl
.
...
...
iris/logic.v
View file @
edff0fe3
...
...
@@ -74,6 +74,7 @@ Instance uPred_entails {M} : SubsetEq (uPred M) := λ P Q, ∀ x n,
Program
Definition
uPred_const
{
M
}
(
P
:
Prop
)
:
uPred
M
:
=
{|
uPred_holds
n
x
:
=
P
|}.
Solve
Obligations
with
done
.
Instance
uPred_inhabited
M
:
Inhabited
(
uPred
M
)
:
=
populate
(
uPred_const
True
).
Program
Definition
uPred_and
{
M
}
(
P
Q
:
uPred
M
)
:
uPred
M
:
=
{|
uPred_holds
n
x
:
=
P
n
x
∧
Q
n
x
|}.
...
...
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