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Iris
Iris
Commits
1109ca07
Commit
1109ca07
authored
Feb 16, 2016
by
Robbert Krebbers
Browse files
Move DRA validity instances into section to avoid ambiguity.
parent
fabb3ff9
Changes
1
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Inline
Side-by-side
algebra/dra.v
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1109ca07
...
...
@@ -12,22 +12,6 @@ Arguments validity_is_valid {_ _} _.
Definition
to_validity
{
A
}
{
P
:
A
→
Prop
}
(
x
:
A
)
:
validity
P
:
=
Validity
x
(
P
x
)
id
.
Instance
validity_valid
{
A
}
(
P
:
A
→
Prop
)
:
Valid
(
validity
P
)
:
=
validity_is_valid
.
Instance
validity_equiv
`
{
Equiv
A
}
(
P
:
A
→
Prop
)
:
Equiv
(
validity
P
)
:
=
λ
x
y
,
(
valid
x
↔
valid
y
)
∧
(
valid
x
→
validity_car
x
≡
validity_car
y
).
Instance
validity_equivalence
`
{
Equiv
A
,!
Equivalence
((
≡
)
:
relation
A
)}
P
:
Equivalence
((
≡
)
:
relation
(
validity
P
)).
Proof
.
split
;
unfold
equiv
,
validity_equiv
.
*
by
intros
[
x
px
?]
;
simpl
.
*
intros
[
x
px
?]
[
y
py
?]
;
naive_solver
.
*
intros
[
x
px
?]
[
y
py
?]
[
z
pz
?]
[?
Hxy
]
[?
Hyz
]
;
simpl
in
*.
split
;
[|
intros
;
transitivity
y
]
;
tauto
.
Qed
.
Instance
validity_valid_proper
`
{
Equiv
A
}
(
P
:
A
→
Prop
)
:
Proper
((
≡
)
==>
iff
)
(
valid
:
validity
P
→
Prop
).
Proof
.
intros
??
[??]
;
naive_solver
.
Qed
.
Definition
dra_included
`
{
Equiv
A
,
Valid
A
,
Disjoint
A
,
Op
A
}
:
=
λ
x
y
,
∃
z
,
y
≡
x
⋅
z
∧
✓
z
∧
x
⊥
z
.
...
...
@@ -64,6 +48,20 @@ Context A `{DRA A}.
Arguments
valid
_
_
!
_
/.
Hint
Immediate
dra_op_proper
:
typeclass_instances
.
Notation
T
:
=
(
validity
(
valid
:
A
→
Prop
)).
Instance
validity_valid
:
Valid
T
:
=
validity_is_valid
.
Instance
validity_equiv
:
Equiv
T
:
=
λ
x
y
,
(
valid
x
↔
valid
y
)
∧
(
valid
x
→
validity_car
x
≡
validity_car
y
).
Instance
validity_equivalence
:
Equivalence
((
≡
)
:
relation
T
).
Proof
.
split
;
unfold
equiv
,
validity_equiv
.
*
by
intros
[
x
px
?]
;
simpl
.
*
intros
[
x
px
?]
[
y
py
?]
;
naive_solver
.
*
intros
[
x
px
?]
[
y
py
?]
[
z
pz
?]
[?
Hxy
]
[?
Hyz
]
;
simpl
in
*.
split
;
[|
intros
;
transitivity
y
]
;
tauto
.
Qed
.
Instance
:
Proper
((
≡
)
==>
(
≡
)
==>
iff
)
(
⊥
).
Proof
.
intros
x1
x2
Hx
y1
y2
Hy
;
split
.
...
...
@@ -83,7 +81,6 @@ Qed.
Hint
Immediate
dra_disjoint_move_l
dra_disjoint_move_r
.
Hint
Unfold
dra_included
.
Notation
T
:
=
(
validity
(
valid
:
A
→
Prop
)).
Lemma
validity_valid_car_valid
(
z
:
T
)
:
✓
z
→
✓
validity_car
z
.
Proof
.
apply
validity_prf
.
Qed
.
Hint
Resolve
validity_valid_car_valid
.
...
...
@@ -106,7 +103,7 @@ Proof.
split
;
intros
(?&?&?)
;
split_ands'
;
first
[
rewrite
?Heq
;
tauto
|
rewrite
-
?Heq
;
tauto
|
tauto
].
*
by
intros
??
[?
Heq
]
;
split
;
[
done
|]
;
simpl
;
intros
?
;
rewrite
Heq
.
*
by
intros
??
->
?
.
*
intros
??
[??]
;
naive_solver
.
*
intros
x1
x2
[?
Hx
]
y1
y2
[?
Hy
]
;
split
;
simpl
;
[|
by
intros
(?&?&?)
;
rewrite
Hx
//
Hy
].
split
;
intros
(?&?&
z
&?&?)
;
split_ands'
;
try
tauto
.
...
...
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