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George Pirlea
Iris
Commits
09ae3206
Commit
09ae3206
authored
Apr 04, 2018
by
Robbert Krebbers
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Separate type class for `■ ⎡P⎤ ⊢ ⎡■ P⎤`.
parent
9a745670
Changes
3
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3 changed files
with
25 additions
and
8 deletions
+25
8
theories/bi/embedding.v
theories/bi/embedding.v
+20
7
theories/bi/monpred.v
theories/bi/monpred.v
+4
0
theories/proofmode/class_instances_sbi.v
theories/proofmode/class_instances_sbi.v
+1
1
No files found.
theories/bi/embedding.v
View file @
09ae3206
...
...
@@ 62,6 +62,13 @@ Class BiEmbedFUpd (PROP1 PROP2 : sbi)
Hint
Mode
BiEmbedFUpd

!



:
typeclass_instances
.
Hint
Mode
BiEmbedFUpd
!




:
typeclass_instances
.
Class
BiEmbedPlainly
(
PROP1
PROP2
:
sbi
)
`
{
BiEmbed
PROP1
PROP2
,
BiPlainly
PROP1
,
BiPlainly
PROP2
}
:
=
{
embed_plainly_2
(
P
:
PROP1
)
:
■
⎡
P
⎤
⊢
(
⎡■
P
⎤
:
PROP2
)
}.
Hint
Mode
BiEmbedPlainly

!



:
typeclass_instances
.
Hint
Mode
BiEmbedPlainly
!




:
typeclass_instances
.
Section
embed_laws
.
Context
`
{
BiEmbed
PROP1
PROP2
}.
Local
Notation
embed
:
=
(
embed
(
A
:
=
PROP1
)
(
B
:
=
PROP2
)).
...
...
@@ 270,6 +277,14 @@ Section sbi_embed.
Lemma
embed_except_0
P
:
⎡◇
P
⎤
⊣
⊢
◇
⎡
P
⎤
.
Proof
.
by
rewrite
embed_or
embed_later
embed_pure
.
Qed
.
(* Not an instance, since it may cause overlap *)
Lemma
bi_embed_plainly_emp
`
{!
BiPlainly
PROP1
,
!
BiPlainly
PROP2
}
:
BiEmbedEmp
PROP1
PROP2
→
BiEmbedPlainly
PROP1
PROP2
.
Proof
.
intros
.
constructor
=>
P
.
rewrite
!
plainly_alt
embed_internal_eq
.
by
rewrite

embed_affinely

embed_emp
embed_interal_inj
.
Qed
.
Lemma
embed_plainly_1
`
{!
BiPlainly
PROP1
,
!
BiPlainly
PROP2
}
P
:
⎡■
P
⎤
⊢
■
⎡
P
⎤
.
Proof
.
assert
(
∀
P
,
<
affine
>
⎡
P
⎤
⊣
⊢
(<
affine
>
⎡
<
affine
>
P
⎤
:
PROP2
))
as
Hhelp
.
...
...
@@ 282,16 +297,14 @@ Section sbi_embed.

apply
bi
.
and_intro
;
auto
using
embed_emp_2
.
}
rewrite
!
plainly_alt
embed_internal_eq
.
by
rewrite
Hhelp

Hemp
!
bi
.
f_equiv
.
Qed
.
Lemma
embed_plainly
`
{!
Bi
EmbedEmp
PROP1
PROP2
,
!
Bi
Plainly
PROP1
,
!
BiPlainly
PROP2
}
P
:
⎡■
P
⎤
⊣
⊢
■
⎡
P
⎤
.
Lemma
embed_plainly
`
{!
Bi
Plainly
PROP1
,
!
BiPlainly
PROP2
,
!
Bi
EmbedPlainly
PROP1
PROP2
}
P
:
⎡■
P
⎤
⊣
⊢
■
⎡
P
⎤
.
Proof
.
apply
(
anti_symm
_
)
;
first
by
apply
embed_plainly_1
.
rewrite
!
plainly_alt
embed_internal_eq
.
by
rewrite

embed_affinely

embed_emp
embed_interal_inj
.
apply
(
anti_symm
_
).
by
apply
embed_plainly_1
.
by
apply
embed_plainly_2
.
Qed
.
Lemma
embed_plainly_if
`
{!
Bi
EmbedEmp
PROP1
PROP2
,
!
Bi
Plainly
PROP1
,
!
BiPlainly
PROP2
}
p
P
:
⎡■
?p
P
⎤
⊣
⊢
■
?p
⎡
P
⎤
.
Lemma
embed_plainly_if
`
{!
Bi
Plainly
PROP1
,
!
BiPlainly
PROP2
,
!
Bi
EmbedPlainly
PROP1
PROP2
}
p
P
:
⎡■
?p
P
⎤
⊣
⊢
■
?p
⎡
P
⎤
.
Proof
.
destruct
p
;
simpl
;
auto
using
embed_plainly
.
Qed
.
Lemma
embed_plainly_if_1
`
{!
BiPlainly
PROP1
,
!
BiPlainly
PROP2
}
p
P
:
⎡■
?p
P
⎤
⊢
■
?p
⎡
P
⎤
.
...
...
theories/bi/monpred.v
View file @
09ae3206
...
...
@@ 933,6 +933,10 @@ Proof.
apply
bi
.
forall_intro
=>?.
by
do
2
f_equiv
.
Qed
.
Global
Instance
monPred_bi_embed_plainly
`
{
BiPlainly
PROP
}
:
BiEmbedPlainly
PROP
monPredSI
.
Proof
.
apply
bi_embed_plainly_emp
,
_
.
Qed
.
Lemma
monPred_plainly_unfold
`
{
BiPlainly
PROP
}
:
plainly
=
λ
P
,
⎡
∀
i
,
■
(
P
i
)
⎤
%
I
.
Proof
.
by
rewrite
monPred_plainly_eq
monPred_embed_eq
.
Qed
.
Lemma
monPred_at_plainly
`
{
BiPlainly
PROP
}
i
P
:
(
■
P
)
i
⊣
⊢
∀
j
,
■
(
P
j
).
...
...
theories/proofmode/class_instances_sbi.v
View file @
09ae3206
...
...
@@ 417,7 +417,7 @@ Global Instance from_modal_plainly `{BiPlainly PROP} P :
Proof
.
by
rewrite
/
FromModal
.
Qed
.
Global
Instance
from_modal_plainly_embed
`
{
BiPlainly
PROP
,
BiPlainly
PROP'
,
BiEmbed
Emp
PROP
PROP'
,
!
SbiEmbed
PROP
PROP'
}
`
(
sel
:
A
)
P
Q
:
BiEmbed
Plainly
PROP
PROP'
,
!
SbiEmbed
PROP
PROP'
}
`
(
sel
:
A
)
P
Q
:
FromModal
modality_plainly
sel
P
Q
→
FromModal
modality_plainly
sel
⎡
P
⎤
⎡
Q
⎤

100
.
Proof
.
rewrite
/
FromModal
/=
=><.
by
rewrite
embed_plainly
.
Qed
.
...
...
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