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662d20dc
Commit
662d20dc
authored
Mar 08, 2016
by
Ralf Jung
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add bidirectional turnstile
parent
3cf0a5fc
Pipeline
#289
passed with stage
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28 deletions
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-28
docs/derived.tex
docs/derived.tex
+3
-3
docs/iris.sty
docs/iris.sty
+1
-0
docs/logic.tex
docs/logic.tex
+24
-25
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docs/derived.tex
View file @
662d20dc
...
...
@@ -8,13 +8,13 @@ We collect here some important and frequently used derived proof rules.
{
\prop
\Ra
\propB
\proves
\prop
\wand
\propB
}
\infer
{}
{
\prop
*
\Exists\var
.
\propB
\
Lra
\Exists\var
.
\prop
*
\propB
}
{
\prop
*
\Exists\var
.
\propB
\
provesIff
\Exists\var
.
\prop
*
\propB
}
\infer
{}
{
\prop
*
\Exists\var
.
\propB
\proves
\Exists\var
.
\prop
*
\propB
}
\infer
{}
{
\always
(
\prop*\propB
)
\
Lra
\always\prop
*
\always\propB
}
{
\always
(
\prop*\propB
)
\
provesIff
\always\prop
*
\always\propB
}
\infer
{}
{
\always
(
\prop
\Ra
\propB
)
\proves
\always\prop
\Ra
\always\propB
}
...
...
@@ -23,7 +23,7 @@ We collect here some important and frequently used derived proof rules.
{
\always
(
\prop
\wand
\propB
)
\proves
\always\prop
\wand
\always\propB
}
\infer
{}
{
\always
(
\prop
\wand
\propB
)
\
Lra
\always
(
\prop
\Ra
\propB
)
}
{
\always
(
\prop
\wand
\propB
)
\
provesIff
\always
(
\prop
\Ra
\propB
)
}
\infer
{}
{
\later
(
\prop
\Ra
\propB
)
\proves
\later\prop
\Ra
\later\propB
}
...
...
docs/iris.sty
View file @
662d20dc
...
...
@@ -184,6 +184,7 @@
\def\Lam
#1.
{
\lambda
#1.
\spac
}
%
\newcommand
{
\proves
}{
\vdash
}
\newcommand
{
\provesIff
}{
\mathrel
{
\dashv\vdash
}}
\newcommand
{
\wand
}{
\;
{{
\mbox
{
---
}}
\!\!
{
*
}}
\;
}
...
...
docs/logic.tex
View file @
662d20dc
...
...
@@ -303,8 +303,7 @@ In writing $\vctx, x:\type$, we presuppose that $x$ is not already declared in $
The judgment
$
\vctx
\mid
\pfctx
\proves
\prop
$
says that with free variables
$
\vctx
$
, proposition
$
\prop
$
holds whenever all assumptions
$
\pfctx
$
hold.
We implicitly assume that an arbitrary variable context,
$
\vctx
$
, is added to every constituent of the rules.
Furthermore, an arbitrary
\emph
{
boxed
}
assertion context
$
\always\pfctx
$
may be added to every constituent.
Axioms
$
\prop
\Ra
\propB
$
stand for judgments
$
\vctx
\mid
\cdot
\proves
\prop
\Ra
\propB
$
with no assumptions.
(Bi-implications are analogous.)
Axioms
$
\vctx
\mid
\prop
\provesIff
\propB
$
indicate that both
$
\vctx
\mid
\prop
\proves
\propB
$
and
$
\vctx
\mid
\propB
\proves
\prop
$
can be derived.
\judgment
{}{
\vctx
\mid
\pfctx
\proves
\prop
}
\paragraph
{
Laws of intuitionistic higher-order logic with equality.
}
...
...
@@ -395,9 +394,9 @@ This is entirely standard.
\paragraph
{
Laws of (affine) bunched implications.
}
\begin{mathpar}
\begin{array}
{
rMcMl
}
\TRUE
*
\prop
&
\
Lra
&
\prop
\\
\prop
*
\propB
&
\
Lra
&
\propB
*
\prop
\\
(
\prop
*
\propB
) *
\propC
&
\
Lra
&
\prop
* (
\propB
*
\propC
)
\TRUE
*
\prop
&
\
provesIff
&
\prop
\\
\prop
*
\propB
&
\
provesIff
&
\propB
*
\prop
\\
(
\prop
*
\propB
) *
\propC
&
\
provesIff
&
\prop
* (
\propB
*
\propC
)
\end{array}
\and
\infer
[$*$-mono]
...
...
@@ -413,14 +412,14 @@ This is entirely standard.
\paragraph
{
Laws for ghosts and physical resources.
}
\begin{mathpar}
\begin{array}
{
rMcMl
}
\ownGGhost
{
\melt
}
*
\ownGGhost
{
\meltB
}
&
\
Lra
&
\ownGGhost
{
\melt
\mtimes
\meltB
}
\\
\ownGGhost
{
\melt
}
&
\
Ra
&
\melt
\in
\mval
\\
\TRUE
&
\
Ra
&
\ownGGhost
{
\munit
}
\ownGGhost
{
\melt
}
*
\ownGGhost
{
\meltB
}
&
\
provesIff
&
\ownGGhost
{
\melt
\mtimes
\meltB
}
\\
\ownGGhost
{
\melt
}
&
\
provesIff
&
\melt
\in
\mval
\\
\TRUE
&
\
proves
&
\ownGGhost
{
\munit
}
\end{array}
\and
\and
\begin{array}
{
c
}
\ownPhys
{
\state
}
*
\ownPhys
{
\state
'
}
\
Ra
\FALSE
\ownPhys
{
\state
}
*
\ownPhys
{
\state
'
}
\
proves
\FALSE
\end{array}
\end{mathpar}
...
...
@@ -439,14 +438,14 @@ This is entirely standard.
{
\later
{
\Exists
x:
\type
.
\prop
}
\proves
\Exists
x:
\type
.
\later\prop
}
\\\\
\begin{array}
[c]
{
rMcMl
}
\later
{
(
\prop
\wedge
\propB
)
}
&
\
Lra
&
\later
{
\prop
}
\wedge
\later
{
\propB
}
\\
\later
{
(
\prop
\vee
\propB
)
}
&
\
Lra
&
\later
{
\prop
}
\vee
\later
{
\propB
}
\\
\later
{
(
\prop
\wedge
\propB
)
}
&
\
provesIff
&
\later
{
\prop
}
\wedge
\later
{
\propB
}
\\
\later
{
(
\prop
\vee
\propB
)
}
&
\
provesIff
&
\later
{
\prop
}
\vee
\later
{
\propB
}
\\
\end{array}
\and
\begin{array}
[c]
{
rMcMl
}
\later
{
\All
x.
\prop
}
&
\
Lra
&
\All
x.
\later\prop
\\
\Exists
x.
\later\prop
&
\
Ra
&
\later
{
\Exists
x.
\prop
}
\\
\later
{
(
\prop
*
\propB
)
}
&
\
Lra
&
\later\prop
*
\later\propB
\later
{
\All
x.
\prop
}
&
\
provesIff
&
\All
x.
\later\prop
\\
\Exists
x.
\later\prop
&
\
proves
&
\later
{
\Exists
x.
\prop
}
\\
\later
{
(
\prop
*
\propB
)
}
&
\
provesIff
&
\later\prop
*
\later\propB
\end{array}
\end{mathpar}
...
...
@@ -487,26 +486,26 @@ This is entirely standard.
{
\always
{
\pfctx
}
\proves
\always
{
\prop
}}
\and
\infer
[$\always$E]
{}
{
\always
{
\prop
}
\
Ra
\prop
}
{
\always
{
\prop
}
\
proves
\prop
}
\and
\begin{array}
[c]
{
rMcMl
}
\always
{
(
\prop
*
\propB
)
}
&
\
Ra
&
\always
{
(
\prop
\land
\propB
)
}
\\
\always
{
\prop
}
*
\propB
&
\
Ra
&
\always
{
\prop
}
\land
\propB
\\
\always
{
\later\prop
}
&
\
Lra
&
\later\always
{
\prop
}
\\
\always
{
(
\prop
*
\propB
)
}
&
\
proves
&
\always
{
(
\prop
\land
\propB
)
}
\\
\always
{
\prop
}
*
\propB
&
\
proves
&
\always
{
\prop
}
\land
\propB
\\
\always
{
\later\prop
}
&
\
provesIff
&
\later\always
{
\prop
}
\\
\end{array}
\and
\begin{array}
[c]
{
rMcMl
}
\always
{
(
\prop
\land
\propB
)
}
&
\
Lra
&
\always
{
\prop
}
\land
\always
{
\propB
}
\\
\always
{
(
\prop
\lor
\propB
)
}
&
\
Lra
&
\always
{
\prop
}
\lor
\always
{
\propB
}
\\
\always
{
\All
x.
\prop
}
&
\
Lra
&
\All
x.
\always
{
\prop
}
\\
\always
{
\Exists
x.
\prop
}
&
\
Lra
&
\Exists
x.
\always
{
\prop
}
\\
\always
{
(
\prop
\land
\propB
)
}
&
\
provesIff
&
\always
{
\prop
}
\land
\always
{
\propB
}
\\
\always
{
(
\prop
\lor
\propB
)
}
&
\
provesIff
&
\always
{
\prop
}
\lor
\always
{
\propB
}
\\
\always
{
\All
x.
\prop
}
&
\
provesIff
&
\All
x.
\always
{
\prop
}
\\
\always
{
\Exists
x.
\prop
}
&
\
provesIff
&
\Exists
x.
\always
{
\prop
}
\\
\end{array}
\and
{
\term
=
_
\type
\term
'
\
Ra
\always
\term
=
_
\type
\term
'
}
{
\term
=
_
\type
\term
'
\
proves
\always
\term
=
_
\type
\term
'
}
\and
{
\knowInv\iname\prop
\
Ra
\always
\knowInv\iname\prop
}
{
\knowInv\iname\prop
\
proves
\always
\knowInv\iname\prop
}
\and
{
\ownGGhost
{
\mcore\melt
}
\
Ra
\always
\ownGGhost
{
\mcore\melt
}}
{
\ownGGhost
{
\mcore\melt
}
\
proves
\always
\ownGGhost
{
\mcore\melt
}}
\end{mathpar}
\paragraph
{
Laws of primitive view shifts.
}
...
...
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