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x10-commits
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From: Vijay S. <vj...@us...> - 2007-07-14 13:33:24
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Update of /cvsroot/x10/x10.man/paper In directory sc8-pr-cvs10.sourceforge.net:/tmp/cvs-serv9478/paper Modified Files: semantics.tex paper.pdf Log Message: Updated to Version 9 of the semantics. Combined T-Inv, T-Constr and T-Var into a single rule. Ensured that |- respects constraint entailment. Added lemmas analogous to the FJ lemmas that we will need to prove for Type Soundness. Added condition in the constraint system about field selection. Index: semantics.tex =================================================================== RCS file: /cvsroot/x10/x10.man/paper/semantics.tex,v retrieving revision 1.6 retrieving revision 1.7 diff -u -d -r1.6 -r1.7 --- semantics.tex 13 Jul 2007 14:11:35 -0000 1.6 +++ semantics.tex 14 Jul 2007 13:33:23 -0000 1.7 @@ -25,7 +25,7 @@ (Class) & L &{::=}& $\tt\class \ C(\bar{T}\ \bar{f}:c)\ \extends\ T\ \{\bar{M}\}$ \\ (Method)& M &{::=}& $\tt T\ m(\bar{T}\ \bar{x}:c)\{\return\ e;\}$\\ (Expr)& e &{::=}& $\tt x \alt e.f \alt e.m(\bar{e})\alt \new\ C(\bar{e})\alt (T)e$\\\quad\\ -(C Terms) & t&{::=}& \tt x\alt \self \alt \this \alt t.f\\ +(C Terms) & t&{::=}& \tt x\alt \self \alt \this \alt t.f \alt \new\ {\tt C($\bar{\tt t}$}\\ (Constraint) & c,d&{::=}&$\tt \true\alt a\alt t=t\alt c,c\alt T\,x;c$\\ (Type)& S,T,U&{::=}& $\tt C(:d)$\\ &&&\\ @@ -39,16 +39,16 @@ $$ \begin{array}{ll} -\Gamma\vdash {\tt T} \subtype {\tt T} + {\tt C} \subtype {\tt C} & \from{\class\ {\tt C(\ldots)}\ \extends\ {\tt D(\ldots)}\{\ldots\}} -\infer{\vdash {\tt C \subtype D}} +\infer{{\tt C} \subtype {\tt D}} \\ & \\ -\from{\Gamma \vdash {\tt C \subtype D} \ \ \ \sigma(\Gamma),{\tt c} \vdash_{\cal C} {\tt d}} +\from{{\tt C} \subtype {\tt D} \ \ \ {\tt D} \subtype {\tt E}} +\infer{{\tt C} \subtype {\tt E}} & +\from{{\tt C} \subtype {\tt D} \ \ \ \Gamma, {\tt C(:c)}\ {\tt x} \vdash +{\tt D(:d)}\ {\tt x}\ \ \ \mbox{({\tt x} new)}} \infer{\Gamma \vdash {\tt C(:c) \subtype D(:d)}} -& -\from{\Gamma \vdash {\tt S \subtype T} \ \ \ \Gamma \vdash {\tt T \subtype U}} -\infer{\Gamma \vdash {\tt S \subtype U}} \end{array} $$ @@ -66,29 +66,25 @@ $$ \begin{array}{ll} -\axname{T-Var} -\axiom{\Gamma, {\tt T}\ {\tt x} \vdash {\tt T}\ {\tt x}} \\ & \\ -\rname{T-Constr} -\from{\Gamma \vdash {\tt S\ x} \ \ \ \Gamma, {\tt S\ x} \vdash_{\cal C} {\tt T\ e}} -\infer{\Gamma \vdash {\tt T\ e}} & -\rname{T-Inv} -\from{\Gamma, {\tt c}, {\tt S\ x} \vdash {\tt T\ e} \ \ \ \ {\tt c}=\inv(T, [{\tt x}/\this])} -\infer{\Gamma, {\tt S\ x} \vdash {\tt T\ e}} +\rname{T-Var} +\from{\sigma(\Gamma, {\tt C(:c)}\ {\tt x}) \vdash_{\cal C} {\tt d}[{\tt x}/\self]} +\infer{\Gamma, {\tt C(:c)\ x} \vdash {\tt C(:d)}\ {\tt x}} & \\ \quad \\ \rname{T-Field} -\from{\Gamma \vdash {\tt T}_0\ {\tt e} \ \ \ fields({\tt T}_0,[{\tt o}/\this])= \bar{\tt U}(:\bar{\tt d}) \bar{\tt f} \ \ \ \mbox{({\tt o} fresh for $\Gamma,{\tt d}_i$)}} -\infer{\Gamma \vdash {\tt U}_i(:{\tt T}_0\ {\tt o;o.f}_i=\self,{\tt d}_i)\, {\tt e.f}_i} +\from{\Gamma \vdash {\tt T}_0\ {\tt e} \ \ \ fields({\tt T}_0,{\tt z}_0)= \bar{\tt U}\ \bar{\tt f} \ \ \ \mbox{(${\tt z}_0$ fresh)}} +\infer{\Gamma \vdash ({\tt T}_0\ {\tt z}_0; {\tt z}_0.{\tt f}=\self;{\tt U}_i)\ {\tt e.f}_i} & -\axname{T-Cast} -\axiom{\Gamma \vdash {\tt T\ (T) e}} \\ +\rname{T-Cast} +\from{\Gamma \vdash {\tt S}\ {\tt e}} +\infer{\Gamma \vdash {\tt T}\ {\tt (T) e}} \\ & \\ \rname{T-Invk} \from{\begin{array}{l} \Gamma \vdash {\tt T}_{0:n} \ {\tt e}_{0:n} \ \ \ -mtype({\tt T}_0,{\tt m},[{\tt z}_0/\this])= \tt {\tt Z}_{1:n}\ {\tt z}_{1:n}:c \rightarrow {\tt S(:d)} \\ -\Gamma, {\tt T}_{0:n}\ {\tt z}_{0:n} \vdash c, {\tt T}_{1:n} \subtype {\tt Z}_{1:n}\ \ \mbox {(${\tt z}_0$ fresh for $\Gamma, {\tt c},{\tt d}$)} +mtype({\tt T}_0,{\tt m},{\tt z}_0)= \tt {\tt Z}_{1:n}\ {\tt z}_{1:n}:c \rightarrow {\tt S} \\ +\Gamma, {\tt T}_{0:n}\ {\tt z}_{0:n} \vdash {\tt c}, {\tt T}_{1:n} \subtype {\tt Z}_{1:n}\ \ \ \mbox {(${\tt z}_{0:n}$ fresh)} \end{array}} -\infer{\Gamma \vdash {\tt S(:T}_{0:n}\ {\tt z}_{0:n}; d)\ {\tt e}_0.{\tt m({\tt e}_{1:n})}}& +\infer{\Gamma \vdash ({\tt T}_{0:n}\ {\tt z}_{0:n}; S)\ {\tt e}_0.{\tt m({\tt e}_{1:n})}}& \rname{T-New} \from{ \begin{array}{l} @@ -175,6 +171,11 @@ $c[\bar{t}/\bar{x}]$. Application of substitutions is extended to types by: ${\tt C(:c)\theta}={\tt C(:c\theta)}$. +All constraint systems are required to satisfy: +$$\new\ {\tt C(\bar{\tt t})}.{\tt f}_i={\tt t}_i $$ +provided that $fields({\tt C})=\bar{\tt T}\ \bar{\tt f}$ (for some +sequence of types $\bar{\tt T}$). + A type assertion {\tt C(:c) x} constrains the variable {\tt x} to contain references to only those objects {\tt o} that are instances of (subclasses of) {\tt C} and for which the constraint {\tt c} is true @@ -210,15 +211,16 @@ \paragraph{Type judgements.} Typing judgements are of the form $\Gamma \vdash {\tt T}\ {\tt e}$ where $\Gamma$ is a multiset of type assertions ${\tt T}\ {\tt x}$ and -constraints ${\tt c}$. The constraint entailment relation -$\vdash_{\cal C}$ is lifted to type assertions through the definition: -$\Gamma \vdash_{\cal C} {\tt D(:d)}\ {\tt x}$ provided that $\{ {\tt -c}[{\tt x}/\self] \alt {\tt C(:c)}\ x \in \Gamma\} \cup -\{{\tt c} \alt {\tt c} \in \Gamma\} \vdash_{\cal C} {\tt d}[{\tt x}/\self] -$ and $\Gamma \vdash {\tt D}\ {\tt x}$. Intuitively, $\Gamma \vdash -\tt D(:d)\ {\tt x}$ if $\Gamma$ constrains {\tt x} to be of type {\tt -D} and there is enough information in the constraints in $\Gamma$ to -entail {\tt d} for {\tt x}. +constraints ${\tt c}$. +%The constraint entailment relation +%$\vdash_{\cal C}$ is lifted to type assertions through the definition: +%$\Gamma \vdash_{\cal C} {\tt D(:d)}\ {\tt x}$ provided that $\{ {\tt +%c}[{\tt x}/\self] \alt {\tt C(:c)}\ x \in \Gamma\} \cup +%\{{\tt c} \alt {\tt c} \in \Gamma\} \vdash_{\cal C} {\tt d}[{\tt x}/\self] +%$ and $\Gamma \vdash {\tt D}\ {\tt x}$. Intuitively, $\Gamma \vdash +%\tt D(:d)\ {\tt x}$ if $\Gamma$ constrains {\tt x} to be of type {\tt +%D} and there is enough information in the constraints in $\Gamma$ to +%entail {\tt d} for {\tt x}. %$\sigma(\Gamma)$ is the set of constraints on the variables whose type %assertions are specified by $\Gamma$, generated by replacing each type @@ -228,11 +230,21 @@ %information about {\tt x} and other variables specified in $\Gamma$. \def\TConstr{\mbox{\sc T-Constr}} \def\TInv{\mbox{\sc T-Inv}} +\def\TVar{\mbox{\sc T-Var}} \def\TField{\mbox{\sc T-Field}} \def\TInvk{\mbox{\sc T-Invk}} \def\TNew{\mbox{\sc T-New}} -\TConstr{} is a form of cut which permits information obtained through -constraint entailment to enrich the type of an expression. +\def\TCast{\mbox{\sc T-Cast}} +\def\TUCast{\mbox{\sc T-UCast}} +\def\TDCast{\mbox{\sc T-DCast}} +\def\TSCast{\mbox{\sc T-SCast}} +%\TConstr{} is a form of cut which permits information obtained through +%constraint entailment to enrich the type of an expression. + +\TVar{} extends the identity rule ($\Gamma, x:C \vdash x:C$) of {\sf FJ} to take into account the constraint entailment relation. + +\TCast{} encapsulates the three inference rules of {\sf FJ} -- +\TUCast{}, \TDCast{} and\TSCast{} for upwards cast, downwards cast, and ``stupid'' cast respectively. \TInv{} is a form of contraction that permits the class invariant {\tt c} of a class {\tt C} to enrich the type of any variable of type {\tt @@ -306,3 +318,20 @@ \end{theorem} +\begin{lemma}[Substitution Lemma] +Assume $\Gamma \vdash \bar{\tt A}\ \bar{\tt d}$, $\Gamma \vdash \bar{\tt A}\subtype \bar{\tt B}$, and $\Gamma, \bar{\tt B}\ \bar{\tt x} \vdash {\tt T}\ {\tt e}$. Then for some type ${\tt S}$ s.t. $\Gamma \vdash {\tt S} \subtype \bar{\tt A}\ \bar{\tt x};{\tt T}$ it is the case that $\Gamma \vdash {\tt S}\ {\tt e}[\bar{\tt d}/\bar{\tt x}]$. +\end{lemma} + +% Unchanged from FJ +\begin{lemma}[Weakening] +If $\Gamma \vdash {\tt T}\ {\tt e}$, then $\Gamma, {\tt S}\ {\tt x}\vdash {\tt T}\ {\tt e}$. +\end{lemma} + +% Unchanged from FJ +\begin{lemma}[Body type] +If $mtype({\tt m}, {\tt T}_0)=\bar{\tt T}\ \bar{\tt x} : {\tt c} +\rightarrow {\tt S}$, and $mbody({\tt m}, {\tt T}_0)=\bar{\tt x}.{\tt +e}$, then for some ${\tt U}_0$ with ${\tt T}_0 \subtype {\tt U}_0$, +there exists ${\tt V}\subtype {\tt S}$ such that +$\bar{\tt T}\ \bar{\tt x},{\tt U}_0\ \this \vdash {\tt V}\ {\tt e}$ +\end{lemma} \ No newline at end of file Index: paper.pdf =================================================================== RCS file: /cvsroot/x10/x10.man/paper/paper.pdf,v retrieving revision 1.6 retrieving revision 1.7 diff -u -d -r1.6 -r1.7 Binary files /tmp/cvsKcEbew and /tmp/cvssK0F3t differ |
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