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@ -50,6 +50,15 @@ |
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\textquotedblleft={ ,150}, % left quotation mark, space from right |
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\textquotedblright={150, }} % right quotation mark, space from left |
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%% |
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%% Theorem environments |
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%% |
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\newtheorem{theorem}{Theorem}[section] |
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\newtheorem{lemma}[theorem]{Lemma} |
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\newtheorem{proposition}[theorem]{Proposition} |
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\newtheorem{corollary}[theorem]{Corollary} |
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\newtheorem{definition}[theorem]{Definition} |
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%% |
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%% Load macros. |
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%% |
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@ -500,11 +509,11 @@ $\TyVarCat$, to be generated by: |
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\mid \lambda x^A .\, M \mid \Lambda \alpha^K .\, M |
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\mid \Record{} \mid \Record{\ell = V;W} \mid (\ell~V)^R \\ |
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& & &\\ |
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\slab{Computations} &M,N \in \CompCat &::= & V\,W \mid V\,A\\ |
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\slab{Computations} &M,N \in \CompCat &::= & V\,W \mid V\,T\\ |
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& &\mid& \Let\; \Record{\ell=x;y} = V \; \In \; N\\ |
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& &\mid& \Case\; V \{\ell~x \mapsto M; y \mapsto N\} \mid \Absurd^C~V\\ |
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& &\mid& \Return~V \mid \Let \; x \revto M \; \In \; N\\ |
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\slab{Terms} &T \in \TermCat &::= & x \mid V \mid M |
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\slab{Terms} &t \in \TermCat &::= & x \mid V \mid M |
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\end{syntax} |
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\caption{Term syntax of \BCalc{}.} |
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@ -552,6 +561,37 @@ kind information (term abstraction, type abstraction, injection, |
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operations, and empty cases). However, we shall omit these annotations |
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whenever they are clear from context. |
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\paragraph{Free variables} We define the function |
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$\FV : \TermCat \to \VarCat$ to compute the free variables of any |
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given term. |
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% |
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\[ |
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\bl |
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\ba[t]{@{~}l@{~}c@{~}l} |
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\begin{eqs} |
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\FV(x) &\defas& \{x\}\\ |
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\FV(\lambda x^A.M) &\defas& \FV(M) \setminus \{x\}\\ |
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\FV(\Lambda \alpha^K.M) &\defas& \FV(M)\\[1.0ex] |
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\FV(V\,W) &\defas& \FV(V) \cup \FV(W)\\ |
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\FV(\Return~V) &\defas& \FV(V)\\ |
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\end{eqs} |
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& \qquad\qquad & |
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\begin{eqs} |
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\FV(\Record{}) &\defas& \emptyset\\ |
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\FV(\Record{\ell = V; W}) &\defas& \FV(V) \cup \FV(W)\\ |
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\FV((\ell~V)^R) &\defas& \FV(V)\\[1.0ex] |
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\FV(V\,T) &\defas& \FV(V)\\ |
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\FV(\Absurd^C~V) &\defas& \FV(V)\\ |
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\end{eqs} |
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\ea\\ |
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\begin{eqs} |
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\FV(\Let\;x \revto M \;\In\;N) &\defas& \FV(M) \cup (\FV(N) \setminus \{x\})\\ |
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\FV(\Let\;\Record{\ell=x;y} = V\;\In\;N) &\defas& \FV(V) \cup (\FV(N) \setminus \{x, y\})\\ |
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\FV(\Case~V~\{\ell\,x \mapsto M; y \mapsto N\} &\defas& \FV(V) \cup (\FV(M) \setminus \{x\}) \cup (\FV(N) \setminus \{y\}) |
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\end{eqs} |
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\el |
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\] |
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\subsection{Typing rules} |
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\label{sec:base-language-type-rules} |
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% |
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@ -660,13 +700,17 @@ of \emph{free type variables} ($\FTV$) to ensure that we do not |
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inadvertently capture a free type variable from the context. |
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% |
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We define $\FTV$ by mutual induction over type environments, $\Gamma$, |
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and the type structure, $T$, as follows. |
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and the type structure, $T$. Note that we always work up to |
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$\alpha$-conversion~\cite{Church32} of types. |
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% |
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\[ |
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\ba[t]{@{~}l@{~~~~~~}c@{~}l} |
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\multicolumn{3}{c}{\begin{eqs} |
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\FTV &:& \TypeCat \to \TyVarCat |
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\end{eqs}}\\ |
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\ba[t]{@{}l} |
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\begin{eqs} |
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\FTV &:& \ValTypeCat \to \TyVarCat\\ |
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% \FTV &:& \ValTypeCat \to \TyVarCat\\ |
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\FTV(\alpha) &\defas& \{\alpha\}\\ |
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\FTV(\forall \alpha^K.C) &\defas& \FTV(C) \setminus \{\alpha\}\\ |
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\FTV(A \to C) &\defas& \FTV(A) \cup \FTV(C)\\ |
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@ -683,12 +727,12 @@ and the type structure, $T$, as follows. |
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% \FTV([R]) &\defas& \FTV(R)\\ |
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% \FTV(\Record{R}) &\defas& \FTV(R)\\ |
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% \FTV(\{R\}) &\defas& \FTV(R)\\ |
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\FTV &:& \RowCat \to \TyVarCat\\ |
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% \FTV &:& \RowCat \to \TyVarCat\\ |
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\FTV(\cdot) &\defas& \emptyset\\ |
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\FTV(\rho) &\defas& \{\rho\}\\ |
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\FTV(l:P;R) &\defas& \FTV(P) \cup \FTV(R)\\[1.0ex] |
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\FTV(l:P;R) &\defas& \FTV(P) \cup \FTV(R)\\ |
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\FTV &:& \PresenceCat \to \TyVarCat\\ |
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% \FTV &:& \PresenceCat \to \TyVarCat\\ |
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\FTV(\theta) &\defas& \{\theta\}\\ |
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\FTV(\Abs) &\defas& \emptyset\\ |
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\FTV(\Pre{A}) &\defas& \FTV(A)\\ |
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@ -747,10 +791,7 @@ defined as follows. |
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-[-/-] &:& \TypeCat \times \TypeCat \times \TyVarCat \to \TypeCat\\ |
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(A \eff E)[B/\beta] &\defas& A[B/\beta] \eff E[B/\beta]\\ |
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(A \to C)[B/\beta] &\defas& A[B/\beta] \to C[B/\beta]\\ |
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(\forall \alpha^K.C)[B/\beta] &\defas& \begin{cases} |
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\forall \alpha^K.C & \text{if } \alpha = \beta\\ |
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\forall \alpha^K.C[B/\beta] & \text{otherwise} |
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\end{cases}\\ |
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(\forall \alpha^K.C)[B/\beta] &\defas& \forall \alpha^K.C[B/\beta] \quad \text{if } \alpha \neq \beta \text{ and } \alpha \notin \FTV(B)\\ |
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\alpha[B/\beta] &\defas& \begin{cases} |
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B & \text{if } \alpha = \beta\\ |
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\alpha & \text{otherwise} |
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@ -848,8 +889,9 @@ rule admits a continuation. |
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% |
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The semantics are based on a substitution model of computation. Thus, |
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before presenting the reduction rules, we define an adequacy |
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substitution function. |
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before presenting the reduction rules, we define an adequate |
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substitution function. As usual we work up to |
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$\alpha$-conversion~\cite{Church32} of terms in $\BCalc{}$. |
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% |
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\paragraph{Term substitution} |
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We write $M[V/x]$ for the substitution of some value $V$ for some |
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@ -870,42 +912,62 @@ and we realise it by pattern matching on the first argument. |
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V & \text{if } x = y\\ |
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x & \text{otherwise } |
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\end{cases}\\ |
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(\lambda x^A.M)[V/y] &\defas& \begin{cases} |
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\lambda x^A.M & \text{if } x = y\\ |
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\lambda x^A.M[V/y] & \text{otherwise} |
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\end{cases}\\ |
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(\lambda x^A.M)[V/y] &\defas& \lambda x^A.M[V/y] \quad \text{if } x \neq y \text{ and } x \notin \FV(V)\\ |
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(\Lambda \alpha^K. M)[V/y] &\defas& \Lambda \alpha^K. M[V/y]\\ |
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\Unit[V/y] &\defas& \Unit\\ |
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\Record{\ell = W; W'}[V/y] &\defas& \Record{\ell = W[V/y]; W'[V/y]}\\ |
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(\ell~W)^R[V/y] &\defas& (\ell~W[V/y])^R\\ |
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(W\,W')[V/y] &\defas& W[V/y]\,W'[V/y]\\ |
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(W\,A)[V/y] &\defas& W[V/y]~A\\ |
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(\Let\;\Record{\ell = x; y} = W\;\In\;N)[V/y] &\defas& \Let\;\Record{\ell = x; y} = W[V/y] \;\In\;N[V/y]\\ |
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(\Case\;(\ell~V)^R\{\ba[t]{@{}l} \ell~x \mapsto M\\ |
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; y \mapsto N \})[V/z]\ea |
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&\defas& \begin{cases} |
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\Case\;(\ell~V)^R\{\ell~x \mapsto M; y \mapsto N \} & \text{if } x = y = z\\ |
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\Case\;(\ell~V)^R\{\ba[t]{@{}l} \ell~x \mapsto M\\ |
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; y \mapsto N[V/z] \}\ea & \text{if } x = z \text{ and } y \neq z\\ |
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\Case\;(\ell~V)^R\{ \ba[t]{@{}l}\ell~x \mapsto M[V/z]\\ |
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; y \mapsto N \}\ea & \text{if } x \neq z \text{ and } y = z\\ |
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\Case\;(\ell~V)^R\{ \ba[t]{@{}l} \ell~x \mapsto M[V/z]\\ |
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; y \mapsto N[V/z] \}\ea & \text{otherwise} |
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\end{cases}\\ |
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(\Let\;x \revto M \;\In\;N)[V/y] &\defas& \begin{cases} |
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\Let\;x \revto M[V/y]\;\In\;N & \text{if } x = y\\ |
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\Let\;x \revto M[V/y]\;\In\;N[V/y] & \text{otherwise} |
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\end{cases} |
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(W\,T)[V/y] &\defas& W[V/y]~T\\ |
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(\ba[t]{@{}l} |
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\Let\;\Record{\ell = x; y} = W\\ |
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\In\;N)[V/z] |
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\ea &\defas& |
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\ba[t]{@{~}l} |
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\Let\;\Record{\ell = x; y} = W[V/z]\\ |
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\In\;N[V/z] |
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\ea \quad |
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\ba[t]{@{}l} |
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\text{if } x \neq z, y \neq z,\\ |
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\text{and } x,y\notin \FV(V) |
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\ea\\ |
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(\Case\;(\ell~W)^R\{ |
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\ba[t]{@{}l} |
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\ell~x \mapsto M\\ |
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; y \mapsto N \})[V/z] |
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\ea |
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&\defas& |
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\Case\;(\ell~W[V/z])^R\{ |
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\ba[t]{@{}l} |
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\ell~x \mapsto M[V/z]\\ |
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; y \mapsto N[V/z] \} |
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\ea\quad |
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\ba[t]{@{}l} |
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\text{if } x \neq z, y \neq z,\\ |
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\text{and } x, y \notin \FV(V) |
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\ea\\ |
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(\Let\;x \revto M \;\In\;N)[V/y] &\defas& \Let\;x \revto M[V/y] \;\In\;N[V/y] \quad\text{if } x \neq y \text{ and } x \notin \FV(V) |
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\end{eqs} |
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\] |
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% |
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% \begin{cases} |
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% \Let\;x \revto M[V/y]\;\In\;N & \text{if } x = y\\ |
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% \Let\;x \revto M[V/y]\;\In\;N[V/y] & \text{otherwise} |
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% \end{cases} |
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% \end{eqs} |
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% |
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We write $t[t_0/x_0,\dots,t_n/x_n]$ to mean the simultaneous |
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substitution of $t_0$ for $x_0$ up to $t_n$ for $x_n$ in $t$. |
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% |
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\paragraph{Reduction semantics} |
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Figure~\ref{fig:base-language-small-step} depicts the reduction |
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rules. The application rules \semlab{App} and \semlab{TyApp} |
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eliminates a lambda and type abstraction, respectively, by |
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substituting the argument for the parameter in their body computation |
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$M$. |
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The reduction relation $\reducesto : \CompCat \pto \CompCat$ is defined |
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on computation terms. Figure~\ref{fig:base-language-small-step} |
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depicts the reduction rules. The application rules \semlab{App} and |
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\semlab{TyApp} eliminates a lambda and type abstraction, respectively, |
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by substituting the argument for the parameter in their body |
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computation $M$. |
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% |
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Record splitting is handled by the \semlab{Split} rule: splitting on |
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some label $\ell$ binds the payload $V$ to $x$ and the remainder $W$ |
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@ -958,6 +1020,22 @@ Thus far we have defined the syntax, static semantics, and dynamic |
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semantics of \BCalc{}. In this section, we finish the definition of |
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\BCalc{} by stating and proving some standard metatheoretic properties |
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about the language. |
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% |
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We begin by showing that type substitutions preserve typability. |
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% |
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\begin{lemma}[Preservation of typing under type substitution] |
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Let $\sigma$ be any type substitution and $V$ be a value and $M$ a |
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computation such that $\typ{\Delta;\Gamma}{V : A}$ and |
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$\typ{\Delta;\Gamma}{M : C}$, then |
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$\typ{\Delta;\sigma~\Gamma}{\sigma~V : \sigma~A}$ and |
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$\typ{\Delta;\sigma~\Gamma}{\sigma~M : \sigma~C}$. |
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\end{lemma} |
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% |
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\begin{proof} |
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By induction on the typing derivations. |
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\end{proof} |
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% |
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\section{Primitive effect: general recursion} |
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\label{sec:base-language-recursion} |
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