Change proofs to the ebproof package
This commit is contained in:
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@ -1,4 +1,5 @@
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# Exclude these environments from syntax checking
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VerbEnvir { pgfpicture tikzpicture mzn nzn grammar proof }
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MathCmd { hypo infer1 infer2 infer3 }
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WipeArg { \mzninline:{} \nzninline:{} \Sem:{} \texttt:{} }
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@ -84,7 +84,7 @@ style=apa,
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% Proof Tree
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\usepackage[nounderscore]{syntax}
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\usepackage{bussproofs}
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\usepackage{ebproof}
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% Half Reif packages (maybe we should get rid of these)
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\usepackage[all]{xy}
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@ -308,57 +308,35 @@ suitable alpha renaming.
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\begin{figure*}
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\centering
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\begin{prooftree}
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\AxiomC{\(\ptinline{F(\(p_1, \ldots, p_k\)) = E;} \in \Prog\) where the \(p_i\)
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are fresh}
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\noLine{}
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\UnaryInfC{\Sem{\(a_i\)}{\Prog, \Env_{i-1}} \(\Rightarrow~ \tuple{v_i,
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\Env_i'}, \ \Env_0=\Env, \Env_i=\Env_i'\cup\{p_i\mapsto v_i\sep[]\}
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\quad \forall
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1 \leq
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i \leq k\)}
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\noLine{}
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\UnaryInfC{\Sem{E}{\Prog, \Env_k}
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\(\Rightarrow ~ \tuple{v, \Env'}\)}
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\RightLabel{(Call)}
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\UnaryInfC{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env} \(\Rightarrow\)
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\(\tuple{v, \Env'}\)}
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\end{prooftree}
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\hypo{\ptinline{F(\(p_1, \ldots, p_k\)) = E;} \in \Prog \text{~where the~} p_i \text{~are fresh}}
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\infer[no rule]1{\Sem{\(a_i\)}{\Prog, \Env_{i-1}} \Rightarrow \tuple{v_i,
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\Env_i'}, \ \Env_0=\Env, \Env_i=\Env_i'\cup\{p_i\mapsto v_i\sep[]\}, \forall 1 \leq i \leq k}
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\infer[no rule]1{\Sem{E}{\Prog, \Env_k} \Rightarrow \tuple{v, \Env'}}
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\infer1[(Call)]{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env_{0}} \Rightarrow \tuple{v, \Env'}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{\(\ptinline{F} \in\) Builtins}
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% \noLine
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\AxiomC{\Sem{\(a_i\)}{\Prog, \Env} \(\Rightarrow~ \tuple{v_i, \Env},
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\forall
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1 \leq
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i \leq k\)}
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% \(\ldots\), \Sem{\(a_k\)}{\Prog, \Env} \(\Rightarrow~\tuple{\Ctx_k,v_k}\)}
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\RightLabel{(CallBuiltin)}
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\BinaryInfC{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env} \(\Rightarrow\)
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\(\tuple{ \mathit{eval}(\ptinline{F}(v_1,\ldots, v_k)), \Env}\)}
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\end{prooftree}
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\hypo{\ptinline{F} \in \text{Builtins}}
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\hypo{\Sem{\(a_i\)}{\Prog, \Env} \Rightarrow \tuple{v_i, \Env}, \forall{} 1 \leq{} i \leq{} k}
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\infer2[(CallBuiltin)]{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env} \Rightarrow \tuple{ \mathit{eval}(\ptinline{F}(v_1,\ldots, v_k)), \Env}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{\(\ptinline{F(\(p_1, \ldots, p_k\));} \in \Prog\)}
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% \noLine
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\AxiomC{\Sem{\(a_i\)}{\Prog, \Env_{i-1}} \(\Rightarrow~ \tuple{v_i, \Env_i},
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~\forall
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1 \leq
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i \leq k\)}
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% \(\ldots\), \Sem{\(a_k\)}{\Prog, \Env} \(\Rightarrow~\tuple{\Ctx_k,v_k}\)}
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\RightLabel{(CallPrimitive)}
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\BinaryInfC{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env_0} \(\Rightarrow\)
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\(\tuple{ x, \{ x \mapsto \ptinline{F}(v_1,\ldots, v_k) \sep [] \} \cup \Env_k}\)}
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\hypo{\ptinline{F(\(p_1, \ldots, p_k\));} \in \Prog}
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\hypo{\Sem{\(a_i\)}{\Prog, \Env_{i-1}} \Rightarrow \tuple{v_i, \Env_i}, \forall 1 \leq i \leq k}
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\infer2[(CallPrimitive)]{\Sem{F(\(a_1, \ldots, a_k\))}{\Prog, \Env_0} \Rightarrow
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\tuple{ x, \{ x \mapsto \ptinline{F}(v_1,\ldots, v_k) \sep [] \} \cup \Env_k}}
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\end{prooftree}
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\caption{\label{fig:4-rewrite-to-nzn-calls} Rewriting rules for partial evaluation
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of \microzinc\ calls to \nanozinc.}
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\end{figure*}
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The rules in \cref{fig:4-rewrite-to-nzn-calls} handle function calls.
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The first rule (Call) evaluates a function call expression in the context of a
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\microzinc\ program \(\Prog\) and \nanozinc\ program \(\Env\). The rule first
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evaluates all actual parameter expressions \(a_i\), creating new contexts where
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the evaluation results are bound to the formal parameters \(p_i\). It then
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evaluates the function body \(\ptinline{E}\) on this context, and returns the
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result.
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The rules in \cref{fig:4-rewrite-to-nzn-calls} handle function calls. The first
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rule (Call) evaluates a function call expression in the context of a \microzinc\
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program \(\Prog\) and \nanozinc\ program \(\Env\). The rule first evaluates all
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actual parameter expressions \(a_i\), creating new contexts where the evaluation
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results are bound to the formal parameters \(p_i\). It then evaluates the
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function body \(\ptinline{E}\) on this context, and returns the result.
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The (CallBuiltin) rule applies to ``built-in'' functions that can be evaluated
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directly, such as arithmetic and Boolean operations on fixed values. The rule
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@ -374,52 +352,41 @@ simply the function applied to the evaluated parameters.
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\begin{figure*}
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\centering
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\begin{prooftree}
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\AxiomC{\Sem{\(\mathbf{I}\)}{\Prog, \Env} \(\Rightarrow (\Ctx, \Env')\)}
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\AxiomC{\Sem{\(E\)}{\Prog, \Env'} \(\Rightarrow \tuple{v, \Env''}\)}
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\AxiomC{\(x\) fresh}
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\RightLabel{(LetC)}
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\TrinaryInfC{\Sem{let \{ \(\mathbf{I}\) \} in \(E\)}{\Prog, \Env} \(\Rightarrow
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\tuple{x, \{ x \mapsto v \sep \Ctx \} \cup \Env''}\)}
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\end{prooftree}
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% \begin{prooftree}
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% \AxiomC{\Sem{\(E_1\)}{\Prog, \Env} \(\Rightarrow \Ctx_1, r_1\)}
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% \AxiomC{\Sem{\(E_2\)}{\Prog, \Env} \(\Rightarrow \Ctx_2, r_2\)}
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% \RightLabel{(Rel)}
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% \BinaryInfC{\Sem{\(E_1 \bowtie E_2\)}{\Prog, \Env} \(\Rightarrow \Ctx_1 \wedge \Ctx_2 \wedge (r_1 \bowtie r_2)\)}
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% \end{prooftree}
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\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env} \Rightarrow (\Ctx, \Env')}
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\hypo{\Sem{\(E\)}{\Prog, \Env'} \Rightarrow \tuple{v, \Env''}}
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\hypo{x \text{~fresh}}
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\infer3[(LetC)]{\Sem{let \{ \(\mathbf{I}\) \} in \(E\)}{\Prog, \Env} \Rightarrow
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\tuple{x, \{ x \mapsto v \sep \Ctx \} \cup \Env''}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{}
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\RightLabel{(Item0)}
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\UnaryInfC{\Sem{\(\epsilon\)}{\Prog, \Env} \(\Rightarrow
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(\ptinline{true}, \Env\))}
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\end{prooftree}
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\hypo{}
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\infer1[(Item0)]{\Sem{\(\epsilon\)}{\Prog, \Env} \Rightarrow \tuple{\ptinline{true}, \Env}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{\Sem{\(\mathbf{I}\)}{\Prog, \Env\ \cup\ \{x \mapsto\ \texttt{mkvar()} \sep\ []\}} \(\Rightarrow (\Ctx, \Env')\)}
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\RightLabel{(ItemT)}
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\UnaryInfC{\Sem{\(t:x, \mathbf{I}\)}{\Prog, \Env} \(\Rightarrow
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(\Ctx, \Env')\)}
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\end{prooftree}
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\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env\ \cup\ \{x \mapsto\ \texttt{mkvar()} \sep\ []\}} \Rightarrow \tuple{\Ctx, \Env'}}
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\infer1[(ItemT)]{\Sem{\(t:x, \mathbf{I}\)}{\Prog, \Env} \Rightarrow \tuple{\Ctx, \Env'}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{\Sem{\(E\)}{\Prog, \Env} \(\Rightarrow \tuple{v, \Env'}\)}
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\AxiomC{\Sem{\(\mathbf{I}\)}{\Prog, \Env' \cup\ \{x \mapsto\ v \sep\ [] \}} \(\Rightarrow (\Ctx, \Env'')\)}
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\RightLabel{(ItemTE)}
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\BinaryInfC{\Sem{\(t:x = E, \mathbf{I}\)}{\Prog, \Env} \(\Rightarrow
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(\Ctx, \Env'')\)}
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\end{prooftree}
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\hypo{\Sem{\(E\)}{\Prog, \Env} \Rightarrow \tuple{v, \Env'}}
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\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env' \cup\ \{x \mapsto\ v \sep\ [] \}} \Rightarrow \tuple{\Ctx, \Env''}}
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\infer2[(ItemTE)]{\Sem{\(t:x = E, \mathbf{I}\)}{\Prog, \Env} \Rightarrow \tuple{\Ctx, \Env''}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{\Sem{\(E\)}{\Prog, \Env} \(\Rightarrow \tuple{\left(v_{1}, \ldots, v_{n}\right), \Env'}\)}
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\noLine{}
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\UnaryInfC{\Sem{\(\mathbf{I}\)}{\Prog, \Env' \cup\ \{x_{1} \mapsto\ v_{1} \sep\ [] \} \cup\ \ldots\ \cup\ \{x_{n} \mapsto\ v_{n} \sep\ [] \}} \(\Rightarrow (\Ctx, \Env'')\)}
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\RightLabel{(ItemTD)}
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\UnaryInfC{\Sem{\(\left(t_{1}: x_{1}, \ldots, t_{n}: x_{n}\right) = E, \mathbf{I}\)}{\Prog, \Env} \(\Rightarrow
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(\Ctx, \Env'')\)}
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\end{prooftree}
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\hypo{\Sem{\(E\)}{\Prog, \Env} \Rightarrow \tuple{\left(v_{1}, \ldots, v_{n}\right), \Env'}}
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\infer[no rule]1{\Sem{\(\mathbf{I}\)}{\Prog, \Env' \cup\ \{x_{1} \mapsto\ v_{1} \sep\ [] \} \cup\ \ldots\ \cup\ \{x_{n} \mapsto\ v_{n} \sep\ [] \}} \Rightarrow \tuple{\Ctx, \Env''}}
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\infer1[(ItemTD)]{\Sem{\(\left(t_{1}: x_{1}, \ldots, t_{n}: x_{n}\right) = E, \mathbf{I}\)}{\Prog, \Env} \Rightarrow
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\tuple{\Ctx, \Env''}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{\Sem{\(C\)}{\Prog, \Env} \(\Rightarrow \tuple{v, \Env'}\)}
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\AxiomC{\Sem{\(\mathbf{I}\)}{\Prog, \Env'} \(\Rightarrow (\Ctx, \Env'')\)}
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\RightLabel{(ItemC)}
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\BinaryInfC{\Sem{\(\mbox{constraint~} C, \mathbf{I}\)}{\Prog, \Env} \(\Rightarrow
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(\{v\}\cup\Ctx, \Env'')\)}
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\hypo{\Sem{\(C\)}{\Prog, \Env} \Rightarrow \tuple{v, \Env'}}
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\hypo{\Sem{\(\mathbf{I}\)}{\Prog, \Env'} \Rightarrow \tuple{\Ctx, \Env''}}
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\infer2[(ItemC)]{\Sem{\(\mbox{constraint~} C, \mathbf{I}\)}{\Prog, \Env} \Rightarrow
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\tuple{\{v\}\cup\Ctx, \Env''}}
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\end{prooftree}
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\caption{\label{fig:4-rewrite-to-nzn-let} Rewriting rules for partial evaluation
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of \microzinc\ let expressions to \nanozinc.}
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@ -439,64 +406,57 @@ is the base case for a list of let items.
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\begin{figure*}
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\centering
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\begin{prooftree}
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\RightLabel{(IdC)}
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\AxiomC{\(x \in \langle ident \rangle\)}
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\AxiomC{\(v \in \langle literal \rangle\)}
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\BinaryInfC{\Sem{\(x\)}{\Prog, \{x \mapsto\ v \sep\ []\} \cup\ \Env} \(\Rightarrow \tuple{v, \{x \mapsto v \sep [] \} \cup \Env}\)}
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\end{prooftree}
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\hypo{x \in \syntax{<ident>}}
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\hypo{v \in \syntax{<literal>}}
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\infer2[(IdC)]{\Sem{\(x\)}{\Prog, \{x \mapsto\ v \sep\ []\} \cup\ \Env} \Rightarrow \tuple{v, \{x \mapsto v \sep [] \} \cup \Env}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\RightLabel{(IdX)}
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\AxiomC{\(x \in \langle ident \rangle\)}
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\AxiomC{\(v\)}
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\AxiomC{\(\phi\) otherwise}
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\TrinaryInfC{\Sem{\(x\)}{\Prog, \{x \mapsto\ v \sep\ \phi\ \} \cup\ \Env} \(\Rightarrow \tuple{x, \{x \mapsto v \sep \phi \} \cup \Env}\)}
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\end{prooftree}
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\hypo{x \in \syntax{<ident>}}
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\hypo{v}
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\hypo{\phi \text{~otherwise}}
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\infer3[(IdX)]{\Sem{\(x\)}{\Prog, \{x \mapsto\ v \sep\ \phi\ \} \cup\ \Env} \Rightarrow \tuple{x, \{x \mapsto v \sep \phi \} \cup \Env}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\RightLabel{(Const)}
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\AxiomC{\(c\) constant}
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\UnaryInfC{\Sem{c}{\Prog, \Env} \(\Rightarrow \tuple{c, \Env}\)}
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\end{prooftree}
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\hypo{c \text{~constant}}
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\infer1[(Const)]{\Sem{c}{\Prog, \Env} \Rightarrow \tuple{c, \Env}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\RightLabel{(Tuple)}
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\AxiomC{\Sem{\(E_{1}\)}{\Prog,\Env} \(\Rightarrow \tuple{v_{1}, \Env^{1}}\)}
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\AxiomC{\ldots}
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\AxiomC{\Sem{\(E_{n}\)}{\Prog,\Env^{n-1}} \(\Rightarrow \tuple{v_{n}, \Env^{n}}\)}
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\TrinaryInfC{\Sem{\(\left(E_{1}, \ldots, E_{n}\right)\)}{\Prog, \Env} \(\Rightarrow \tuple{x, \{x \mapsto \left(v_{1}, \ldots, v_{n}\right) \sep [] \} \cup \Env^{n})}\)}
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\end{prooftree}
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\hypo{\Sem{\(E_{i}\)}{\Prog,\Env^{i-1}} \Rightarrow \tuple{v_{i}, \Env^{i}}, \forall{} 1 \leq{} i \leq{} k}
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\infer1[(Tuple)]{\Sem{\(\left(E_{1}, \ldots, E_{k}\right)\)}{\Prog, \Env^{0}} \Rightarrow \tuple{x, \{x \mapsto \left(v_{1}, \ldots, v_{n}\right) \sep [] \} \cup \Env^{k}}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\RightLabel{(If\(_T\))}
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\AxiomC{\Sem{\(C\)}{\Prog, \Env} \(\Rightarrow\) \(\tuple{\ptinline{true}, \_}\)}
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\UnaryInfC{\Sem{if \(C\) then \(E_t\) else \(E_e\) endif}{\Prog, \Env} \(\Rightarrow\) \Sem{\(E_t\)}{\Prog, \Env}}
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\end{prooftree}
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\hypo{\Sem{\(C\)}{\Prog, \Env} \Rightarrow \tuple{\ptinline{true}, \_}}
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\infer1[(If\(_T\))]{\Sem{if \(C\) then \(E_t\) else \(E_e\) endif}{\Prog, \Env} \Rightarrow \Sem{\(E_t\)}{\Prog, \Env}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\RightLabel{(If\(_F\))}
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\AxiomC{\Sem{\(C\)}{\Prog, \Env} \(\Rightarrow\) \(\tuple{\ptinline{false}, \_}\)}
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\UnaryInfC{\Sem{if \(C\) then \(E_t\) else \(E_e\) endif}{\Prog, \Env} \(\Rightarrow\) \Sem{\(E_e\)}{\Prog, \Env}}
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\end{prooftree}
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\hypo{\Sem{\(C\)}{\Prog, \Env} \Rightarrow \tuple{\ptinline{false}, \_}}
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\infer1[(If\(_F\))]{\Sem{if \(C\) then \(E_t\) else \(E_e\) endif}{\Prog, \Env} \Rightarrow \Sem{\(E_e\)}{\Prog, \Env}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{\Sem{G}{\Prog,\Env} \(\Rightarrow \tuple{ \ptinline{true}, \_}\)}
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\AxiomC{\Sem{\(E\)}{\Prog, \Env} \(\Rightarrow \tuple {v, \Env'}\)}
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\RightLabel{(WhereT)}
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\BinaryInfC{\Sem{\([E ~|~ \mbox{where~} G]\)}{\Prog, \Env} \(\Rightarrow
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\tuple{[v], \Env'}\)}
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\end{prooftree}
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\hypo{\Sem{G}{\Prog,\Env} \Rightarrow \tuple{ \ptinline{true}, \_}}
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\hypo{\Sem{\(E\)}{\Prog, \Env} \Rightarrow \tuple {v, \Env'}}
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\infer2[(WhereT)]{\Sem{\([E ~|~ \mbox{where~} G]\)}{\Prog, \Env} \Rightarrow \tuple{[v], \Env'}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{\Sem{G}{\Prog,\Env} \(\Rightarrow \tuple{ \ptinline{false}, \_}\)}
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\RightLabel{(WhereF)}
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\UnaryInfC{\Sem{\([E ~|~ \mbox{where~} G]\)}{\Prog, \Env} \(\Rightarrow
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\tuple{[], \Env}\)}
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\end{prooftree}
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\hypo{\Sem{G}{\Prog,\Env} \Rightarrow \tuple{\ptinline{false}, \_}}
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\infer1[(WhereF)]{\Sem{\([E ~|~ \mbox{where~} G]\)}{\Prog, \Env} \Rightarrow \tuple{[], \Env}}
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\end{prooftree} \\
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\medskip
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\begin{prooftree}
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\AxiomC{\Sem{\(\mathit{PE}\)}{\Prog,\Env} \(\Rightarrow \tuple{S, \_}\)}
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\noLine\
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\UnaryInfC{\Sem{\([E ~|~ \mathit{GE} \mbox{~where~} G]\)}{\Prog, \Env\ \cup\ \{ x\mapsto\
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\hypo{\Sem{\(\mathit{PE}\)}{\Prog,\Env} \Rightarrow \tuple{S, \_}}
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\infer[no rule]1{\Sem{\([E ~|~ \mathit{GE} \mbox{~where~} G]\)}{\Prog, \Env\ \cup\ \{ x\mapsto\
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v \sep\ []\}}}
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\noLine\
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\UnaryInfC{\hspace{8em} \(\Rightarrow \tuple{A_v, \Env \cup \{x \mapsto v \sep []\} \cup \Env_v}, \forall
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v \in S \)}
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\RightLabel{(ListG)}
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\UnaryInfC{\Sem{\([E~|~ x \mbox{~in~} \mathit{PE}, \mathit{GE} \mbox{~where~} G]\)}{\Prog, \Env} \(\Rightarrow
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\tuple{\mbox{concat} [ A_v ~|~ v \in S ], \Env \cup \bigcup_{v \in S} \Env_v}\)}
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\infer[no rule]1{\hspace{8em} \Rightarrow \tuple{A_v, \Env \cup \{x \mapsto v \sep []\} \cup \Env_v}, \forall
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v \in S }
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\infer1[(ListG)]{\Sem{\([E~|~ x \mbox{~in~} \mathit{PE}, \mathit{GE} \mbox{~where~} G]\)}{\Prog, \Env} \Rightarrow
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\tuple{\mbox{concat} [ A_v ~|~ v \in S ], \Env \cup \bigcup_{v \in S} \Env_v}}
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\end{prooftree}
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\caption{\label{fig:4-rewrite-to-nzn-other} Rewriting rules for partial evaluation
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of other \microzinc\ expressions to \nanozinc.}
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