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[cc1516.git] / deliverables / report / infRules.tex

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+\paragraph{Op2 expressions}
+Equation~\ref{eq:inferOp2} shows the inference rule for op2 expressions. 
+\textit{op2type} is a function which takes an operator and returns the 
+corresponding function type, i.e. $\textit{op2type}(\&\&) = Bool \rightarrow
+Bool \rightarrow Bool$. 
+
+\begin{equation} \label{eq:inferOp2}
+       \infer[Op2]
+               {\Gamma \vdash e_1 \oplus e_2 \Rightarrow 
+                       (\Gamma^{\star}, \star, \alpha^\star, (e_1' \oplus e_2')^\star)}
+               {\Gamma \vdash e_1 \Rightarrow (\Gamma^{\star_1},\star_1, \tau_1, e_1') 
+               &\Gamma^{\star_1} \vdash e_2 \Rightarrow (\Gamma^{\star_2},
+                       \star_2, \tau_2, e_2') 
+               &(\tau_1 \rightarrow \tau_2 \rightarrow \alpha) \unif
+                       \textit{op2type}(\oplus) = \star_3
+               &\star = \star_3 \cdot \star_2 \cdot \star_1
+               }
+\end{equation}
+
+\paragraph{Op1 expressions}
+\begin{equation} \label{eq:inferOp1}
+       \infer[Op1]
+               {\Gamma \vdash \ominus e \Rightarrow 
+                       (\Gamma^{\star}, \star, \alpha^\star, (\ominus e')^\star)}
+               {\Gamma \vdash e \Rightarrow (\star_1, \tau, e_1') 
+               &(\tau \rightarrow \alpha) \unif
+                       \textit{op1type}(\ominus) = \star_2
+               &\star = \star_2 \cdot \star_1
+               }
+\end{equation}
+
+\paragraph{Tuples}
+\begin{equation} \label{eq:tuples}
+       \infer[Tuple]
+               {\Gamma \vdash (e_1, e_2) \Rightarrow 
+                       (\Gamma^{\star}, \star, (\tau_1, \tau_2)^\star, 
+                               (e_1', e_2')^\star)}
+               {\Gamma \vdash e_1 \Rightarrow (\Gamma^{\star_1},\star_1, \tau_1, e_1') 
+               &\Gamma^{\star_1} \vdash e_2 \Rightarrow (\Gamma^{\star_2},
+                       \star_2, \tau_2, e_2') 
+               &\star = \star_3 \cdot \star_2 \cdot \star_1
+               }
+\end{equation}
+
+\paragraph{Literals}
+
+\begin{equation}
+       \infer[Int]{\Gamma \vdash n \Rightarrow (\Gamma, \star_0, Int, n)}{}
+\end{equation}
+
+\begin{equation}
+       \infer[Bool]{\Gamma \vdash b \Rightarrow (\Gamma, \star_0, Bool, b)}{}
+\end{equation}
+
+\begin{equation}
+       \infer[Char]{\Gamma \vdash c \Rightarrow (\Gamma, \star_0, Char, c)}{}
+\end{equation}
+
+\begin{equation}
+       \infer[\emptyset]
+               {\Gamma \vdash [] \Rightarrow (\Gamma, \star_0, \alpha, [])}{}
+\end{equation}
+
+\paragraph{If statements}
+\begin{equation}
+       \infer[If]
+               {\Gamma \vdash \underline{\textrm{if }} e 
+                       \underline{\textrm{ then }} s_1 
+                       \underline{\textrm{ else }} s_2
+                       \Rightarrow
+                       (\Gamma^\star, \star, \tau_e^\star, \underline{\textrm{if }} e' 
+                       \underline{\textrm{ then }} s_1' 
+                       \underline{\textrm{ else }} s_2')
+               }
+               {\Gamma \vdash e \Rightarrow (\Gamma^{\star_1}, \star_1, \tau_1, e')
+               &\tau_1 \unif \textrm{Bool} = \star_2
+               &\Gamma^{\star_2\cdot \star_1} \vdash s_1 \Rightarrow 
+                       (\Gamma^{\star_3}, \star_3, \tau_t, s_1')
+               &\Gamma^{\star_3} \vdash s_2 \Rightarrow 
+                       (\Gamma^{\star_4}, \star_4, \tau_e, s_2')
+               &\tau_t \unif \tau_e = \star_5
+               }
+\end{equation}
+
+\paragraph{While statements}
+\begin{equation}
+       \infer[While]
+               {\Gamma \vdash \underline{\textrm{while }} e \textrm{ } s 
+                       \Rightarrow
+                       (\Gamma^\star, \star, \tau_s^\star, 
+                               \underline{\textrm{while }} e' \textrm{ } s_1')
+               }
+               {\Gamma \vdash e \Rightarrow (\Gamma^{\star_1}, \star_1, \tau_1, e')
+               &\tau_1 \unif \textrm{Bool} = \star_2
+               &\Gamma^{\star_2\cdot \star_1} \vdash s \Rightarrow 
+                       (\Gamma^{\star_3}, \star_3, \tau_t, s')
+               &\star = \star_3 \cdot \star_2 \cdot \star_1
+               }
+\end{equation}
+
+\paragraph{Function Statements}
+\begin{equation} \label{eq:inferStmtApp0}
+       \infer[Stmt App 0]
+               {\Gamma \vdash \textit{id}().\textit{fs}^* \Rightarrow
+                       ((\Gamma^\star, \star, \textrm{Void}, 
+                               \textit{id}().\textit{fs}^*)}
+               {\Gamma(id) = \lfloor \tau^e \rfloor
+               &\texttt{fold apfs } \tau \textit{ fs}^* = \tau^r
+               }
+\end{equation}
+
+\begin{equation} \label{eq:inferAppN}
+       \infer[Stmt AppN]
+               {\Gamma \vdash \textit{id}(e^*).\textit{fs}^* \Rightarrow
+                       ((\Gamma^\star, \star, \textrm{Void}, 
+                               \textit{id}({e^*}').\textit{fs}^*)}
+               {\Gamma(id) = \lfloor \tau^e \rfloor
+               &\Gamma \vdash e^* \Rightarrow 
+                       (\Gamma^{\star_1}, \star_1, \tau^*, {e^*}')
+               &(\texttt{fold } (\rightarrow) \texttt{ } \alpha \texttt{ } \tau^*) 
+                       \unif \tau^e = \star_2
+               &\star = \star_2 \cdot \star_1
+               &\texttt{fold apfs } \tau \textit{ fs}^* = \tau^e
+               }
+\end{equation}
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