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cartesianproducts [2015/04/21 17:40] wikiadmin |
cartesianproducts [2015/04/21 17:43] wikiadmin |
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| ...would have their corresponding sets: | ...would have their corresponding sets: | ||
| - | \[ ENode \subseteq Node \] | + | \begin{equation*} ENode \subseteq Node \end{equation*} |
| - | \[ ELeaf \subseteq Leaf \] | + | \begin{equation*} ELeaf \subseteq Leaf \end{equation*} |
| Now, a user could want to indicate that ENode not only matches Node's, but actually matches **all** of them (same for ELeaf of Leaf's). He would **need** to specify the following constraints: | Now, a user could want to indicate that ENode not only matches Node's, but actually matches **all** of them (same for ELeaf of Leaf's). He would **need** to specify the following constraints: | ||
| - | \[ ENode \supseteq Node \] | + | \begin{equation*} ENode \supseteq Node \end{equation*} |
| - | \[ ELeaf \supseteq Leaf \] | + | \begin{equation*} ELeaf \supseteq Leaf \end{equation*} |
| (Note that we then have equality between the sets of the extractors and their corresponding types. This is the situation we have with case classes.) | (Note that we then have equality between the sets of the extractors and their corresponding types. This is the situation we have with case classes.) | ||
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| This is readily expressible thanks to the set operators: | This is readily expressible thanks to the set operators: | ||
| - | \[EvenNode \cap OddNode = \emptyset \] | + | \begin{equation*}EvenNode \cap OddNode = \emptyset \end{equation*} |
| - | \[EvenNode \cup OddNode = Node \] | + | \begin{equation*}EvenNode \cup OddNode = Node \end{equation*} |
| ==== Analyzing patterns ==== | ==== Analyzing patterns ==== | ||
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| The set of values for the scrutinee which **will** be matched by a given pattern can now be seen as a subset of $s$ expressed by constraints. Recall that each extractor $ex$ has a corresponding set $EX$ which defines all values which it matches. A pattern will successfully match a value if all extractors in it match what they are submitted with. In other words, for a pattern with calls to extractors $ex_1,ex_2,\ldots$ is a cartesian product of the same dimension as $s$, where the positions corresponding to the extractors are "replaced" with the proper sets: | The set of values for the scrutinee which **will** be matched by a given pattern can now be seen as a subset of $s$ expressed by constraints. Recall that each extractor $ex$ has a corresponding set $EX$ which defines all values which it matches. A pattern will successfully match a value if all extractors in it match what they are submitted with. In other words, for a pattern with calls to extractors $ex_1,ex_2,\ldots$ is a cartesian product of the same dimension as $s$, where the positions corresponding to the extractors are "replaced" with the proper sets: | ||
| - | \[ EX_1 \times B \times EX_2 \times \ldots \]. Note that by construction of the types $A,B,\ldots$, this is always a subset of $s$. | + | \begin{equation*} EX_1 \times B \times EX_2 \times \ldots \end{equation*}. Note that by construction of the types $A,B,\ldots$, this is always a subset of $s$. |
| === Completeness, disjointness and the like === | === Completeness, disjointness and the like === | ||