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sav08:untyped_lambda_calculus [2009/03/05 12:45] vkuncak |
sav08:untyped_lambda_calculus [2009/03/05 12:45] vkuncak |
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| \end{array}\] | \end{array}\] | ||
| That is, an expression is a variable, a lambda expression defining a function, or an application of one expression to another. | That is, an expression is a variable, a lambda expression defining a function, or an application of one expression to another. | ||
| + | |||
| ===== Beta reduction ===== | ===== Beta reduction ===== | ||
| Line 33: | Line 34: | ||
| The main rule of lambda calculus is beta reduction, which formalizes function application: | The main rule of lambda calculus is beta reduction, which formalizes function application: | ||
| \begin{equation*} | \begin{equation*} | ||
| - | (\lambda x.\, s) t\ \to\limits^{\beta}\ s[x:=t] | + | (\lambda x.\, s) t\ \mathop{\to}\limits^{\beta}\ s[x:=t] |
| \end{equation*} | \end{equation*} | ||
| where we assume capture-avoiding substitution. In general, we identify terms that differ only by renaming of fresh bound variables. | where we assume capture-avoiding substitution. In general, we identify terms that differ only by renaming of fresh bound variables. | ||