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Beta normal form : ウィキペディア英語版
Beta normal form
In the lambda calculus, a term is in beta normal form if no ''beta reduction'' is possible. A term is in beta-eta normal form if neither a beta reduction nor an ''eta reduction'' is possible. A term is in head normal form if there is no ''beta-redex in head position''.
==Beta reduction==
In the lambda calculus, a beta redex is a term of the form
: ((\mathbf x . A(x)) t)
where A(x) is a term (possibly) involving variable x.
A ''beta reduction'' is an application of the following rewrite rule to a beta redex
: ((\mathbf x . A(x)) t) \rightarrow A(t)
where A(t) is the result of substituting the term t for the variable x in the term A(x).
A beta reduction is in head position if it is of the following form:
* \lambda x_0 \ldots \lambda x_ . (\lambda x_i . A(x_i)) M_1 M_2 \ldots M_n \rightarrow
\lambda x_0 \ldots \lambda x_ . A(M_1) M_2 \ldots M_n , where i \geq 0, n \geq 1 .
Any reduction not in this form is an internal beta reduction.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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