3.13

证明：若在系统 $\lambda \beta$ 中加入下述公理 $\text{(A)}\quad \lambda xy.x = \lambda xy. y,$ 则对任何的 $M, N \in \Lambda$， $\lambda \beta+(\text{A}) \vdash M = N$

证明

【方法一】

$\lambda \beta + (\text{A}) \vdash \lambda xy.x = \lambda xy. y$

$\Rightarrow (\lambda xy. x) MN = (\lambda xy. y)MN\cdots (\nu)$

$M = N \cdots(\beta)$

【方法二】

即证形式系统 $\lambda\beta + (\text{A})$ 矛盾，viz. $\left\{ \begin{array}{l} \lambda\beta + (\text{A}) \vdash \lambda x.x = \lambda x. xx \\ \lambda\beta \nvdash \lambda x.x =\lambda x. xx \end{array}\right.$ $\lambda xy.x = \lambda xy. y\cdots(\text{A})$

$\Rightarrow (\lambda xy. x)(yy) = (\lambda xy. y)(yy) \cdots(\nu)$

$\Rightarrow \lambda y. yy = (\lambda xy. y)(yy) \cdots(\beta)$

$\Rightarrow \lambda y. yy = \lambda y. y \cdots(\beta)$