常见定义

布尔和条件

$$\text{True} = \lambda x.\lambda y.x$$

$$\text{False} = \lambda x.\lambda y.y$$

$$\text{If} = \lambda b.\lambda x.\lambda y. b\ x\ y$$

例如，将 $\text{True}\ M\ N$ 归约：

$$\text{If}\ \text{True}\ M\ N \\ \downarrow_{\beta} \\ (\lambda b.\lambda x.\lambda y. b\ x\ y)\ \text{True}\ M\ N \\ \downarrow_{\beta} \\ (\lambda x.\lambda y. \text{True}\ x\ y)\ M\ N \\ \downarrow_{\beta} \\ (\lambda y. \text{True}\ M\ y)\ N \\ \downarrow_{\beta} \\ \text{True}\ M\ N \\ \downarrow_{\beta} \\ (\lambda x.\lambda y.x)\ M\ N \\ \downarrow_{\beta} \\ (\lambda y.M)\ N \\ \downarrow_{\beta} \\ M$$