Confluence proof for a simple rewriting system
Okay, so let's consider the case that $r = \mathtt{if}\: t_1 \:\mathtt{then}\: t_2 \:\mathtt{else}\: t_3$, $s$ has been derived by applying the E-If rule and $t$ has been derived by applying the E-Funny rule: So $s = \mathtt{if}\: t&#39;_1 \:\mathtt{then}\: t_2 \:\mathtt{else}\: t_3$ where $t_1 \rightarrow t&#39;_1$ and $t = \mathtt{if}\: t_1 \:\mathtt{then}\: t&#39;_2 \:\mathtt{else}\: t_3$ where $t_2 \rightarrow t&#39;_2$.

The $u$ we're looking for is $u = \mathtt{if}\: t&#39;_1 \:\mathtt{then}\: t&#39;_2 \:\mathtt{else}\: t_3$. $s \rightarrow u$ follows from rule E-Funny and $t \rightarrow u$ follows from rule E-If.