# How to linearize the following logical constraints?

I am having trouble linearizing the following logical constraints.

$$x,y,z$$ are non negative continuous variables such that $$x=y+z$$, and $$A$$ is a positive parameter. I would like to linearize $$y= \left\{ \begin{array}{ll} x & \mbox{if } x \le A \\ A & \mbox{otherwise} \end{array} \right.$$ and $$z= \left\{ \begin{array}{ll} 0 & \mbox{if } x \le A \\ x-A & \mbox{otherwise} \end{array} \right.$$

Assume $$0 \le x \le U$$ for some constant $$U$$. Introduce binary decision variable $$\delta$$ and impose $$x=y+z$$ and indicator constraints: \begin{align} \delta=0 &\implies (0 \le x \le A \land y = x) \tag1\label1\\ \delta=1 &\implies (A \le x \le U \land y = A) \tag2\label2 \end{align}
You can linearize \eqref{1} and \eqref{2} as follows: \begin{align} A \delta \le x &\le A(1-\delta)+ U \delta \\ (A - U) \delta \le y - x &\le 0 \\ A \delta \le y &\le A \end{align}
Note that you can express the desired relationship as $$y=\min(x,A)$$. If you instead wanted to enforce only $$y\le\min(x,A)$$, you would omit $$\delta$$ and just impose \begin{align} y &\le x \\ y &\le A \end{align}