Correct notational use of $:=$

Question

Suppose I use the notation $:=$ to define some symbol the first time it is introduced, e.g. $A:=\Im(f+f^2)$. If later on I make use/remind the reader of the symbol definition, e.g. $A=\Im(f+f^2)=\ldots$, should I use the notation "$:=$", or simply write "$=$" as I did above?

Answer 1

If you're hellbent on using "$:=$" to define symbols, then use it only when you define symbols. After the symbol is introduced, use $=$ as you normally would; else it looks like you're redefining the symbol, which temporarily confuses the reader, particularly when the expressions on the right hand side of "$:=$" are not the exact same thing.

Answer 2

This is more like a long comment.

We can view $:=$ as an imperative (its a "command"; it does something). In this case, it probably should not be repeated. We write $x := \tau$ once, and once only, since this has the effect of defining $x$. After this, we always write: "Since $x = \tau...$"

We can view $:=$ as a strengthening of equality ($x := \tau$ means its true that $x$ is defined to equal $\tau$.) In this case, we can actually reason about statements involving $:=.$ For example, supposing $x:=y$, we can deduce $x=y$. But assuming $x=y$, we cannot deduce $x:=y$. Under this interpretation, repeating $:=$ multiple times is perfectly reasonable.