Explain concepts used in papers

[allo-]

Like

"Let x be ..." means x will be defined there, either explicit (you can code it directly) or as an element from an not fully known set, which should be explored in the remainder of the theorem.

"... then there exists a y ..." means y must exist, if the preconditions are met, but we do not neccessarly know how to find y.

And metaconcepts like:

In a paper the author can define anything he wants. There are many papers, which define "if CONDITION is met, the element is called admissable", but there is no general concept of admissable.
Authors also can redefine existing concepts like "in the following we assume all vectors are normed" (this means you need to make sure in your code, your vectors are normed). Some will even admit some easy going notation like "with a bit abusive notation we can write ..."