I think you're right on the formalism, and I shouldn't have brought that in. I've been out of grad school a while and should clearly brush up on my theory.
My main line of reasoning is:
1. If I say "this algorithm time-complexity is O(X)", I mean that all possible cases are in O(X).
2. If all possible cases are in O(X), then the worst case is in O(X).
3. If the worst case is in O(X), then any other case will also be in O(X). This follows from our definition of "worst case".
4. Therefore, "all possible cases are in O(X)" <=> "worst case is in O(X)"
5. Therefore, "this algorithm time-complexity is O(X)" <=> "this algorithm time-complexity is worst-case O(X)"
If your thought is that 1) isn't a formal definition, fair enough. It may have been a convention we adopted within the department. But aside from that, I think the reasoning is solid.
My main line of reasoning is:
1. If I say "this algorithm time-complexity is O(X)", I mean that all possible cases are in O(X).
2. If all possible cases are in O(X), then the worst case is in O(X).
3. If the worst case is in O(X), then any other case will also be in O(X). This follows from our definition of "worst case".
4. Therefore, "all possible cases are in O(X)" <=> "worst case is in O(X)"
5. Therefore, "this algorithm time-complexity is O(X)" <=> "this algorithm time-complexity is worst-case O(X)"
If your thought is that 1) isn't a formal definition, fair enough. It may have been a convention we adopted within the department. But aside from that, I think the reasoning is solid.