If I finish something and the object gets transformed into something else during this process, can this be expressed in english by a sentence similar to the following one?
I have completed my plan into/to a solution to the problem. (by adding some further steps that final manage to get the thing done)
Is into or to correct or does the formulation not work at all?
Neither formulation sounds correct to my ears. The problem is with 'completed'. Here are some suggestions.
I have expanded my plan into a solution to the problem.
(The plan was insufficient, perhaps because it was too 'small', and you have enlarged it enough so that it is now a solution to the problem.)
I have developed my plan into a solution to the problem.
(The plan has been changed in a more ambiguous fashion than above.)
I have adapted my plan into a solution to the problem.
(This one implies the plan was a solution to some other problem, and you have made modifications to it so that it is now a solution to this problem.)
There are many more. As you can see and as you said, when an object (the plan) is transformed by some process (developing, modifying, and so on) the correct particle is into.
"Completed into" and "completed to" are grammatically incorrect. The verb "complete" takes the form "X completes Y."
You are looking for something like "I have turned my plan into a solution...".
