MATH.1:1 - Problem frame

Use this pattern when you know elementary steps and their permitted connections, but still need mathematical objects for their finite combinations. Typical questions are: which steps form a permitted sequence, how can two sequences be joined, and which distinctions must remain available for a later operation?

The construction below makes paths from generating arrows. An arrow has a starting object and an ending object; these can be states, types or mathematical objects. A path is a finite ordered list of arrows whose adjacent endpoints agree. The resulting structure can describe formal words as well as possible routes through a process. Its mathematical laws come from the construction.

Start by writing two steps you want to join and their endpoints. If the first ends where the second starts, their ordered pair is already a useful first path. If the written endpoint hides a condition that changes whether the second step is available, refine the endpoint before joining them.

You need elementary sets, ordered lists and equality. If an existing structure already supplies the combinations and distinctions your question needs, use its operations directly. This pattern is useful when constructing or changing those operations is itself the difficulty.