Lower Bound: Mental Model

Start with the idea

Begin with the concrete question: Return the first index whose value is at least 4. For the visible input [1,2,4,4,7], the stated result is 2. Before reading code, name the smallest piece of information that must survive from one step to the next. Binary search maintains an interval that is still capable of containing the answer. State precisely why the discarded half cannot contain it. That is the mental model for Lower Bound; it gives you a reason for each update instead of a line of code to memorise.

The core mechanism is this: Maintain the first position that may still hold a value greater than or equal to target. Treat it as a claim to test after every operation. Write down the state before the first operation, update it once, and ask which part of the claim is still true. If an item is skipped, inserted, or removed, explain why that action cannot destroy an answer that should be retained. A useful trace has columns for the current input position, the state before the step, the action, and the state afterward. Use the visible sample until the expected result becomes inevitable rather than merely plausible.

Now inspect the Rust sketch, not just its final expression. The supplied Rust sketch is a starting point. Identify what it stores, how one iteration changes it, and whether it already solves the exact task or only demonstrates a related operation.