MATH.4:4.3 - Construct the step and strengthen what it needs

Take one constructor with smaller inputs. Assume that each smaller input already has an output satisfying the stated property. Give an operation that uses those outputs to build the required output for the whole input.

Show why the construction preserves the property. Check each branch that changes the returned object. When the step needs information absent from the proposed output, strengthen the result or generalize a fixed parameter, then revisit the base cases and affected steps.

For example, a tree-coloring step may need to choose either color for a subtree’s root. A construction that promises only a root of color zero leaves that step unsupported. A construction parameterized by the required root color supplies what the step uses.

To obtain an executable construction, each case distinction must be decidable from the supplied data, and each operation used to build an output must itself return. A step that says only “choose a suitable object” identifies further mathematical work unless a way to make that choice is already supplied.