MATH.4:11 - SoTA-Echoing

Question: how can a specification over inductively formed inputs produce a witness-building operation and a reusable correctness argument?

Adopt the induction and computation rules from Egbert Rijke’s Introduction to Homotopy Type Theory, §3.1, printed pp.19-22. Section 4.6, pp.33-34, supplies dependent pairs and their projections as one formal presentation of an output with dependent information. The pattern uses these construction questions in ordinary mathematical language; it does not require univalent foundations for the examples.

The current Theorem Proving in Lean 4, §8.3 gives an operative comparison: recursive definitions and induction follow the input constructors, with recursive calls on smaller constituent terms. Adapt that organization here by foregrounding the wanted witness and strengthening the result when a step cannot be built. The integer and tree constructions are authored examples of the method.

The book’s axiom-of-choice discussion shows why formal existence and executable witness production require separate attention: definitions that manufacture data through classical choice are noncomputable in that setting. This makes recovering the output rule consequential; classical reasoning about an independently computable operation can still be useful.

Direct enumeration or a supplied operation can answer a small instance with less construction work. Reopen the selected method when the input is no longer finite and inductively formed, a required choice lacks an obtaining operation, or execution cost calls for another algorithm. The set example adds the representative-independence question supplied by MATH.2: producing a witness from each expression and defining one function on identified inputs require different arguments.