MATH.12:11 - SoTA-Echoing

Adopt the proof/construction correspondence explained by Philip Wadler in Propositions as Types, especially §3 and the paired proof and computation rules in §§6-7. Its useful contribution here is to reconstruct operations from proof structure rather than treat a logical consequence as a finished obtaining procedure. Sections :4.2-:4.3 and the function case use that distinction. The selected correspondence concerns a stated logic and typed computation; a different calculus can require different rules. Author’s paper.

Use Egbert Rijke’s Introduction to Homotopy Type Theory, §2.2 and §4.6, for function application and dependent pairs with their projections. These give :4.3 a way to retain the witness together with its dependent property. The pattern presents those constructions without requiring the broader univalent foundation. Book.

Compare current Lean’s Axioms and Computation with a blanket interpretation of every existence proof as executable witness production. Its distinction between proof erasure, computation and classical choice changes :4.4: trace the data-producing operation and the evaluation actually claimed. Classical reasoning can remain in a correctness argument for an independently computable function. Lean documentation.

For an implementation using program extraction, Rocq’s Program extraction explains how logical content, informative axioms and supplied realizations affect the extracted code. This supports checking the actual extraction assumptions rather than reading a successful proof as validation of arbitrary inserted implementation code. Such implementation work becomes useful when the receiver needs executable code; it is not required for the hand calculations here. Rocq 9.1 documentation.

Reopen the chosen method when a needed proof rule has no available computational interpretation, a representation prevents a required decision, a source result changes that limit, or a more affordable construction provides the same wanted output.