Formal proofs, especially in support of Mathematics, is not something that I work with - (I use a lot of intuition in analyzing my code). I found that the articles from The American Mathematical Society A Special Issue on Formal Proof are fairly good introductions to the subject of using Proof Assistants in Formalizing 100 Math Theorems.

Traditional mathematical proofs are written in a way to make them easily understood by mathematicians. Routine logical steps are omitted. An enormous amount of context is assumed on the part of the reader. Proofs, especially in topology and geometry, rely on intuitive arguments in situations where a trained mathematician would be capable of translating those intuitive arguments into a more rigorous argument.

A formal proof is a proof in which every logical inference has been checked all the way back to the fundamental axioms of mathematics. All the intermediate logical steps are supplied, without exception. No appeal is made to intuition, even if the translation from intuition to logic is routine. Thus, a formal proof is less intuitive, and yet less susceptible to logical errors.

Though the articles are focused on the application to mathematical proofs, they do give a background on languages that are continually mentioned on LtU (HOL, Coq, Isabelle, etc...).

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