# Does anyone actually publish "structured proofs"?

In 1995, Leslie Lamport published an essay in the American Mathematical Monthly titled "How to write a proof". In the essay, Lamport introduced the concept of a structured proof, in which the traditional high-level proof is augmented by a sequence of lower levels. Each level of proof expands each step of the higher level into substeps. The amount of detail at the lowest level is rather extreme -- Lamport's proof of the irrationality of the square root of 2 runs to 1.5 pages.

The essay has over 250 citations according to Google Scholar, but I have never seen a proof published in this format. In a PDF or on paper, the extreme detail could be overwhelming; but I think modern web publishing platforms could accommodate it very well (with hierarchical collapsible subsections for each part of the proof).

In any case, are there examples of proofs published in the format suggested by Lamport?

• I like the collapsible idea! Have you asked about this over at matheducators.stackexchange.com? It seems like this would be especially useful for students. Aug 31 '15 at 2:59
• Somewhere on the internet is a whole book full of structured proofs (à la Lamport, and maybe even with his coauthorship). As far as I remember, its goal is to prove the correctness of some software, but a good part of it is basic mathematics; can anyone find it? Sep 12 '15 at 14:25
• Ah, found that book! Thomas L. Rodeheffer, The Naiad Clock Protocol: Specification, Model Checking, and Correctness Proof, research.microsoft.com/apps/pubs/?id=183826 . Sep 12 '15 at 14:41
• Apparently research.microsoft.com/en-us/um/people/lamport/tla/… is a good introduction into structured proofs, with lots of examples. Sep 12 '15 at 14:44
• Everything so far in the comments and the one current answer indicates that you can find plenty of structured proofs, and every last one of them has Lamport's hand in it. So, apparently, nobody but Lamport and his coworkers at MS use it. Sep 12 '15 at 22:41

• Modern Coq proofs written in ssreflect (e.g., github.com/hivert/Coq-Combi ) are actually not that long! The problems, as far as I can discern them from (unfortunately) somewhat afar are (1) that the Coq format does not easily support human readability (it is very hard to follow a Coq proof on pen and paper), (2) that lack of function extensionality and (lack of) good support for setoids make a lot of mathematical constructions forbiddingly hard to deal with (forget about encoding a tuple of elements of $A$ as a map $\left\{1,2,\ldots,n\right\} \to A$; the ... Sep 12 '15 at 14:47