I read a paper which mentioned routine induction many times. But when I google it, there are nothing showed up.

I think routine induction means the induction on the structure, analyzing possible cases in proofs. am I right?

Just to make sure, I am asking it here.

  • $\begingroup$ Could you refer to the paper that you read? $\endgroup$ – nekketsuuu Mar 14 '17 at 6:39
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    $\begingroup$ It might be that routine is used in the non-technical sense. A routine induction is the same as a pedestrian induction. $\endgroup$ – Yuval Filmus Mar 14 '17 at 7:38
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    $\begingroup$ @nekketsuuu The paper is at the link gabbay.org.uk/papers/nomu-jv.pdf. $\endgroup$ – alim Mar 14 '17 at 8:14
  • $\begingroup$ @YuvalFilmus could you give an example of pedestrian induction? $\endgroup$ – alim Mar 14 '17 at 8:15
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    $\begingroup$ @alim It's just plain induction. Take any proof by induction in an introduction to proofs course. $\endgroup$ – Yuval Filmus Mar 14 '17 at 8:35

I'm going to give a bit more nuanced answer.

To begin, if you are going to read many programming language theory and related research papers, get used to the phrase "by structural induction". It's not uncommon that these three words are the entirety of a proof sketch in a paper. The majority of proofs in this area are very boring but also tedious. So boring and tedious that we've written programs to write these proofs for us. If a computer can quickly find these proofs fully formally with a one-line proof script, this elliptical style is moderately justified.

Sometimes an author will say that you can find a proof by "following your nose". This means that the structure of the theorem strongly suggests one (or at most a few) next steps. This (combined with some searching) is what's behind Coq's auto tactic, Agda's Agsy tool, and the Djinn tool. Structural induction is the "follow your nose" step you get when faced with the problem of proving something about all members of an inductively defined type like an abstract syntax tree. So an elliptical "by structural induction" is really the statement of "do the obvious thing; there are no surprises".

  • $\begingroup$ "very boring but also tedious" Aren't "boring" and "tedious" synonyms? $\endgroup$ – David Richerby Mar 14 '17 at 16:36
  • $\begingroup$ @DavidRicherby "Tedious" also means "time consuming", though arguably it subsumes "boring". I guess I could have said "but also long". Boring and long is a bad fit for a research paper which is why such proofs are effectively omitted. $\endgroup$ – Derek Elkins left SE Mar 14 '17 at 22:40

The word routine is used here in its non-technical sense: normal, usual, plain. It just means a proof by induction where nothing special happens.

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    $\begingroup$ With an implication of, "We're not going to write it down because, if you think about it for a moment or too, you'll see it." $\endgroup$ – David Richerby Mar 14 '17 at 8:58

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