# Proof by definition substitution, or by rewriting?

In a previous post, I discussed what I called “Proof by definition”. However, some authors reserve that term for the solution of problems such as  “Prove that every even number is divisible by two.”

Given that I was talking about something a little less trivial, it might have been better to call the kind of proof I was discussing Proof by definition substitution or some other description indicating that the solution becomes clear once you have substituted in the definitions.

On the page http://www.abstractmath.org/MM/MMFormsProof.htm Charles Wells describes this as “proof by rewriting according to the definition of the words in the theorem”. So perhaps proof by rewriting would be a short alternative. This term is, however, already used in Functional Programming, or at least I have found it used on the page
http://www.seas.upenn.edu/~cis500/current/sf/html/Basics.html,
so it might be necessary to check that there is no clash of ideas there.

Meanwhile, I have added some new questions to my “routine” question sheet
More practice with de finitions, proofs and examples
available from
http://www.maths.nott.ac.uk/personal/jff/Teaching/More-Proofs.pdf

Joel

### One response to “Proof by definition substitution, or by rewriting?”

1. David

To slightly play devil’s advocate (perhaps I’m just asking a question I haven’t yet decided is a valid one) is this nomenclature of “proof of defintion/rewriting” a helpful one? Do I often come across a proof where I don’t have to know what the technical terms used in the question mean, or at least some equivalent definition that’s useful in this context?

Are these really just proofs which requires only very simple logic once technical terms have had their meaning substituted? It strikes me that this distinction isn’t one which makes it a different sort of proof, like a direct proof by contradiction or a proof by induction are different. Both of these latter proof mechanisms are big ideas which can be considered as candidates for proving something, but might NOT work. Starting out by substituting in a definition of a term or two is never going to leave you with an approach which you can’t progress (okay, never might be a little strong, it might lead you up a long garden path very occasionally), but trying an inductive argument for a question for which it’s inappropriate will never work.

As a final comparison, a proof by induction might be very hard because it’s not clear why P(n) implies P(n+1), in fact it may even by that we need to do something exciting like P(n) implies P(n+5). Equally, however, the proof of P(n) implying P(n+1) might be as simple as one of our proofs by definition above. It strike me that the proof by definition ‘idea’ is more like the idea of factorizing a polynomial, i.e. a step in an argument, rather than a full blown proof approach which is where we normally use phrases like “Proof by XXXX”.

Perhaps I’m not arguing with any particular view here as I don’t think you’ve expressly suggested that “Proof by Definition” would be taught as a Section after the “Proof by induction” Section. But if he issue is one of choosing how to name an idea which is useful when we’re thinking about proving results I think I’m suggesting that “Proof by anything” is probably not a good name for this definition substitution.

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