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I think such proof is quite standard however I can't make them as the case of addition and multiplication functions. (which can be done by first prove $\mathsf{Q}\vdash x+/*y=z$ then show such $x +/*y = z$ represents addition/multiplication in Robinson's $\mathsf{Q}$).

However, I get stuck on the case of $/$ and $\%$, should they follow the same procedure?

Noah Schweber
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qwerty
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  • Are you familiar with the completeness theorem, and how it lets you prove facts about provability in a theory (and so representability) by reasoning about models? If you are, that makes a lot of arguments much easier. – Noah Schweber Mar 21 '21 at 21:05
  • Yeah, I know we should first define a proper formula and to prove Q|- such formula. And this formula should be <-> to y = f(x) where f is the function I need to prove. However, the reminder and quot function, is quite hard to define their related formula, both of which should have the form like y = xq + r where certainly (y and x) will be the variable for the formula, and for each function, either q or r will be the third variable. Then if I can prove such Q can lead to such formula, the remains is obvious. However, how to state this clearly and formally makes me confused. – qwerty Mar 21 '21 at 21:51

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