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I wrote a formal proof in a software that might no support strong induction. So now I have to rewrite my proof. As I saw the principles of strong induction and simple induction are equivalent I would like to know if there is an standard procedure to transform strong induction proofs into simple induction proofs.

user1868607
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    You prove that simple induction implies that strong induction holds, and then you tack that on to the beginning of your proof using strong induction. In a system like Agda, where proofs are expressed as programs, this amounts to implementing strong induction in terms of simple induction and then using the resulting program in your proof. – Derek Elkins left SE Mar 25 '17 at 23:56
  • @DerekElkins: I think the OP wants a recipe for rewriting a formula $P(x)$ that can be proved for all natural numbers $x$ by strong induction into a formula $Q(x)$ that can be proved by regular induction. Of course this recipe is essentially spelled out in proving the validity of strong induction using regular induction. – hardmath Mar 27 '17 at 14:39
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    @hardmath I was being (very slightly) tongue-in-cheek which was why it was a comment and not an answer. What you describe corresponds to inlining the implementation of strong induction in terms of simple induction into the program corresponding to the proof of $P(x)$. In proof-theoretic terms, this would be a step of cut-elimination. – Derek Elkins left SE Mar 27 '17 at 14:51

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Just unpack and apply the proof of strong induction to the particular instance you need, which will give you a simple induction proof, where the predicate on which you are applying simple induction will have an extra quantifier as compared to if you use strong induction.

For example if you want to perform strong induction on a $1$-parameter predicate $P$, you would need to instead perform simple induction on $Q = ( \mathbb{N}\ n \mapsto \forall x \in \mathbb{N}_{<n}\ ( P(x) ))$. It is clear that $Q(0)$ is true, and I leave it to you to appropriately adjust the rest of your proof.

user21820
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  • Note that this particular formulation of strong induction using strict "$<$" and no "$+1$" can be generalized elegantly to transfinite induction, which is described here. – user21820 Mar 27 '17 at 14:37