Is there a procedure to transform strong induction proofs into simple induction proofs?

Question

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.

Answer 1

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.