I've just finished reading the paper Practical Coinduction by Kozen and Silva. What is the difference between induction over $\mathbb N$ and coinduction over $\mathbb N_\infty$?
From the paper, it seems that coinduction is almost the same as induction, but with more restrictions. In particular, it appears that most (or all) coinduction proofs over $\mathbb N_\infty$ are also valid induction proofs over $\mathbb N$.
The main restrictions the paper lists are opacity and guardedness. What examples are there of induction proofs over $\mathbb N$ that satisfy these properties (and are therefore valid coinduction proofs)? And what examples are there of induction proofs that don't satisfy these properties (and are therefore not valid coinductions)?
