The style of coinductive reasoning is very different.
In general, in induction, one has a base case and an inductive case. Hence, you can only build "finite" objects (since you start with the base case and you can apply the induction rule a finite number of times)
In coinduction, you start with an infinite object, and you show that you can "progress" - that is, take a step in your computation. So, this shows that you can _reach_ states in finite time, even if the full structure is inaccessible, since you show progress on each state.
The phrase that I've heard is "induction is to prove what's true, coinduction is to prove what's not false"
Please do Google "practical coinduction" it has a wealth of examples.
In general, in induction, one has a base case and an inductive case. Hence, you can only build "finite" objects (since you start with the base case and you can apply the induction rule a finite number of times)
In coinduction, you start with an infinite object, and you show that you can "progress" - that is, take a step in your computation. So, this shows that you can _reach_ states in finite time, even if the full structure is inaccessible, since you show progress on each state.
The phrase that I've heard is "induction is to prove what's true, coinduction is to prove what's not false"
Please do Google "practical coinduction" it has a wealth of examples.