I wan to use the destruct tactic to prove a statement by cases. I have read a couple of examples online and I'm confused. Could someone explain it better?
Here is a small example (there are other ways to solve it but try using destruct):
Inductive three := zero
| one
| two.
Lemma has2b2: forall a:three, a<>zero /\ a<>one -> a=two.
Now some examples online suggest doing the following:
intros. destruct a.
In which case I get:
3 subgoals H : zero <> zero /\ zero <> one
______________________________________(1/3)
zero = two
______________________________________(2/3)
one = two
______________________________________(3/3)
two = two
So, I want to prove that the first two cases are impossible. But the machine lists them as subgoals and wants me to PROVE them... which is impossible.
Summary: How to exactly discard the impossible cases?
I have seen some examples using inversion but I don't understand the procedure.
Finally, what happens if my lemma depends on several inductive types and I still want to cover ALL cases?