Proof by Contrapositive Is Wrong
Proof by contrapositive is wrong. There is a true, false and undecidable case also.
Let me explain how it works.
p. =>. q. does not necessarily imply not q ==> not p. There are three cases here. If not q is true, then p can also be undecidable. If p is undecidable in that case, then p ==> q is false. The meaning of the statement p ==> q is that p is never true without q also being true. You have to first prove that p is not undecidable. This is not being done in many proof by contra-positive based proofs in mathematics. First prove that p is not undecidable is not a step in Analysis I by Terence Tao for instance. Which letter of "if" do you not understand?