如何在 Coq 中证明 (~Q -> ~P) - > (P -> Q)

Question

I am trying to prove (~Q -> ~P) - > (P -> Q) in coq, which is the reverse of the contrapositive theorem (P-> Q) (~Q -> ~P). Currently I was thinking about using the same logic of proving the contrapositive theorem like this:

unfold not. intros P Q. intros AB C. apply B. apply A. apply C.

However I am stuck at the beginning. Maybe I need additional axioms to prove the reverse of the contrapositive theorem. Can anyone help me?

Answer 1

Require Import Classical.

Lemma myproof :  forall (P Q : Prop), (~Q -> ~P) -> (P -> Q) . 
unfold not. intros P Q. intros A B.
destruct (classic Q) as [Q_holds|NQ_holds].
apply Q_holds.
apply False_ind.
apply A.
apply NQ_holds.
apply B.
Qed.

Answer 2

Yep, you need some extra power (classical logic) to do this. Put

Require Import Classical.

at the beginning of your file. Now when you have a proposition Q with

destruct (classic Q) as [Q_holds|NQ_holds].

it creates two subgoals: one when Q holds and one when ~Q holds.

Together with the theorem False_ind (that let you prove anything from False ) it should be enough to conclude your proof.