contraposition in intuitionistic logic

Question

I know that a sentence does not imply its contrapositive in institutionistic logic. I tried very hard to come up with a counter model to prove that but I failed. Can someone help me please? Any hints will be appreciated. Thanks so much.

Answer 1

Proof :

1) $A \to B$ --- premise

2) $\lnot B$ --- assumed [a]

3) $A$ --- assumed [b]

4) $B$ --- from 3) and 1) by $\to$-elim

5) $\bot$ --- from 4) and 2) by $\to$-elim

6) $\lnot A$ --- from 3) and 5) by $\to$-intro, discharging [b]

7) $\lnot B \to \lnot A$ --- from 2) and 6) by $\to$-intro, discharging [a].