Prove the all cheaters will be killed

Question

In a city there is a law against infidelity. It is postulated that there are women who have cheated on their husbands. Also, if a woman cheated on her husband, all the people in the city except her husband know this fact. Now husbands should just think logically without asking others or investigating on their own. Everyone understands the law that if a husband's wife cheated on him, then he must kill her by the next morning. Prove that eventually, after several days, all the wives who cheated will be killed.

My try : I proved it intuitive for two pairs of people but I can't change it to algebraic forms and prove it using induction.

Answer 1

I think, that the law should be

if husband finds out, that his wife cheated on him, then he must kill her by the next morning

Then the reasoning is easy:

• if there is one cheating wife, then her husband sees, that every other wifes are faithful, so his wife have to the one cheating. He kills her in first night.
• if there are $n>1$ cheating wifes, the cheated husbands see $n-1$ cheating wifes and don't have to think, that their wife is cheating on them. Also they think, that the cheating wives will be killed in $(n-1)$th night. But after that night nobody was killed (because cheated husbands thought, that only the other $n-1$ were cheated), so there have to be $n$th cheating wife. So in the $n$th night all $n$ wives are killed.

There is also more brutal way to kill all cheating wifes - burn all the witches! ;)

Edit

Interesting thing is that if there were no cheating wives (what is 'against the law', that there should be at least one), all innocent women would be killed by their husbands in the first night. So one wife have to 'sacrifice' herself and cheat on her husband to save all the other women.