(* Title: HOL/Isar_Examples/Drinker.thy
Author: Makarius
*)
section \<open>The Drinker's Principle\<close>
theory Drinker
imports Main
begin
text \<open>
Here is another example of classical reasoning: the Drinker's Principle says
that for some person, if he is drunk, everybody else is drunk!
We first prove a classical part of de-Morgan's law.
\<close>
lemma de_Morgan:
assumes "\<not> (\<forall>x. P x)"
shows "\<exists>x. \<not> P x"
proof (rule classical)
assume "\<nexists>x. \<not> P x"
have "\<forall>x. P x"
proof
fix x show "P x"
proof (rule classical)
assume "\<not> P x"
then have "\<exists>x. \<not> P x" ..
with \<open>\<nexists>x. \<not> P x\<close> show ?thesis by contradiction
qed
qed
with \<open>\<not> (\<forall>x. P x)\<close> show ?thesis by contradiction
qed
theorem Drinker's_Principle: "\<exists>x. drunk x \<longrightarrow> (\<forall>x. drunk x)"
proof cases
assume "\<forall>x. drunk x"
then have "drunk a \<longrightarrow> (\<forall>x. drunk x)" for a ..
then show ?thesis ..
next
assume "\<not> (\<forall>x. drunk x)"
then have "\<exists>x. \<not> drunk x" by (rule de_Morgan)
then obtain a where "\<not> drunk a" ..
have "drunk a \<longrightarrow> (\<forall>x. drunk x)"
proof
assume "drunk a"
with \<open>\<not> drunk a\<close> show "\<forall>x. drunk x" by contradiction
qed
then show ?thesis ..
qed
end