(* Title: HOL/Isar_examples/Drinker.thy
ID: $Id$
Author: Makarius
*)
header {* The Drinker's Principle *}
theory Drinker imports Main begin
text {*
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.
*}
lemma deMorgan:
assumes "\<not> (\<forall>x. P x)"
shows "\<exists>x. \<not> P x"
using prems
proof (rule contrapos_np)
assume a: "\<not> (\<exists>x. \<not> P x)"
show "\<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 a show ?thesis by contradiction
qed
qed
qed
theorem Drinker's_Principle: "\<exists>x. drunk x \<longrightarrow> (\<forall>x. drunk x)"
proof cases
fix a assume "\<forall>x. drunk x"
then have "drunk a \<longrightarrow> (\<forall>x. drunk x)" ..
then show ?thesis ..
next
assume "\<not> (\<forall>x. drunk x)"
then have "\<exists>x. \<not> drunk x" by (rule deMorgan)
then obtain a where a: "\<not> drunk a" ..
have "drunk a \<longrightarrow> (\<forall>x. drunk x)"
proof
assume "drunk a"
with a show "\<forall>x. drunk x" by (contradiction)
qed
then show ?thesis ..
qed
end