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 de_Morgan:
assumes "¬ (∀x. P x)"
shows "∃x. ¬ P x"
proof (rule classical)
assume "¬ (∃x. ¬ P x)"
have "∀x. P x"
proof
fix x show "P x"
proof (rule classical)
assume "¬ P x"
then have "∃x. ¬ P x" ..
with `¬ (∃x. ¬ P x)` show ?thesis by contradiction
qed
qed
with `¬ (∀x. P x)` show ?thesis by contradiction
qed
theorem Drinker's_Principle: "∃x. drunk x --> (∀x. drunk x)"
proof cases
fix a assume "∀x. drunk x"
then have "drunk a --> (∀x. drunk x)" ..
then show ?thesis ..
next
assume "¬ (∀x. drunk x)"
then have "∃x. ¬ drunk x" by (rule de_Morgan)
then obtain a where a: "¬ drunk a" ..
have "drunk a --> (∀x. drunk x)"
proof
assume "drunk a"
with a show "∀x. drunk x" by contradiction
qed
then show ?thesis ..
qed
end