header {* Cantor's Theorem *}
theory Cantor
imports Main
begin
text_raw {*
\footnote{This is an Isar version of the final example of the
Isabelle/HOL manual \cite{isabelle-HOL}.}
*}
text {*
Cantor's Theorem states that every set has more subsets than it has
elements. It has become a favorite basic example in pure
higher-order logic since it is so easily expressed: \[\all{f::\alpha
\To \alpha \To \idt{bool}} \ex{S::\alpha \To \idt{bool}}
\all{x::\alpha} f \ap x \not= S\]
Viewing types as sets, $\alpha \To \idt{bool}$ represents the
powerset of $\alpha$. This version of the theorem states that for
every function from $\alpha$ to its powerset, some subset is outside
its range. The Isabelle/Isar proofs below uses HOL's set theory,
with the type $\alpha \ap \idt{set}$ and the operator
$\idt{range}::(\alpha \To \beta) \To \beta \ap \idt{set}$.
*}
theorem "EX S. S ~: range (f :: 'a => 'a set)"
proof
let ?S = "{x. x ~: f x}"
show "?S ~: range f"
proof
assume "?S : range f"
then obtain y where "?S = f y" ..
then show False
proof (rule equalityCE)
assume "y : f y"
assume "y : ?S" then have "y ~: f y" ..
with `y : f y` show ?thesis by contradiction
next
assume "y ~: ?S"
assume "y ~: f y" then have "y : ?S" ..
with `y ~: ?S` show ?thesis by contradiction
qed
qed
qed
text {*
How much creativity is required? As it happens, Isabelle can prove
this theorem automatically using best-first search. Depth-first
search would diverge, but best-first search successfully navigates
through the large search space. The context of Isabelle's classical
prover contains rules for the relevant constructs of HOL's set
theory.
*}
theorem "EX S. S ~: range (f :: 'a => 'a set)"
by best
text {*
While this establishes the same theorem internally, we do not get
any idea of how the proof actually works. There is currently no way
to transform internal system-level representations of Isabelle
proofs back into Isar text. Writing intelligible proof documents
really is a creative process, after all.
*}
end