(* Title: HOL/Isar_Examples/Cantor.thy
Author: Markus Wenzel, TU Muenchen
*)
header \<open>Cantor's Theorem\<close>
theory Cantor
imports Main
begin
text_raw \<open>\footnote{This is an Isar version of the final example of
the Isabelle/HOL manual @{cite "isabelle-HOL"}.}\<close>
text \<open>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
$\idt{range}::(\alpha \To \beta) \To \beta \ap \idt{set}$.\<close>
theorem "\<exists>S. S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
proof
let ?S = "{x. x \<notin> f x}"
show "?S \<notin> range f"
proof
assume "?S \<in> range f"
then obtain y where "?S = f y" ..
then show False
proof (rule equalityCE)
assume "y \<in> f y"
assume "y \<in> ?S"
then have "y \<notin> f y" ..
with \<open>y : f y\<close> show ?thesis by contradiction
next
assume "y \<notin> ?S"
assume "y \<notin> f y"
then have "y \<in> ?S" ..
with \<open>y \<notin> ?S\<close> show ?thesis by contradiction
qed
qed
qed
text \<open>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.\<close>
theorem "\<exists>S. S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
by best
text \<open>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.\<close>
end