author wenzelm Sat, 26 Dec 2015 12:28:47 +0100 changeset 61930 380cbe15cca5 parent 61929 b8e242e52c97 child 61931 ed47044ee6a6
modernized example;
```--- a/src/HOL/Isar_Examples/Cantor.thy	Thu Dec 24 09:42:49 2015 +0100
+++ b/src/HOL/Isar_Examples/Cantor.thy	Sat Dec 26 12:28:47 2015 +0100
@@ -1,5 +1,5 @@
(*  Title:      HOL/Isar_Examples/Cantor.thy
-    Author:     Markus Wenzel, TU Muenchen
+    Author:     Makarius
*)

section \<open>Cantor's Theorem\<close>
@@ -8,12 +8,49 @@
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 there is no surjection from
+  a set to its powerset.  The proof works by diagonalization.  E.g.\ see
+  \<^item> @{url "http://mathworld.wolfram.com/CantorDiagonalMethod.html"}
+  \<^item> @{url "https://en.wikipedia.org/wiki/Cantor's_diagonal_argument"}
+\<close>
+
+theorem Cantor: "\<not> (\<exists>f :: 'a \<Rightarrow> 'a set. \<forall>A :: 'a set. \<exists>x :: 'a. f x = A)"
+proof
+  assume "\<exists>f :: 'a \<Rightarrow> 'a set. \<forall>A. \<exists>x. f x = A"
+  then obtain f :: "'a \<Rightarrow> 'a set" where *: "\<forall>A. \<exists>x. f x = A" ..
+  let ?D = "{x. x \<notin> f x}"
+  from * obtain a where "f a = ?D" by blast
+  moreover have "a \<in> ?D \<longleftrightarrow> a \<notin> f a" by blast
+  ultimately show False by blast
+qed
+

-text \<open>Cantor's Theorem states that every set has more subsets than
-  it has elements.  It has become a favourite basic example in pure
-  higher-order logic since it is so easily expressed:
+text \<open>
+  Here are shorter proofs via automated proof tools, but without any hints why
+  and how it works.
+\<close>
+
+theorem "\<not> (\<exists>f :: 'a \<Rightarrow> 'a set. \<forall>A. \<exists>x. f x = A)"
+  by best
+
+theorem "\<not> (\<exists>f :: 'a \<Rightarrow> 'a set. \<forall>A. \<exists>x. f x = A)"
+  by force
+
+
+subsection \<open>Classic Isabelle/HOL example\<close>
+
+text \<open>
+  The following treatment of Cantor's Theorem follows the classic from the
+  early 1990s, e.g.\ see the file @{verbatim "92/HOL/ex/set.ML"} in Isabelle92
+  or @{cite \<open>\S18.7\<close> "paulson-isa-book"}. The old tactic scripts synthesize
+  key information of the proof by refinement of schematic goal states. In
+  contrast, the Isar proof needs to say explicitly what is proven.
+
+  \<^bigskip>
+  Cantor's Theorem states that every set has more subsets than it has
+  elements. It has become a favourite basic example in pure higher-order logic
+  since it is so easily expressed:

@{text [display]
\<open>\<forall>f::\<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool. \<exists>S::\<alpha> \<Rightarrow> bool. \<forall>x::\<alpha>. f x \<noteq> S\<close>}
@@ -21,8 +58,9 @@
Viewing types as sets, \<open>\<alpha> \<Rightarrow> bool\<close> represents the powerset of \<open>\<alpha>\<close>. This
version of the theorem states that for every function from \<open>\<alpha>\<close> to its
powerset, some subset is outside its range. The Isabelle/Isar proofs below
-  uses HOL's set theory, with the type \<open>\<alpha> set\<close> and the operator
-  \<open>range :: (\<alpha> \<Rightarrow> \<beta>) \<Rightarrow> \<beta> set\<close>.\<close>
+  uses HOL's set theory, with the type \<open>\<alpha> set\<close> and the operator \<open>range :: (\<alpha> \<Rightarrow>
+  \<beta>) \<Rightarrow> \<beta> set\<close>.
+\<close>

theorem "\<exists>S. S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
proof
@@ -36,7 +74,7 @@
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
+      with \<open>y \<in> f y\<close> show ?thesis by contradiction
next
assume "y \<notin> ?S"
assume "y \<notin> f y"
@@ -46,19 +84,15 @@
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>
+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```
```--- a/src/HOL/Isar_Examples/document/root.bib	Thu Dec 24 09:42:49 2015 +0100
+++ b/src/HOL/Isar_Examples/document/root.bib	Sat Dec 26 12:28:47 2015 +0100
@@ -65,6 +65,13 @@
title		= {The {Isabelle} Reference Manual},
institution	= CUCL}

+@Book{paulson-isa-book,
+  author	= {Lawrence C. Paulson},
+  title		= {Isabelle: A Generic Theorem Prover},
+  publisher	= {Springer},
+  year		= 1994,
+  note		= {LNCS 828}}
+
@TechReport{paulson-mutilated-board,
author = 	 {Lawrence C. Paulson},
title = 	 {A Simple Formalization and Proof for the Mutilated Chess Board},```