src/HOL/Isar_Examples/Cantor.thy
 author hoelzl Tue Mar 26 12:20:58 2013 +0100 (2013-03-26) changeset 51526 155263089e7b parent 37671 fa53d267dab3 child 55640 abc140f21caa permissions -rw-r--r--
move SEQ.thy and Lim.thy to Limits.thy
     1 (*  Title:      HOL/Isar_Examples/Cantor.thy

     2     Author:     Markus Wenzel, TU Muenchen

     3 *)

     4

     5 header {* Cantor's Theorem *}

     6

     7 theory Cantor

     8 imports Main

     9 begin

    10

    11 text_raw {* \footnote{This is an Isar version of the final example of

    12   the Isabelle/HOL manual \cite{isabelle-HOL}.}  *}

    13

    14 text {* Cantor's Theorem states that every set has more subsets than

    15   it has elements.  It has become a favorite basic example in pure

    16   higher-order logic since it is so easily expressed: $\all{f::\alpha   17 \To \alpha \To \idt{bool}} \ex{S::\alpha \To \idt{bool}}   18 \all{x::\alpha} f \ap x \not= S$

    19

    20   Viewing types as sets, $\alpha \To \idt{bool}$ represents the

    21   powerset of $\alpha$.  This version of the theorem states that for

    22   every function from $\alpha$ to its powerset, some subset is outside

    23   its range.  The Isabelle/Isar proofs below uses HOL's set theory,

    24   with the type $\alpha \ap \idt{set}$ and the operator

    25   $\idt{range}::(\alpha \To \beta) \To \beta \ap \idt{set}$. *}

    26

    27 theorem "EX S. S ~: range (f :: 'a => 'a set)"

    28 proof

    29   let ?S = "{x. x ~: f x}"

    30   show "?S ~: range f"

    31   proof

    32     assume "?S : range f"

    33     then obtain y where "?S = f y" ..

    34     then show False

    35     proof (rule equalityCE)

    36       assume "y : f y"

    37       assume "y : ?S" then have "y ~: f y" ..

    38       with y : f y show ?thesis by contradiction

    39     next

    40       assume "y ~: ?S"

    41       assume "y ~: f y" then have "y : ?S" ..

    42       with y ~: ?S show ?thesis by contradiction

    43     qed

    44   qed

    45 qed

    46

    47 text {* How much creativity is required?  As it happens, Isabelle can

    48   prove this theorem automatically using best-first search.

    49   Depth-first search would diverge, but best-first search successfully

    50   navigates through the large search space.  The context of Isabelle's

    51   classical prover contains rules for the relevant constructs of HOL's

    52   set theory.  *}

    53

    54 theorem "EX S. S ~: range (f :: 'a => 'a set)"

    55   by best

    56

    57 text {* While this establishes the same theorem internally, we do not

    58   get any idea of how the proof actually works.  There is currently no

    59   way to transform internal system-level representations of Isabelle

    60   proofs back into Isar text.  Writing intelligible proof documents

    61   really is a creative process, after all. *}

    62

    63 end