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src/HOL/Isar_examples/Cantor.thy

author | wenzelm |

Sat, 30 Oct 1999 20:20:48 +0200 | |

changeset 7982 | d534b897ce39 |

parent 7955 | f30e08579481 |

child 9474 | b0ce3b7c9c26 |

permissions | -rw-r--r-- |

improved presentation;

(* Title: HOL/Isar_examples/Cantor.thy ID: $Id$ Author: Markus Wenzel, TU Muenchen *) header {* Cantor's Theorem *}; theory Cantor = Main:; 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 $\idt{range}::(\alpha \To \beta) \To \beta \ap \idt{set}$. \bigskip We first consider a slightly awkward version of the proof, with the innermost reasoning expressed quite naively. *}; 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"; thus False; proof; fix y; assume "?S = f y"; thus ?thesis; proof (rule equalityCE); assume in_S: "y : ?S"; assume in_fy: "y : f y"; from in_S; have notin_fy: "y ~: f y"; ..; from notin_fy in_fy; show ?thesis; by contradiction; next; assume notin_S: "y ~: ?S"; assume notin_fy: "y ~: f y"; from notin_S; have in_fy: "y : f y"; ..; from notin_fy in_fy; show ?thesis; by contradiction; qed; qed; qed; qed; text {* The following version of the proof essentially does the same reasoning, only that it is expressed more neatly. In particular, we change the order of assumptions introduced in the two cases of rule \name{equalityCE}, streamlining the flow of intermediate facts and avoiding explicit naming.\footnote{In general, neither the order of assumptions as introduced by \isacommand{assume}, nor the order of goals as solved by \isacommand{show} is of any significance. The basic logical structure has to be left intact, though. In particular, assumptions ``belonging'' to some goal have to be introduced \emph{before} its corresponding \isacommand{show}.} *}; 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"; thus False; proof; fix y; assume "?S = f y"; thus ?thesis; proof (rule equalityCE); assume "y : f y"; assume "y : ?S"; hence "y ~: f y"; ..; thus ?thesis; by contradiction; next; assume "y ~: f y"; assume "y ~: ?S"; hence "y : f y"; ..; thus ?thesis; by contradiction; qed; qed; qed; qed; text {* How much creativity is required? As it happens, Isabelle can prove this theorem automatically. The context of Isabelle's classical prover contains rules for most of the constructs of HOL's set theory. We must augment it with \name{equalityCE} to break up set equalities, and then apply best-first search. Depth-first search would diverge, but best-first search successfully navigates through the large search space. *}; theorem "EX S. S ~: range (f :: 'a => 'a set)"; by (best elim: equalityCE); 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;