Theory Document

theory Document
  imports Main
begin

section ‹Some section›

subsection ‹Some subsection›

subsection ‹Some subsubsection›

subsubsection ‹Some subsubsubsection›

paragraph ‹A paragraph.›

text ‹Informal bla bla.›

definition "foo = True"  ― ‹side remark on constfoo

definition "bar = False"  ― ‹side remark on constbar

lemma foo unfolding foo_def ..


paragraph ‹Another paragraph.›

text ‹See also cite‹\S3› in "isabelle-system".›


section ‹Formal proof of Cantor's theorem›

text_raw ‹\isakeeptag{proof}›
text ‹
  Cantor's Theorem states that there is no surjection from
  a set to its powerset.  The proof works by diagonalization.  E.g.\ see
   🌐‹http://mathworld.wolfram.com/CantorDiagonalMethod.html›
   🌐‹https://en.wikipedia.org/wiki/Cantor's_diagonal_argument›

theorem Cantor: "f :: 'a  'a set. A. x. A = f x"
proof
  assume "f :: 'a  'a set. A. x. A = f x"
  then obtain f :: "'a  'a set" where *: "A. x. A = f x" ..
  let ?D = "{x. x  f x}"
  from * obtain a where "?D = f a" by blast
  moreover have "a  ?D  a  f a" by blast
  ultimately show False by blast
qed


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›

end