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 \<^const>‹foo››

definition "bar = False"  ― ‹side remark on \<^const>‹bar››

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