theory Document
imports Main
begin
section \<open>Some section\<close>
subsection \<open>Some subsection\<close>
subsection \<open>Some subsubsection\<close>
subsubsection \<open>Some subsubsubsection\<close>
paragraph \<open>A paragraph.\<close>
text \<open>Informal bla bla.\<close>
definition "foo = True" \<comment> \<open>side remark on \<^const>\<open>foo\<close>\<close>
definition "bar = False" \<comment> \<open>side remark on \<^const>\<open>bar\<close>\<close>
lemma foo unfolding foo_def ..
paragraph \<open>Another paragraph.\<close>
text \<open>See also \<^cite>\<open>\<open>\S3\<close> in "isabelle-system"\<close>.\<close>
section \<open>Formal proof of Cantor's theorem\<close>
text_raw \<open>\isakeeptag{proof}\<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>\<open>http://mathworld.wolfram.com/CantorDiagonalMethod.html\<close>
\<^item> \<^url>\<open>https://en.wikipedia.org/wiki/Cantor's_diagonal_argument\<close>
\<close>
theorem Cantor: "\<nexists>f :: 'a \<Rightarrow> 'a set. \<forall>A. \<exists>x. A = f x"
proof
assume "\<exists>f :: 'a \<Rightarrow> 'a set. \<forall>A. \<exists>x. A = f x"
then obtain f :: "'a \<Rightarrow> 'a set" where *: "\<forall>A. \<exists>x. A = f x" ..
let ?D = "{x. x \<notin> f x}"
from * obtain a where "?D = f a" by blast
moreover have "a \<in> ?D \<longleftrightarrow> a \<notin> f a" by blast
ultimately show False by blast
qed
subsection \<open>Lorem ipsum dolor\<close>
text \<open>
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec id ipsum
sapien. Vivamus malesuada enim nibh, a tristique nisi sodales ac. Praesent
ut sem consectetur, interdum tellus ac, sodales nulla. Quisque vel diam at
risus tempus tempor eget a tortor. Suspendisse potenti. Nulla erat lacus,
dignissim sed volutpat nec, feugiat non leo. Nunc blandit et justo sed
venenatis. Donec scelerisque placerat magna, et congue nulla convallis vel.
Cras tristique dolor consequat dolor tristique rutrum. Suspendisse ultrices
sem nibh, et suscipit felis ultricies at. Aliquam venenatis est vel nulla
efficitur ornare. Lorem ipsum dolor sit amet, consectetur adipiscing elit.
\<close>
end