theory HelloWorld
imports Main
begin
section{*Playing around with Isabelle*}
text{* creating a lemma with the name hello_world*}
lemma hello_world: "True" by simp
(*inspecting it*)
thm hello_world
text{* defining a string constant HelloWorld *}
definition HelloWorld :: "string" where
"HelloWorld \<equiv> ''Hello World!''"
(*reversing HelloWorld twice yilds HelloWorld again*)
theorem "rev (rev HelloWorld) = HelloWorld"
by (fact List.rev_rev_ident)
text{*now we delete the already proven List.rev_rev_ident lema and show it by hand*}
declare List.rev_rev_ident[simp del]
hide_fact List.rev_rev_ident
(*It's trivial since we can just 'execute' it*)
corollary "rev (rev HelloWorld) = HelloWorld"
apply(simp add: HelloWorld_def)
done
text{*does it hold in general?*}
theorem rev_rev_ident:"rev (rev l) = l"
proof(induction l)
case Nil thus ?case by simp
next
case (Cons l ls)
assume IH: "rev (rev ls) = ls"
have "rev (l#ls) = (rev ls) @ [l]" by simp
hence "rev (rev (l#ls)) = rev ((rev ls) @ [l])" by simp
also have "\<dots> = [l] @ rev (rev ls)" by simp
finally show "rev (rev (l#ls)) = l#ls" using IH by simp
qed
corollary "\<forall>(l::string). rev (rev l) = l" by(fastforce intro: rev_rev_ident)
end