header {* Various examples for transfer procedure *}
theory Transfer_Ex
imports Complex_Main
begin
(* nat to int *)
lemma ex1: "(x::nat) + y = y + x"
by auto
thm ex1 [transferred]
lemma ex2: "(a::nat) div b * b + a mod b = a"
by (rule mod_div_equality)
thm ex2 [transferred]
lemma ex3: "ALL (x::nat). ALL y. EX z. z >= x + y"
by auto
thm ex3 [transferred natint]
lemma ex4: "(x::nat) >= y \<Longrightarrow> (x - y) + y = x"
by auto
thm ex4 [transferred]
lemma ex5: "(2::nat) * (SUM i <= n. i) = n * (n + 1)"
by (induct n rule: nat_induct, auto)
thm ex5 [transferred]
theorem ex6: "0 <= (n::int) \<Longrightarrow> 2 * \<Sum>{0..n} = n * (n + 1)"
by (rule ex5 [transferred])
thm ex6 [transferred]
thm ex5 [transferred, transferred]
end