(* Title: HOL/Complex/ex/Arithmetic_Series_Complex
ID: $Id$
Author: Benjamin Porter, 2006
*)
header {* Arithmetic Series for Reals *}
theory Arithmetic_Series_Complex
imports Complex_Main Arithmetic_Series
begin
lemma arith_series_real:
"(2::real) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
of_nat n * (a + (a + of_nat(n - 1)*d))"
proof -
have
"((1::real) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat(i)*d) =
of_nat(n) * (a + (a + of_nat(n - 1)*d))"
by (rule arith_series_general)
thus ?thesis by simp
qed
end