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theory Arithmetic_Series_Complex(* Title: HOL/Complex/ex/Arithmetic_Series_Complex ID: $Id: Arithmetic_Series_Complex.thy,v 1.2 2006/04/26 05:01:33 kleing Exp $ Author: Benjamin Porter, 2006 *) header {* Arithmetic Series for Reals *} theory Arithmetic_Series_Complex imports Complex_Main begin lemma arith_series_real: "(2::real) * (∑i∈{..<n}. a + of_nat i * d) = of_nat n * (a + (a + of_nat(n - 1)*d))" proof - have "((1::real) + 1) * (∑i∈{..<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
lemma arith_series_real:
2 * (∑i<n. a + real_of_nat i * d) =
real_of_nat n * (a + (a + real_of_nat (n - 1) * d))