theory Real

imports RComplete RealVector

begin

lemma field_le_epsilon:

  fixes x y :: "'a:: {number_ring,division_by_zero,linordered_field}"

  assumes e: "(!!e. 0 < e ==> x \<le> y + e)"

  shows "x \<le> y"

proof (rule ccontr)

  assume xy: "\<not> x \<le> y"

  hence "(x-y)/2 > 0"

    by (metis half_gt_zero le_iff_diff_le_0 linorder_not_le local.xy)

  hence "x \<le> y + (x - y) / 2"

    by (rule e [of "(x-y)/2"])

  also have "... = (x - y + 2*y)/2"

    by auto

       (metis add_less_cancel_left add_numeral_0_right class_semiring.add_c xy e

           diff_add_cancel gt_half_sum less_half_sum linorder_not_le number_of_Pls)

  also have "... = (x + y) / 2" 

    by auto

  also have "... < x" using xy 

    by auto

  finally have "x<x" .

  thus False

    by auto 

qed

end