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theory BigO_Complex(* Title: HOL/Complex/ex/BigO_Complex.thy ID: $Id: BigO_Complex.thy,v 1.3 2008/05/07 08:57:22 berghofe Exp $ Authors: Jeremy Avigad and Kevin Donnelly *) header {* Big O notation -- continued *} theory BigO_Complex imports BigO Complex begin text {* Additional lemmas that require the \texttt{HOL-Complex} logic image. *} lemma bigo_LIMSEQ1: "f =o O(g) ==> g ----> 0 ==> f ----> (0::real)" apply (simp add: LIMSEQ_def bigo_alt_def) apply clarify apply (drule_tac x = "r / c" in spec) apply (drule mp) apply (erule divide_pos_pos) apply assumption apply clarify apply (rule_tac x = no in exI) apply (rule allI) apply (drule_tac x = n in spec)+ apply (rule impI) apply (drule mp) apply assumption apply (rule order_le_less_trans) apply assumption apply (rule order_less_le_trans) apply (subgoal_tac "c * abs(g n) < c * (r / c)") apply assumption apply (erule mult_strict_left_mono) apply assumption apply simp done lemma bigo_LIMSEQ2: "f =o g +o O(h) ==> h ----> 0 ==> f ----> a ==> g ----> (a::real)" apply (drule set_plus_imp_minus) apply (drule bigo_LIMSEQ1) apply assumption apply (simp only: fun_diff_def) apply (erule LIMSEQ_diff_approach_zero2) apply assumption done end
lemma bigo_LIMSEQ1:
[| f ∈ O(g); g ----> 0 |] ==> f ----> 0
lemma bigo_LIMSEQ2:
[| f ∈ g +o O(h); h ----> 0; f ----> a |] ==> g ----> a