src/HOL/Library/BigO.thy

 (*  Title:      HOL/Library/BigO.thy
-    ID:		$Id$
     Authors:    Jeremy Avigad and Kevin Donnelly
 *)
 
 header {* Big O notation *}
 
 theory BigO
-imports Plain "~~/src/HOL/Presburger" SetsAndFunctions
+imports Complex_Main SetsAndFunctions
 begin
 
 text {*
 This library is designed to support asymptotic ``big O'' calculations,
 i.e.~reasoning with expressions of the form $f = O(g)$ and $f = g +

   apply assumption
   apply (force split: split_max)
   apply (auto split: split_max simp add: func_plus)
   done
 
+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