src/HOL/Library/BigO.thy
author wenzelm
Wed Jan 12 17:14:27 2011 +0100 (2011-01-12)
changeset 41528 276078f01ada
parent 38622 86fc906dcd86
child 41865 4e8483cc2cc5
permissions -rw-r--r--
eliminated global prems;
tuned proofs;
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(*  Title:      HOL/Library/BigO.thy
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    Authors:    Jeremy Avigad and Kevin Donnelly
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*)
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header {* Big O notation *}
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theory BigO
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imports Complex_Main Function_Algebras Set_Algebras
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begin
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text {*
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This library is designed to support asymptotic ``big O'' calculations,
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i.e.~reasoning with expressions of the form $f = O(g)$ and $f = g +
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O(h)$.  An earlier version of this library is described in detail in
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\cite{Avigad-Donnelly}.
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The main changes in this version are as follows:
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\begin{itemize}
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\item We have eliminated the @{text O} operator on sets. (Most uses of this seem
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  to be inessential.)
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\item We no longer use @{text "+"} as output syntax for @{text "+o"}
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\item Lemmas involving @{text "sumr"} have been replaced by more general lemmas 
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  involving `@{text "setsum"}.
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\item The library has been expanded, with e.g.~support for expressions of
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  the form @{text "f < g + O(h)"}.
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\end{itemize}
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See \verb,Complex/ex/BigO_Complex.thy, for additional lemmas that
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require the \verb,HOL-Complex, logic image.
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Note also since the Big O library includes rules that demonstrate set
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inclusion, to use the automated reasoners effectively with the library
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one should redeclare the theorem @{text "subsetI"} as an intro rule,
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rather than as an @{text "intro!"} rule, for example, using
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\isa{\isakeyword{declare}}~@{text "subsetI [del, intro]"}.
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*}
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subsection {* Definitions *}
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definition
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  bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set"  ("(1O'(_'))") where
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  "O(f::('a => 'b)) =
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      {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
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lemma bigo_pos_const: "(EX (c::'a::linordered_idom). 
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0")
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  apply simp
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  apply (rule_tac x = "1" in exI)
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  apply simp
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  apply (rule_tac x = "abs c" in exI)
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  apply auto
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  apply (subgoal_tac "c * abs(f x) <= abs c * abs (f x)")
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  apply (erule_tac x = x in allE)
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  apply force
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  apply (rule mult_right_mono)
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  apply (rule abs_ge_self)
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  apply (rule abs_ge_zero)
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  done
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lemma bigo_alt_def: "O(f) = 
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    {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
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  by (auto simp add: bigo_def bigo_pos_const)
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lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
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  apply (auto simp add: bigo_alt_def)
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  apply (rule_tac x = "ca * c" in exI)
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  apply (rule conjI)
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  apply (rule mult_pos_pos)
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  apply (assumption)+
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  apply (rule allI)
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  apply (drule_tac x = "xa" in spec)+
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  apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))")
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  apply (erule order_trans)
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  apply (simp add: mult_ac)
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  apply (rule mult_left_mono, assumption)
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  apply (rule order_less_imp_le, assumption)
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  done
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lemma bigo_refl [intro]: "f : O(f)"
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  apply(auto simp add: bigo_def)
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  apply(rule_tac x = 1 in exI)
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  apply simp
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  done
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lemma bigo_zero: "0 : O(g)"
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  apply (auto simp add: bigo_def func_zero)
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  apply (rule_tac x = 0 in exI)
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  apply auto
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  done
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lemma bigo_zero2: "O(%x.0) = {%x.0}"
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  apply (auto simp add: bigo_def) 
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  apply (rule ext)
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  apply auto
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  done
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lemma bigo_plus_self_subset [intro]: 
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  "O(f) \<oplus> O(f) <= O(f)"
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  apply (auto simp add: bigo_alt_def set_plus_def)
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  apply (rule_tac x = "c + ca" in exI)
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  apply auto
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  apply (simp add: ring_distribs func_plus)
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  apply (rule order_trans)
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  apply (rule abs_triangle_ineq)
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  apply (rule add_mono)
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  apply force
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  apply force
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done
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lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
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  apply (rule equalityI)
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  apply (rule bigo_plus_self_subset)
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  apply (rule set_zero_plus2) 
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  apply (rule bigo_zero)
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  done
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lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) \<oplus> O(g)"
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  apply (rule subsetI)
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  apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
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  apply (subst bigo_pos_const [symmetric])+
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  apply (rule_tac x = 
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    "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
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  apply (rule conjI)
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  apply (rule_tac x = "c + c" in exI)
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  apply (clarsimp)
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  apply (auto)
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  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
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  apply (erule_tac x = xa in allE)
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  apply (erule order_trans)
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  apply (simp)
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  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
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  apply (erule order_trans)
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  apply (simp add: ring_distribs)
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  apply (rule mult_left_mono)
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  apply assumption
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  apply (simp add: order_less_le)
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  apply (rule mult_left_mono)
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  apply (simp add: abs_triangle_ineq)
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  apply (simp add: order_less_le)
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  apply (rule mult_nonneg_nonneg)
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  apply (rule add_nonneg_nonneg)
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  apply auto
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  apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0" 
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     in exI)
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  apply (rule conjI)
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  apply (rule_tac x = "c + c" in exI)
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  apply auto
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  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
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  apply (erule_tac x = xa in allE)
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  apply (erule order_trans)
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  apply (simp)
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  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
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  apply (erule order_trans)
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  apply (simp add: ring_distribs)
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  apply (rule mult_left_mono)
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  apply (simp add: order_less_le)
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  apply (simp add: order_less_le)
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  apply (rule mult_left_mono)
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  apply (rule abs_triangle_ineq)
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  apply (simp add: order_less_le)
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  apply (rule mult_nonneg_nonneg)
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  apply (rule add_nonneg_nonneg)
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  apply (erule order_less_imp_le)+
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  apply simp
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  apply (rule ext)
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  apply (auto simp add: if_splits linorder_not_le)
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  done
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lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A \<oplus> B <= O(f)"
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  apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> O(f)")
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  apply (erule order_trans)
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  apply simp
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  apply (auto del: subsetI simp del: bigo_plus_idemp)
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  done
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lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> 
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    O(f + g) = O(f) \<oplus> O(g)"
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  apply (rule equalityI)
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  apply (rule bigo_plus_subset)
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  apply (simp add: bigo_alt_def set_plus_def func_plus)
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  apply clarify
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  apply (rule_tac x = "max c ca" in exI)
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  apply (rule conjI)
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  apply (subgoal_tac "c <= max c ca")
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  apply (erule order_less_le_trans)
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  apply assumption
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  apply (rule le_maxI1)
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  apply clarify
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  apply (drule_tac x = "xa" in spec)+
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  apply (subgoal_tac "0 <= f xa + g xa")
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  apply (simp add: ring_distribs)
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  apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
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  apply (subgoal_tac "abs(a xa) + abs(b xa) <= 
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      max c ca * f xa + max c ca * g xa")
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  apply (force)
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  apply (rule add_mono)
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  apply (subgoal_tac "c * f xa <= max c ca * f xa")
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  apply (force)
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  apply (rule mult_right_mono)
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  apply (rule le_maxI1)
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  apply assumption
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  apply (subgoal_tac "ca * g xa <= max c ca * g xa")
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  apply (force)
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  apply (rule mult_right_mono)
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  apply (rule le_maxI2)
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  apply assumption
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  apply (rule abs_triangle_ineq)
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  apply (rule add_nonneg_nonneg)
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  apply assumption+
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  done
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lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
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    f : O(g)" 
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  apply (auto simp add: bigo_def)
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  apply (rule_tac x = "abs c" in exI)
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  apply auto
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  apply (drule_tac x = x in spec)+
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  apply (simp add: abs_mult [symmetric])
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  done
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lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> 
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    f : O(g)" 
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  apply (erule bigo_bounded_alt [of f 1 g])
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  apply simp
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  done
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lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
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    f : lb +o O(g)"
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  apply (rule set_minus_imp_plus)
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  apply (rule bigo_bounded)
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  apply (auto simp add: diff_minus fun_Compl_def func_plus)
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  apply (drule_tac x = x in spec)+
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  apply force
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  apply (drule_tac x = x in spec)+
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  apply force
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  done
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lemma bigo_abs: "(%x. abs(f x)) =o O(f)" 
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  apply (unfold bigo_def)
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  apply auto
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  apply (rule_tac x = 1 in exI)
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  apply auto
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  done
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lemma bigo_abs2: "f =o O(%x. abs(f x))"
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  apply (unfold bigo_def)
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  apply auto
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  apply (rule_tac x = 1 in exI)
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  apply auto
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  done
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lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
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  apply (rule equalityI)
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  apply (rule bigo_elt_subset)
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  apply (rule bigo_abs2)
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  apply (rule bigo_elt_subset)
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  apply (rule bigo_abs)
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  done
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lemma bigo_abs4: "f =o g +o O(h) ==> 
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    (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
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  apply (drule set_plus_imp_minus)
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  apply (rule set_minus_imp_plus)
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  apply (subst fun_diff_def)
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proof -
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  assume a: "f - g : O(h)"
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  have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
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    by (rule bigo_abs2)
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  also have "... <= O(%x. abs (f x - g x))"
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    apply (rule bigo_elt_subset)
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    apply (rule bigo_bounded)
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    apply force
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    apply (rule allI)
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    apply (rule abs_triangle_ineq3)
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    done
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  also have "... <= O(f - g)"
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    apply (rule bigo_elt_subset)
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    apply (subst fun_diff_def)
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    apply (rule bigo_abs)
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    done
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  also from a have "... <= O(h)"
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    by (rule bigo_elt_subset)
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  finally show "(%x. abs (f x) - abs (g x)) : O(h)".
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qed
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lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" 
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  by (unfold bigo_def, auto)
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lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) \<oplus> O(h)"
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proof -
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  assume "f : g +o O(h)"
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  also have "... <= O(g) \<oplus> O(h)"
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    by (auto del: subsetI)
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  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
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    apply (subst bigo_abs3 [symmetric])+
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    apply (rule refl)
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    done
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  also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
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    by (rule bigo_plus_eq [symmetric], auto)
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  finally have "f : ...".
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  then have "O(f) <= ..."
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    by (elim bigo_elt_subset)
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  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
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    by (rule bigo_plus_eq, auto)
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  finally show ?thesis
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    by (simp add: bigo_abs3 [symmetric])
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qed
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lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)"
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  apply (rule subsetI)
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  apply (subst bigo_def)
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  apply (auto simp add: bigo_alt_def set_times_def func_times)
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  apply (rule_tac x = "c * ca" in exI)
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  apply(rule allI)
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  apply(erule_tac x = x in allE)+
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  apply(subgoal_tac "c * ca * abs(f x * g x) = 
avigad@16908
   320
      (c * abs(f x)) * (ca * abs(g x))")
avigad@16908
   321
  apply(erule ssubst)
avigad@16908
   322
  apply (subst abs_mult)
avigad@16908
   323
  apply (rule mult_mono)
avigad@16908
   324
  apply assumption+
avigad@16908
   325
  apply (rule mult_nonneg_nonneg)
avigad@16908
   326
  apply auto
avigad@16908
   327
  apply (simp add: mult_ac abs_mult)
wenzelm@22665
   328
  done
avigad@16908
   329
avigad@16908
   330
lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
avigad@16908
   331
  apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
avigad@16908
   332
  apply (rule_tac x = c in exI)
avigad@16908
   333
  apply auto
avigad@16908
   334
  apply (drule_tac x = x in spec)
avigad@16908
   335
  apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
avigad@16908
   336
  apply (force simp add: mult_ac)
avigad@16908
   337
  apply (rule mult_left_mono, assumption)
avigad@16908
   338
  apply (rule abs_ge_zero)
wenzelm@22665
   339
  done
avigad@16908
   340
avigad@16908
   341
lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
avigad@16908
   342
  apply (rule subsetD)
avigad@16908
   343
  apply (rule bigo_mult)
avigad@16908
   344
  apply (erule set_times_intro, assumption)
wenzelm@22665
   345
  done
avigad@16908
   346
avigad@16908
   347
lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
avigad@16908
   348
  apply (drule set_plus_imp_minus)
avigad@16908
   349
  apply (rule set_minus_imp_plus)
avigad@16908
   350
  apply (drule bigo_mult3 [where g = g and j = g])
nipkow@29667
   351
  apply (auto simp add: algebra_simps)
wenzelm@22665
   352
  done
avigad@16908
   353
wenzelm@41528
   354
lemma bigo_mult5:
wenzelm@41528
   355
  assumes "ALL x. f x ~= 0"
wenzelm@41528
   356
  shows "O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
wenzelm@41528
   357
proof
wenzelm@41528
   358
  fix h
wenzelm@41528
   359
  assume "h : O(f * g)"
wenzelm@41528
   360
  then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
wenzelm@41528
   361
    by auto
wenzelm@41528
   362
  also have "... <= O((%x. 1 / f x) * (f * g))"
wenzelm@41528
   363
    by (rule bigo_mult2)
wenzelm@41528
   364
  also have "(%x. 1 / f x) * (f * g) = g"
wenzelm@41528
   365
    apply (simp add: func_times) 
wenzelm@41528
   366
    apply (rule ext)
wenzelm@41528
   367
    apply (simp add: assms nonzero_divide_eq_eq mult_ac)
wenzelm@41528
   368
    done
wenzelm@41528
   369
  finally have "(%x. (1::'b) / f x) * h : O(g)" .
wenzelm@41528
   370
  then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
wenzelm@41528
   371
    by auto
wenzelm@41528
   372
  also have "f * ((%x. (1::'b) / f x) * h) = h"
wenzelm@41528
   373
    apply (simp add: func_times) 
wenzelm@41528
   374
    apply (rule ext)
wenzelm@41528
   375
    apply (simp add: assms nonzero_divide_eq_eq mult_ac)
wenzelm@41528
   376
    done
wenzelm@41528
   377
  finally show "h : f *o O(g)" .
avigad@16908
   378
qed
avigad@16908
   379
avigad@16908
   380
lemma bigo_mult6: "ALL x. f x ~= 0 ==>
haftmann@35028
   381
    O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)"
avigad@16908
   382
  apply (rule equalityI)
avigad@16908
   383
  apply (erule bigo_mult5)
avigad@16908
   384
  apply (rule bigo_mult2)
wenzelm@22665
   385
  done
avigad@16908
   386
avigad@16908
   387
lemma bigo_mult7: "ALL x. f x ~= 0 ==>
haftmann@35028
   388
    O(f * g) <= O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
avigad@16908
   389
  apply (subst bigo_mult6)
avigad@16908
   390
  apply assumption
avigad@16908
   391
  apply (rule set_times_mono3)
avigad@16908
   392
  apply (rule bigo_refl)
wenzelm@22665
   393
  done
avigad@16908
   394
avigad@16908
   395
lemma bigo_mult8: "ALL x. f x ~= 0 ==>
haftmann@35028
   396
    O(f * g) = O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
avigad@16908
   397
  apply (rule equalityI)
avigad@16908
   398
  apply (erule bigo_mult7)
avigad@16908
   399
  apply (rule bigo_mult)
wenzelm@22665
   400
  done
avigad@16908
   401
avigad@16908
   402
lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
berghofe@26814
   403
  by (auto simp add: bigo_def fun_Compl_def)
avigad@16908
   404
avigad@16908
   405
lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
avigad@16908
   406
  apply (rule set_minus_imp_plus)
avigad@16908
   407
  apply (drule set_plus_imp_minus)
avigad@16908
   408
  apply (drule bigo_minus)
avigad@16908
   409
  apply (simp add: diff_minus)
wenzelm@22665
   410
  done
avigad@16908
   411
avigad@16908
   412
lemma bigo_minus3: "O(-f) = O(f)"
wenzelm@41528
   413
  by (auto simp add: bigo_def fun_Compl_def)
avigad@16908
   414
avigad@16908
   415
lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
avigad@16908
   416
proof -
avigad@16908
   417
  assume a: "f : O(g)"
avigad@16908
   418
  show "f +o O(g) <= O(g)"
avigad@16908
   419
  proof -
avigad@16908
   420
    have "f : O(f)" by auto
berghofe@26814
   421
    then have "f +o O(g) <= O(f) \<oplus> O(g)"
avigad@16908
   422
      by (auto del: subsetI)
berghofe@26814
   423
    also have "... <= O(g) \<oplus> O(g)"
avigad@16908
   424
    proof -
avigad@16908
   425
      from a have "O(f) <= O(g)" by (auto del: subsetI)
avigad@16908
   426
      thus ?thesis by (auto del: subsetI)
avigad@16908
   427
    qed
wenzelm@41528
   428
    also have "... <= O(g)" by simp
avigad@16908
   429
    finally show ?thesis .
avigad@16908
   430
  qed
avigad@16908
   431
qed
avigad@16908
   432
avigad@16908
   433
lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
avigad@16908
   434
proof -
avigad@16908
   435
  assume a: "f : O(g)"
avigad@16908
   436
  show "O(g) <= f +o O(g)"
avigad@16908
   437
  proof -
avigad@16908
   438
    from a have "-f : O(g)" by auto
avigad@16908
   439
    then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
avigad@16908
   440
    then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
avigad@16908
   441
    also have "f +o (-f +o O(g)) = O(g)"
avigad@16908
   442
      by (simp add: set_plus_rearranges)
avigad@16908
   443
    finally show ?thesis .
avigad@16908
   444
  qed
avigad@16908
   445
qed
avigad@16908
   446
avigad@16908
   447
lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
avigad@16908
   448
  apply (rule equalityI)
avigad@16908
   449
  apply (erule bigo_plus_absorb_lemma1)
avigad@16908
   450
  apply (erule bigo_plus_absorb_lemma2)
wenzelm@22665
   451
  done
avigad@16908
   452
avigad@16908
   453
lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
avigad@16908
   454
  apply (subgoal_tac "f +o A <= f +o O(g)")
avigad@16908
   455
  apply force+
wenzelm@22665
   456
  done
avigad@16908
   457
avigad@16908
   458
lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
avigad@16908
   459
  apply (subst set_minus_plus [symmetric])
avigad@16908
   460
  apply (subgoal_tac "g - f = - (f - g)")
avigad@16908
   461
  apply (erule ssubst)
avigad@16908
   462
  apply (rule bigo_minus)
avigad@16908
   463
  apply (subst set_minus_plus)
avigad@16908
   464
  apply assumption
avigad@16908
   465
  apply  (simp add: diff_minus add_ac)
wenzelm@22665
   466
  done
avigad@16908
   467
avigad@16908
   468
lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
avigad@16908
   469
  apply (rule iffI)
avigad@16908
   470
  apply (erule bigo_add_commute_imp)+
wenzelm@22665
   471
  done
avigad@16908
   472
avigad@16908
   473
lemma bigo_const1: "(%x. c) : O(%x. 1)"
wenzelm@22665
   474
  by (auto simp add: bigo_def mult_ac)
avigad@16908
   475
avigad@16908
   476
lemma bigo_const2 [intro]: "O(%x. c) <= O(%x. 1)"
avigad@16908
   477
  apply (rule bigo_elt_subset)
avigad@16908
   478
  apply (rule bigo_const1)
wenzelm@22665
   479
  done
avigad@16908
   480
haftmann@35028
   481
lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
avigad@16908
   482
  apply (simp add: bigo_def)
avigad@16908
   483
  apply (rule_tac x = "abs(inverse c)" in exI)
avigad@16908
   484
  apply (simp add: abs_mult [symmetric])
wenzelm@22665
   485
  done
avigad@16908
   486
haftmann@35028
   487
lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
wenzelm@22665
   488
  by (rule bigo_elt_subset, rule bigo_const3, assumption)
avigad@16908
   489
haftmann@35028
   490
lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==> 
avigad@16908
   491
    O(%x. c) = O(%x. 1)"
wenzelm@22665
   492
  by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
avigad@16908
   493
avigad@16908
   494
lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
avigad@16908
   495
  apply (simp add: bigo_def)
avigad@16908
   496
  apply (rule_tac x = "abs(c)" in exI)
avigad@16908
   497
  apply (auto simp add: abs_mult [symmetric])
wenzelm@22665
   498
  done
avigad@16908
   499
avigad@16908
   500
lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
wenzelm@22665
   501
  by (rule bigo_elt_subset, rule bigo_const_mult1)
avigad@16908
   502
haftmann@35028
   503
lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%x. c * f x)"
avigad@16908
   504
  apply (simp add: bigo_def)
avigad@16908
   505
  apply (rule_tac x = "abs(inverse c)" in exI)
avigad@16908
   506
  apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
wenzelm@22665
   507
  done
avigad@16908
   508
haftmann@35028
   509
lemma bigo_const_mult4: "(c::'a::linordered_field) ~= 0 ==> 
avigad@16908
   510
    O(f) <= O(%x. c * f x)"
wenzelm@22665
   511
  by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
avigad@16908
   512
haftmann@35028
   513
lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==> 
avigad@16908
   514
    O(%x. c * f x) = O(f)"
wenzelm@22665
   515
  by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
avigad@16908
   516
haftmann@35028
   517
lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==> 
avigad@16908
   518
    (%x. c) *o O(f) = O(f)"
avigad@16908
   519
  apply (auto del: subsetI)
avigad@16908
   520
  apply (rule order_trans)
avigad@16908
   521
  apply (rule bigo_mult2)
avigad@16908
   522
  apply (simp add: func_times)
wenzelm@41528
   523
  apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
avigad@16908
   524
  apply (rule_tac x = "%y. inverse c * x y" in exI)
avigad@16908
   525
  apply (simp add: mult_assoc [symmetric] abs_mult)
avigad@16908
   526
  apply (rule_tac x = "abs (inverse c) * ca" in exI)
avigad@16908
   527
  apply (rule allI)
avigad@16908
   528
  apply (subst mult_assoc)
avigad@16908
   529
  apply (rule mult_left_mono)
avigad@16908
   530
  apply (erule spec)
avigad@16908
   531
  apply force
wenzelm@22665
   532
  done
avigad@16908
   533
avigad@16908
   534
lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
wenzelm@41528
   535
  apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
avigad@16908
   536
  apply (rule_tac x = "ca * (abs c)" in exI)
avigad@16908
   537
  apply (rule allI)
avigad@16908
   538
  apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
avigad@16908
   539
  apply (erule ssubst)
avigad@16908
   540
  apply (subst abs_mult)
avigad@16908
   541
  apply (rule mult_left_mono)
avigad@16908
   542
  apply (erule spec)
avigad@16908
   543
  apply simp
avigad@16908
   544
  apply(simp add: mult_ac)
wenzelm@22665
   545
  done
avigad@16908
   546
avigad@16908
   547
lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
avigad@16908
   548
proof -
avigad@16908
   549
  assume "f =o O(g)"
avigad@16908
   550
  then have "(%x. c) * f =o (%x. c) *o O(g)"
avigad@16908
   551
    by auto
avigad@16908
   552
  also have "(%x. c) * f = (%x. c * f x)"
avigad@16908
   553
    by (simp add: func_times)
avigad@16908
   554
  also have "(%x. c) *o O(g) <= O(g)"
avigad@16908
   555
    by (auto del: subsetI)
avigad@16908
   556
  finally show ?thesis .
avigad@16908
   557
qed
avigad@16908
   558
avigad@16908
   559
lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
avigad@16908
   560
by (unfold bigo_def, auto)
avigad@16908
   561
avigad@16908
   562
lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o 
avigad@16908
   563
    O(%x. h(k x))"
berghofe@26814
   564
  apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def
avigad@16908
   565
      func_plus)
avigad@16908
   566
  apply (erule bigo_compose1)
avigad@16908
   567
done
avigad@16908
   568
wenzelm@22665
   569
avigad@16908
   570
subsection {* Setsum *}
avigad@16908
   571
avigad@16908
   572
lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==> 
avigad@16908
   573
    EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
avigad@16908
   574
      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"  
avigad@16908
   575
  apply (auto simp add: bigo_def)
avigad@16908
   576
  apply (rule_tac x = "abs c" in exI)
wenzelm@17199
   577
  apply (subst abs_of_nonneg) back back
avigad@16908
   578
  apply (rule setsum_nonneg)
avigad@16908
   579
  apply force
ballarin@19279
   580
  apply (subst setsum_right_distrib)
avigad@16908
   581
  apply (rule allI)
avigad@16908
   582
  apply (rule order_trans)
avigad@16908
   583
  apply (rule setsum_abs)
avigad@16908
   584
  apply (rule setsum_mono)
avigad@16908
   585
  apply (rule order_trans)
avigad@16908
   586
  apply (drule spec)+
avigad@16908
   587
  apply (drule bspec)+
avigad@16908
   588
  apply assumption+
avigad@16908
   589
  apply (drule bspec)
avigad@16908
   590
  apply assumption+
avigad@16908
   591
  apply (rule mult_right_mono) 
avigad@16908
   592
  apply (rule abs_ge_self)
avigad@16908
   593
  apply force
wenzelm@22665
   594
  done
avigad@16908
   595
avigad@16908
   596
lemma bigo_setsum1: "ALL x y. 0 <= h x y ==> 
avigad@16908
   597
    EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
avigad@16908
   598
      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
avigad@16908
   599
  apply (rule bigo_setsum_main)
avigad@16908
   600
  apply force
avigad@16908
   601
  apply clarsimp
avigad@16908
   602
  apply (rule_tac x = c in exI)
avigad@16908
   603
  apply force
wenzelm@22665
   604
  done
avigad@16908
   605
avigad@16908
   606
lemma bigo_setsum2: "ALL y. 0 <= h y ==> 
avigad@16908
   607
    EX c. ALL y. abs(f y) <= c * (h y) ==>
avigad@16908
   608
      (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
wenzelm@22665
   609
  by (rule bigo_setsum1, auto)  
avigad@16908
   610
avigad@16908
   611
lemma bigo_setsum3: "f =o O(h) ==>
avigad@16908
   612
    (%x. SUM y : A x. (l x y) * f(k x y)) =o
avigad@16908
   613
      O(%x. SUM y : A x. abs(l x y * h(k x y)))"
avigad@16908
   614
  apply (rule bigo_setsum1)
avigad@16908
   615
  apply (rule allI)+
avigad@16908
   616
  apply (rule abs_ge_zero)
avigad@16908
   617
  apply (unfold bigo_def)
avigad@16908
   618
  apply auto
avigad@16908
   619
  apply (rule_tac x = c in exI)
avigad@16908
   620
  apply (rule allI)+
avigad@16908
   621
  apply (subst abs_mult)+
avigad@16908
   622
  apply (subst mult_left_commute)
avigad@16908
   623
  apply (rule mult_left_mono)
avigad@16908
   624
  apply (erule spec)
avigad@16908
   625
  apply (rule abs_ge_zero)
wenzelm@22665
   626
  done
avigad@16908
   627
avigad@16908
   628
lemma bigo_setsum4: "f =o g +o O(h) ==>
avigad@16908
   629
    (%x. SUM y : A x. l x y * f(k x y)) =o
avigad@16908
   630
      (%x. SUM y : A x. l x y * g(k x y)) +o
avigad@16908
   631
        O(%x. SUM y : A x. abs(l x y * h(k x y)))"
avigad@16908
   632
  apply (rule set_minus_imp_plus)
berghofe@26814
   633
  apply (subst fun_diff_def)
avigad@16908
   634
  apply (subst setsum_subtractf [symmetric])
avigad@16908
   635
  apply (subst right_diff_distrib [symmetric])
avigad@16908
   636
  apply (rule bigo_setsum3)
berghofe@26814
   637
  apply (subst fun_diff_def [symmetric])
avigad@16908
   638
  apply (erule set_plus_imp_minus)
wenzelm@22665
   639
  done
avigad@16908
   640
avigad@16908
   641
lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==> 
avigad@16908
   642
    ALL x. 0 <= h x ==>
avigad@16908
   643
      (%x. SUM y : A x. (l x y) * f(k x y)) =o
avigad@16908
   644
        O(%x. SUM y : A x. (l x y) * h(k x y))" 
avigad@16908
   645
  apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) = 
avigad@16908
   646
      (%x. SUM y : A x. abs((l x y) * h(k x y)))")
avigad@16908
   647
  apply (erule ssubst)
avigad@16908
   648
  apply (erule bigo_setsum3)
avigad@16908
   649
  apply (rule ext)
avigad@16908
   650
  apply (rule setsum_cong2)
avigad@16908
   651
  apply (subst abs_of_nonneg)
avigad@16908
   652
  apply (rule mult_nonneg_nonneg)
avigad@16908
   653
  apply auto
wenzelm@22665
   654
  done
avigad@16908
   655
avigad@16908
   656
lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
avigad@16908
   657
    ALL x. 0 <= h x ==>
avigad@16908
   658
      (%x. SUM y : A x. (l x y) * f(k x y)) =o
avigad@16908
   659
        (%x. SUM y : A x. (l x y) * g(k x y)) +o
avigad@16908
   660
          O(%x. SUM y : A x. (l x y) * h(k x y))" 
avigad@16908
   661
  apply (rule set_minus_imp_plus)
berghofe@26814
   662
  apply (subst fun_diff_def)
avigad@16908
   663
  apply (subst setsum_subtractf [symmetric])
avigad@16908
   664
  apply (subst right_diff_distrib [symmetric])
avigad@16908
   665
  apply (rule bigo_setsum5)
berghofe@26814
   666
  apply (subst fun_diff_def [symmetric])
avigad@16908
   667
  apply (drule set_plus_imp_minus)
avigad@16908
   668
  apply auto
wenzelm@22665
   669
  done
wenzelm@22665
   670
avigad@16908
   671
avigad@16908
   672
subsection {* Misc useful stuff *}
avigad@16908
   673
avigad@16908
   674
lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
berghofe@26814
   675
  A \<oplus> B <= O(f)"
avigad@16908
   676
  apply (subst bigo_plus_idemp [symmetric])
avigad@16908
   677
  apply (rule set_plus_mono2)
avigad@16908
   678
  apply assumption+
wenzelm@22665
   679
  done
avigad@16908
   680
avigad@16908
   681
lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
avigad@16908
   682
  apply (subst bigo_plus_idemp [symmetric])
avigad@16908
   683
  apply (rule set_plus_intro)
avigad@16908
   684
  apply assumption+
wenzelm@22665
   685
  done
avigad@16908
   686
  
haftmann@35028
   687
lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 ==> 
avigad@16908
   688
    (%x. c) * f =o O(h) ==> f =o O(h)"
avigad@16908
   689
  apply (rule subsetD)
avigad@16908
   690
  apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
avigad@16908
   691
  apply assumption
avigad@16908
   692
  apply (rule bigo_const_mult6)
avigad@16908
   693
  apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
avigad@16908
   694
  apply (erule ssubst)
avigad@16908
   695
  apply (erule set_times_intro2)
nipkow@23413
   696
  apply (simp add: func_times)
wenzelm@22665
   697
  done
avigad@16908
   698
avigad@16908
   699
lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
avigad@16908
   700
    f =o O(h)"
avigad@16908
   701
  apply (simp add: bigo_alt_def)
avigad@16908
   702
  apply auto
avigad@16908
   703
  apply (rule_tac x = c in exI)
avigad@16908
   704
  apply auto
avigad@16908
   705
  apply (case_tac "x = 0")
avigad@16908
   706
  apply simp
avigad@16908
   707
  apply (rule mult_nonneg_nonneg)
avigad@16908
   708
  apply force
avigad@16908
   709
  apply force
avigad@16908
   710
  apply (subgoal_tac "x = Suc (x - 1)")
wenzelm@17199
   711
  apply (erule ssubst) back
avigad@16908
   712
  apply (erule spec)
avigad@16908
   713
  apply simp
wenzelm@22665
   714
  done
avigad@16908
   715
avigad@16908
   716
lemma bigo_fix2: 
avigad@16908
   717
    "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==> 
avigad@16908
   718
       f 0 = g 0 ==> f =o g +o O(h)"
avigad@16908
   719
  apply (rule set_minus_imp_plus)
avigad@16908
   720
  apply (rule bigo_fix)
berghofe@26814
   721
  apply (subst fun_diff_def)
berghofe@26814
   722
  apply (subst fun_diff_def [symmetric])
avigad@16908
   723
  apply (rule set_plus_imp_minus)
avigad@16908
   724
  apply simp
berghofe@26814
   725
  apply (simp add: fun_diff_def)
wenzelm@22665
   726
  done
wenzelm@22665
   727
avigad@16908
   728
avigad@16908
   729
subsection {* Less than or equal to *}
avigad@16908
   730
wenzelm@19736
   731
definition
haftmann@35028
   732
  lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)"
wenzelm@21404
   733
    (infixl "<o" 70) where
wenzelm@19736
   734
  "f <o g = (%x. max (f x - g x) 0)"
avigad@16908
   735
avigad@16908
   736
lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
avigad@16908
   737
    g =o O(h)"
avigad@16908
   738
  apply (unfold bigo_def)
avigad@16908
   739
  apply clarsimp
avigad@16908
   740
  apply (rule_tac x = c in exI)
avigad@16908
   741
  apply (rule allI)
avigad@16908
   742
  apply (rule order_trans)
avigad@16908
   743
  apply (erule spec)+
wenzelm@22665
   744
  done
avigad@16908
   745
avigad@16908
   746
lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
avigad@16908
   747
      g =o O(h)"
avigad@16908
   748
  apply (erule bigo_lesseq1)
avigad@16908
   749
  apply (rule allI)
avigad@16908
   750
  apply (drule_tac x = x in spec)
avigad@16908
   751
  apply (rule order_trans)
avigad@16908
   752
  apply assumption
avigad@16908
   753
  apply (rule abs_ge_self)
wenzelm@22665
   754
  done
avigad@16908
   755
avigad@16908
   756
lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
wenzelm@22665
   757
    g =o O(h)"
avigad@16908
   758
  apply (erule bigo_lesseq2)
avigad@16908
   759
  apply (rule allI)
avigad@16908
   760
  apply (subst abs_of_nonneg)
avigad@16908
   761
  apply (erule spec)+
wenzelm@22665
   762
  done
avigad@16908
   763
avigad@16908
   764
lemma bigo_lesseq4: "f =o O(h) ==>
avigad@16908
   765
    ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
avigad@16908
   766
      g =o O(h)"
avigad@16908
   767
  apply (erule bigo_lesseq1)
avigad@16908
   768
  apply (rule allI)
avigad@16908
   769
  apply (subst abs_of_nonneg)
avigad@16908
   770
  apply (erule spec)+
wenzelm@22665
   771
  done
avigad@16908
   772
avigad@16908
   773
lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
avigad@16908
   774
  apply (unfold lesso_def)
avigad@16908
   775
  apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
avigad@16908
   776
  apply (erule ssubst)
avigad@16908
   777
  apply (rule bigo_zero)
avigad@16908
   778
  apply (unfold func_zero)
avigad@16908
   779
  apply (rule ext)
avigad@16908
   780
  apply (simp split: split_max)
wenzelm@22665
   781
  done
avigad@16908
   782
avigad@16908
   783
lemma bigo_lesso2: "f =o g +o O(h) ==>
avigad@16908
   784
    ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
avigad@16908
   785
      k <o g =o O(h)"
avigad@16908
   786
  apply (unfold lesso_def)
avigad@16908
   787
  apply (rule bigo_lesseq4)
avigad@16908
   788
  apply (erule set_plus_imp_minus)
avigad@16908
   789
  apply (rule allI)
avigad@16908
   790
  apply (rule le_maxI2)
avigad@16908
   791
  apply (rule allI)
berghofe@26814
   792
  apply (subst fun_diff_def)
avigad@16908
   793
  apply (case_tac "0 <= k x - g x")
avigad@16908
   794
  apply simp
avigad@16908
   795
  apply (subst abs_of_nonneg)
wenzelm@17199
   796
  apply (drule_tac x = x in spec) back
nipkow@29667
   797
  apply (simp add: algebra_simps)
avigad@16908
   798
  apply (subst diff_minus)+
avigad@16908
   799
  apply (rule add_right_mono)
avigad@16908
   800
  apply (erule spec)
avigad@16908
   801
  apply (rule order_trans) 
avigad@16908
   802
  prefer 2
avigad@16908
   803
  apply (rule abs_ge_zero)
nipkow@29667
   804
  apply (simp add: algebra_simps)
wenzelm@22665
   805
  done
avigad@16908
   806
avigad@16908
   807
lemma bigo_lesso3: "f =o g +o O(h) ==>
avigad@16908
   808
    ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
avigad@16908
   809
      f <o k =o O(h)"
avigad@16908
   810
  apply (unfold lesso_def)
avigad@16908
   811
  apply (rule bigo_lesseq4)
avigad@16908
   812
  apply (erule set_plus_imp_minus)
avigad@16908
   813
  apply (rule allI)
avigad@16908
   814
  apply (rule le_maxI2)
avigad@16908
   815
  apply (rule allI)
berghofe@26814
   816
  apply (subst fun_diff_def)
avigad@16908
   817
  apply (case_tac "0 <= f x - k x")
avigad@16908
   818
  apply simp
avigad@16908
   819
  apply (subst abs_of_nonneg)
wenzelm@17199
   820
  apply (drule_tac x = x in spec) back
nipkow@29667
   821
  apply (simp add: algebra_simps)
avigad@16908
   822
  apply (subst diff_minus)+
avigad@16908
   823
  apply (rule add_left_mono)
avigad@16908
   824
  apply (rule le_imp_neg_le)
avigad@16908
   825
  apply (erule spec)
avigad@16908
   826
  apply (rule order_trans) 
avigad@16908
   827
  prefer 2
avigad@16908
   828
  apply (rule abs_ge_zero)
nipkow@29667
   829
  apply (simp add: algebra_simps)
wenzelm@22665
   830
  done
avigad@16908
   831
haftmann@35028
   832
lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::linordered_field) ==>
avigad@16908
   833
    g =o h +o O(k) ==> f <o h =o O(k)"
avigad@16908
   834
  apply (unfold lesso_def)
avigad@16908
   835
  apply (drule set_plus_imp_minus)
wenzelm@17199
   836
  apply (drule bigo_abs5) back
berghofe@26814
   837
  apply (simp add: fun_diff_def)
avigad@16908
   838
  apply (drule bigo_useful_add)
avigad@16908
   839
  apply assumption
wenzelm@17199
   840
  apply (erule bigo_lesseq2) back
avigad@16908
   841
  apply (rule allI)
nipkow@29667
   842
  apply (auto simp add: func_plus fun_diff_def algebra_simps
avigad@16908
   843
    split: split_max abs_split)
wenzelm@22665
   844
  done
avigad@16908
   845
avigad@16908
   846
lemma bigo_lesso5: "f <o g =o O(h) ==>
avigad@16908
   847
    EX C. ALL x. f x <= g x + C * abs(h x)"
avigad@16908
   848
  apply (simp only: lesso_def bigo_alt_def)
avigad@16908
   849
  apply clarsimp
avigad@16908
   850
  apply (rule_tac x = c in exI)
avigad@16908
   851
  apply (rule allI)
avigad@16908
   852
  apply (drule_tac x = x in spec)
avigad@16908
   853
  apply (subgoal_tac "abs(max (f x - g x) 0) = max (f x - g x) 0")
nipkow@29667
   854
  apply (clarsimp simp add: algebra_simps) 
avigad@16908
   855
  apply (rule abs_of_nonneg)
avigad@16908
   856
  apply (rule le_maxI2)
wenzelm@22665
   857
  done
avigad@16908
   858
avigad@16908
   859
lemma lesso_add: "f <o g =o O(h) ==>
avigad@16908
   860
      k <o l =o O(h) ==> (f + k) <o (g + l) =o O(h)"
avigad@16908
   861
  apply (unfold lesso_def)
avigad@16908
   862
  apply (rule bigo_lesseq3)
avigad@16908
   863
  apply (erule bigo_useful_add)
avigad@16908
   864
  apply assumption
avigad@16908
   865
  apply (force split: split_max)
avigad@16908
   866
  apply (auto split: split_max simp add: func_plus)
wenzelm@22665
   867
  done
avigad@16908
   868
haftmann@29786
   869
lemma bigo_LIMSEQ1: "f =o O(g) ==> g ----> 0 ==> f ----> (0::real)"
huffman@31337
   870
  apply (simp add: LIMSEQ_iff bigo_alt_def)
haftmann@29786
   871
  apply clarify
haftmann@29786
   872
  apply (drule_tac x = "r / c" in spec)
haftmann@29786
   873
  apply (drule mp)
haftmann@29786
   874
  apply (erule divide_pos_pos)
haftmann@29786
   875
  apply assumption
haftmann@29786
   876
  apply clarify
haftmann@29786
   877
  apply (rule_tac x = no in exI)
haftmann@29786
   878
  apply (rule allI)
haftmann@29786
   879
  apply (drule_tac x = n in spec)+
haftmann@29786
   880
  apply (rule impI)
haftmann@29786
   881
  apply (drule mp)
haftmann@29786
   882
  apply assumption
haftmann@29786
   883
  apply (rule order_le_less_trans)
haftmann@29786
   884
  apply assumption
haftmann@29786
   885
  apply (rule order_less_le_trans)
haftmann@29786
   886
  apply (subgoal_tac "c * abs(g n) < c * (r / c)")
haftmann@29786
   887
  apply assumption
haftmann@29786
   888
  apply (erule mult_strict_left_mono)
haftmann@29786
   889
  apply assumption
haftmann@29786
   890
  apply simp
haftmann@29786
   891
done
haftmann@29786
   892
haftmann@29786
   893
lemma bigo_LIMSEQ2: "f =o g +o O(h) ==> h ----> 0 ==> f ----> a 
haftmann@29786
   894
    ==> g ----> (a::real)"
haftmann@29786
   895
  apply (drule set_plus_imp_minus)
haftmann@29786
   896
  apply (drule bigo_LIMSEQ1)
haftmann@29786
   897
  apply assumption
haftmann@29786
   898
  apply (simp only: fun_diff_def)
haftmann@29786
   899
  apply (erule LIMSEQ_diff_approach_zero2)
haftmann@29786
   900
  apply assumption
haftmann@29786
   901
done
haftmann@29786
   902
avigad@16908
   903
end