src/HOL/Library/BigO.thy
author haftmann
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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\item Lemmas involving @{text "sumr"} have been replaced by more general lemmas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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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 bigo :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(1O'(_'))")
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  where "O(f:: 'a \<Rightarrow> 'b) = {h. \<exists>c. \<forall>x. abs (h x) \<le> c * abs (f x)}"
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lemma bigo_pos_const:
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  "(\<exists>c::'a::linordered_idom. \<forall>x. abs (h x) \<le> c * abs (f x)) \<longleftrightarrow>
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    (\<exists>c. 0 < c \<and> (\<forall>x. abs (h x) \<le> 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) \<le> 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) = {h. \<exists>c. 0 < c \<and> (\<forall>x. abs (h x) \<le> 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 \<in> O(g) \<Longrightarrow> O(f) \<le> 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 simp
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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) \<le> ca * (c * abs (g xa))")
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  apply (erule order_trans)
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  apply (simp add: ac_simps)
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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 \<in> 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 \<in> 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(\<lambda>x. 0) = {\<lambda>x. 0}"
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  by (auto simp add: bigo_def)
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lemma bigo_plus_self_subset [intro]: "O(f) + O(f) \<subseteq> 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) + 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) \<subseteq> O(f) + 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 = "\<lambda>n. if abs (g n) \<le> (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 (subgoal_tac "c * abs (f xa + g xa) \<le> (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) \<le> 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: abs_triangle_ineq)
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  apply (simp add: order_less_le)
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  apply (rule_tac x = "\<lambda>n. if (abs (f n)) < abs (g 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 auto
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  apply (subgoal_tac "c * abs (f xa + g xa) \<le> (c + c) * abs (g xa)")
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  apply (erule_tac x = xa in allE)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   133
  apply (erule order_trans)
55821
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wenzelm
parents: 54863
diff changeset
   134
  apply simp
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   135
  apply (subgoal_tac "c * abs (f xa + g xa) \<le> c * (abs (f xa) + abs (g xa))")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   136
  apply (erule order_trans)
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   137
  apply (simp add: ring_distribs)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   138
  apply (rule mult_left_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   139
  apply (rule abs_triangle_ineq)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   140
  apply (simp add: order_less_le)
22665
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wenzelm
parents: 21404
diff changeset
   141
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   142
55821
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wenzelm
parents: 54863
diff changeset
   143
lemma bigo_plus_subset2 [intro]: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   144
  apply (subgoal_tac "A + B \<subseteq> O(f) + O(f)")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   145
  apply (erule order_trans)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   146
  apply simp
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   147
  apply (auto del: subsetI simp del: bigo_plus_idemp)
22665
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wenzelm
parents: 21404
diff changeset
   148
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   149
55821
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wenzelm
parents: 54863
diff changeset
   150
lemma bigo_plus_eq: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   151
  apply (rule equalityI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   152
  apply (rule bigo_plus_subset)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   153
  apply (simp add: bigo_alt_def set_plus_def func_plus)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   154
  apply clarify
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   155
  apply (rule_tac x = "max c ca" in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   156
  apply (rule conjI)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   157
  apply (subgoal_tac "c \<le> max c ca")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   158
  apply (erule order_less_le_trans)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   159
  apply assumption
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54230
diff changeset
   160
  apply (rule max.cobounded1)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   161
  apply clarify
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   162
  apply (drule_tac x = "xa" in spec)+
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   163
  apply (subgoal_tac "0 \<le> f xa + g xa")
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   164
  apply (simp add: ring_distribs)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   165
  apply (subgoal_tac "abs (a xa + b xa) \<le> abs (a xa) + abs (b xa)")
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   166
  apply (subgoal_tac "abs (a xa) + abs (b xa) \<le> max c ca * f xa + max c ca * g xa")
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   167
  apply force
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   168
  apply (rule add_mono)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   169
  apply (subgoal_tac "c * f xa \<le> max c ca * f xa")
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   170
  apply force
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   171
  apply (rule mult_right_mono)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54230
diff changeset
   172
  apply (rule max.cobounded1)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   173
  apply assumption
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   174
  apply (subgoal_tac "ca * g xa \<le> max c ca * g xa")
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   175
  apply force
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   176
  apply (rule mult_right_mono)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54230
diff changeset
   177
  apply (rule max.cobounded2)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   178
  apply assumption
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   179
  apply (rule abs_triangle_ineq)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   180
  apply (rule add_nonneg_nonneg)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   181
  apply assumption+
22665
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wenzelm
parents: 21404
diff changeset
   182
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   183
55821
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wenzelm
parents: 54863
diff changeset
   184
lemma bigo_bounded_alt: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> c * g x \<Longrightarrow> f \<in> O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   185
  apply (auto simp add: bigo_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   186
  apply (rule_tac x = "abs c" in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   187
  apply auto
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   188
  apply (drule_tac x = x in spec)+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   189
  apply (simp add: abs_mult [symmetric])
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   190
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   191
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   192
lemma bigo_bounded: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> g x \<Longrightarrow> f \<in> O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   193
  apply (erule bigo_bounded_alt [of f 1 g])
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   194
  apply simp
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   195
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   196
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   197
lemma bigo_bounded2: "\<forall>x. lb x \<le> f x \<Longrightarrow> \<forall>x. f x \<le> lb x + g x \<Longrightarrow> f \<in> lb +o O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   198
  apply (rule set_minus_imp_plus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   199
  apply (rule bigo_bounded)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 47445
diff changeset
   200
  apply (auto simp add: fun_Compl_def func_plus)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   201
  apply (drule_tac x = x in spec)+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   202
  apply force
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   203
  apply (drule_tac x = x in spec)+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   204
  apply force
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   205
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   206
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   207
lemma bigo_abs: "(\<lambda>x. abs (f x)) =o O(f)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   208
  apply (unfold bigo_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   209
  apply auto
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   210
  apply (rule_tac x = 1 in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   211
  apply auto
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   212
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   213
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   214
lemma bigo_abs2: "f =o O(\<lambda>x. abs (f x))"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   215
  apply (unfold bigo_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   216
  apply auto
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   217
  apply (rule_tac x = 1 in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   218
  apply auto
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   219
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   220
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   221
lemma bigo_abs3: "O(f) = O(\<lambda>x. abs (f x))"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   222
  apply (rule equalityI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   223
  apply (rule bigo_elt_subset)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   224
  apply (rule bigo_abs2)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   225
  apply (rule bigo_elt_subset)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   226
  apply (rule bigo_abs)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   227
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   228
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   229
lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. abs (f x)) =o (\<lambda>x. abs (g x)) +o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   230
  apply (drule set_plus_imp_minus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   231
  apply (rule set_minus_imp_plus)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   232
  apply (subst fun_diff_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   233
proof -
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   234
  assume a: "f - g \<in> O(h)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   235
  have "(\<lambda>x. abs (f x) - abs (g x)) =o O(\<lambda>x. abs (abs (f x) - abs (g x)))"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   236
    by (rule bigo_abs2)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   237
  also have "\<dots> \<subseteq> O(\<lambda>x. abs (f x - g x))"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   238
    apply (rule bigo_elt_subset)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   239
    apply (rule bigo_bounded)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   240
    apply force
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   241
    apply (rule allI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   242
    apply (rule abs_triangle_ineq3)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   243
    done
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   244
  also have "\<dots> \<subseteq> O(f - g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   245
    apply (rule bigo_elt_subset)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   246
    apply (subst fun_diff_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   247
    apply (rule bigo_abs)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   248
    done
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   249
  also from a have "\<dots> \<subseteq> O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   250
    by (rule bigo_elt_subset)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   251
  finally show "(\<lambda>x. abs (f x) - abs (g x)) \<in> O(h)".
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   252
qed
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   253
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   254
lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. abs (f x)) =o O(g)"
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   255
  by (unfold bigo_def, auto)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   256
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   257
lemma bigo_elt_subset2 [intro]: "f \<in> g +o O(h) \<Longrightarrow> O(f) \<subseteq> O(g) + O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   258
proof -
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   259
  assume "f \<in> g +o O(h)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   260
  also have "\<dots> \<subseteq> O(g) + O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   261
    by (auto del: subsetI)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   262
  also have "\<dots> = O(\<lambda>x. abs (g x)) + O(\<lambda>x. abs (h x))"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   263
    apply (subst bigo_abs3 [symmetric])+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   264
    apply (rule refl)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   265
    done
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   266
  also have "\<dots> = O((\<lambda>x. abs (g x)) + (\<lambda>x. abs (h x)))"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   267
    by (rule bigo_plus_eq [symmetric]) auto
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   268
  finally have "f \<in> \<dots>" .
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   269
  then have "O(f) \<subseteq> \<dots>"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   270
    by (elim bigo_elt_subset)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   271
  also have "\<dots> = O(\<lambda>x. abs (g x)) + O(\<lambda>x. abs (h x))"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   272
    by (rule bigo_plus_eq, auto)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   273
  finally show ?thesis
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   274
    by (simp add: bigo_abs3 [symmetric])
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   275
qed
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   276
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   277
lemma bigo_mult [intro]: "O(f)*O(g) \<subseteq> O(f * g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   278
  apply (rule subsetI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   279
  apply (subst bigo_def)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   280
  apply (auto simp add: bigo_alt_def set_times_def func_times)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   281
  apply (rule_tac x = "c * ca" in exI)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   282
  apply (rule allI)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   283
  apply (erule_tac x = x in allE)+
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   284
  apply (subgoal_tac "c * ca * abs (f x * g x) = (c * abs (f x)) * (ca * abs (g x))")
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   285
  apply (erule ssubst)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   286
  apply (subst abs_mult)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   287
  apply (rule mult_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   288
  apply assumption+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   289
  apply auto
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   290
  apply (simp add: ac_simps abs_mult)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   291
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   292
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   293
lemma bigo_mult2 [intro]: "f *o O(g) \<subseteq> O(f * g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   294
  apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   295
  apply (rule_tac x = c in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   296
  apply auto
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   297
  apply (drule_tac x = x in spec)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   298
  apply (subgoal_tac "abs (f x) * abs (b x) \<le> abs (f x) * (c * abs (g x))")
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   299
  apply (force simp add: ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   300
  apply (rule mult_left_mono, assumption)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   301
  apply (rule abs_ge_zero)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   302
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   303
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   304
lemma bigo_mult3: "f \<in> O(h) \<Longrightarrow> g \<in> O(j) \<Longrightarrow> f * g \<in> O(h * j)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   305
  apply (rule subsetD)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   306
  apply (rule bigo_mult)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   307
  apply (erule set_times_intro, assumption)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   308
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   309
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   310
lemma bigo_mult4 [intro]: "f \<in> k +o O(h) \<Longrightarrow> g * f \<in> (g * k) +o O(g * h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   311
  apply (drule set_plus_imp_minus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   312
  apply (rule set_minus_imp_plus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   313
  apply (drule bigo_mult3 [where g = g and j = g])
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 27487
diff changeset
   314
  apply (auto simp add: algebra_simps)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   315
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   316
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   317
lemma bigo_mult5:
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   318
  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   319
  assumes "\<forall>x. f x \<noteq> 0"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   320
  shows "O(f * g) \<subseteq> f *o O(g)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   321
proof
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   322
  fix h
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   323
  assume "h \<in> O(f * g)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   324
  then have "(\<lambda>x. 1 / (f x)) * h \<in> (\<lambda>x. 1 / f x) *o O(f * g)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   325
    by auto
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   326
  also have "\<dots> \<subseteq> O((\<lambda>x. 1 / f x) * (f * g))"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   327
    by (rule bigo_mult2)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   328
  also have "(\<lambda>x. 1 / f x) * (f * g) = g"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   329
    apply (simp add: func_times)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   330
    apply (rule ext)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   331
    apply (simp add: assms nonzero_divide_eq_eq ac_simps)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   332
    done
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   333
  finally have "(\<lambda>x. (1::'b) / f x) * h \<in> O(g)" .
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   334
  then have "f * ((\<lambda>x. (1::'b) / f x) * h) \<in> f *o O(g)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   335
    by auto
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   336
  also have "f * ((\<lambda>x. (1::'b) / f x) * h) = h"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   337
    apply (simp add: func_times)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   338
    apply (rule ext)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   339
    apply (simp add: assms nonzero_divide_eq_eq ac_simps)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   340
    done
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   341
  finally show "h \<in> f *o O(g)" .
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   342
qed
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   343
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   344
lemma bigo_mult6:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   345
  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   346
  shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   347
  apply (rule equalityI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   348
  apply (erule bigo_mult5)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   349
  apply (rule bigo_mult2)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   350
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   351
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   352
lemma bigo_mult7:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   353
  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   354
  shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   355
  apply (subst bigo_mult6)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   356
  apply assumption
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   357
  apply (rule set_times_mono3)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   358
  apply (rule bigo_refl)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   359
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   360
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   361
lemma bigo_mult8:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   362
  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   363
  shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   364
  apply (rule equalityI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   365
  apply (erule bigo_mult7)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   366
  apply (rule bigo_mult)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   367
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   368
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   369
lemma bigo_minus [intro]: "f \<in> O(g) \<Longrightarrow> - f \<in> O(g)"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   370
  by (auto simp add: bigo_def fun_Compl_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   371
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   372
lemma bigo_minus2: "f \<in> g +o O(h) \<Longrightarrow> - f \<in> -g +o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   373
  apply (rule set_minus_imp_plus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   374
  apply (drule set_plus_imp_minus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   375
  apply (drule bigo_minus)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 47445
diff changeset
   376
  apply simp
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   377
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   378
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   379
lemma bigo_minus3: "O(- f) = O(f)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   380
  by (auto simp add: bigo_def fun_Compl_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   381
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   382
lemma bigo_plus_absorb_lemma1: "f \<in> O(g) \<Longrightarrow> f +o O(g) \<subseteq> O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   383
proof -
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   384
  assume a: "f \<in> O(g)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   385
  show "f +o O(g) \<subseteq> O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   386
  proof -
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   387
    have "f \<in> O(f)" by auto
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   388
    then have "f +o O(g) \<subseteq> O(f) + O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   389
      by (auto del: subsetI)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   390
    also have "\<dots> \<subseteq> O(g) + O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   391
    proof -
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   392
      from a have "O(f) \<subseteq> O(g)" by (auto del: subsetI)
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56544
diff changeset
   393
      then show ?thesis by (auto del: subsetI)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   394
    qed
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   395
    also have "\<dots> \<subseteq> O(g)" by simp
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   396
    finally show ?thesis .
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   397
  qed
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   398
qed
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   399
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   400
lemma bigo_plus_absorb_lemma2: "f \<in> O(g) \<Longrightarrow> O(g) \<subseteq> f +o O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   401
proof -
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   402
  assume a: "f \<in> O(g)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   403
  show "O(g) \<subseteq> f +o O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   404
  proof -
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   405
    from a have "- f \<in> O(g)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   406
      by auto
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   407
    then have "- f +o O(g) \<subseteq> O(g)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   408
      by (elim bigo_plus_absorb_lemma1)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   409
    then have "f +o (- f +o O(g)) \<subseteq> f +o O(g)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   410
      by auto
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   411
    also have "f +o (- f +o O(g)) = O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   412
      by (simp add: set_plus_rearranges)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   413
    finally show ?thesis .
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   414
  qed
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   415
qed
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   416
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   417
lemma bigo_plus_absorb [simp]: "f \<in> O(g) \<Longrightarrow> f +o O(g) = O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   418
  apply (rule equalityI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   419
  apply (erule bigo_plus_absorb_lemma1)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   420
  apply (erule bigo_plus_absorb_lemma2)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   421
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   422
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   423
lemma bigo_plus_absorb2 [intro]: "f \<in> O(g) \<Longrightarrow> A \<subseteq> O(g) \<Longrightarrow> f +o A \<subseteq> O(g)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   424
  apply (subgoal_tac "f +o A \<subseteq> f +o O(g)")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   425
  apply force+
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   426
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   427
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   428
lemma bigo_add_commute_imp: "f \<in> g +o O(h) \<Longrightarrow> g \<in> f +o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   429
  apply (subst set_minus_plus [symmetric])
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   430
  apply (subgoal_tac "g - f = - (f - g)")
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   431
  apply (erule ssubst)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   432
  apply (rule bigo_minus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   433
  apply (subst set_minus_plus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   434
  apply assumption
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   435
  apply (simp add: ac_simps)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   436
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   437
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   438
lemma bigo_add_commute: "f \<in> g +o O(h) \<longleftrightarrow> g \<in> f +o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   439
  apply (rule iffI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   440
  apply (erule bigo_add_commute_imp)+
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   441
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   442
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   443
lemma bigo_const1: "(\<lambda>x. c) \<in> O(\<lambda>x. 1)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   444
  by (auto simp add: bigo_def ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   445
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   446
lemma bigo_const2 [intro]: "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   447
  apply (rule bigo_elt_subset)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   448
  apply (rule bigo_const1)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   449
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   450
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   451
lemma bigo_const3:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   452
  fixes c :: "'a::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   453
  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   454
  apply (simp add: bigo_def)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   455
  apply (rule_tac x = "abs (inverse c)" in exI)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   456
  apply (simp add: abs_mult [symmetric])
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   457
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   458
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   459
lemma bigo_const4:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   460
  fixes c :: "'a::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   461
  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   462
  apply (rule bigo_elt_subset)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   463
  apply (rule bigo_const3)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   464
  apply assumption
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   465
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   466
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   467
lemma bigo_const [simp]:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   468
  fixes c :: "'a::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   469
  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   470
  apply (rule equalityI)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   471
  apply (rule bigo_const2)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   472
  apply (rule bigo_const4)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   473
  apply assumption
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   474
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   475
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   476
lemma bigo_const_mult1: "(\<lambda>x. c * f x) \<in> O(f)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   477
  apply (simp add: bigo_def)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   478
  apply (rule_tac x = "abs c" in exI)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   479
  apply (auto simp add: abs_mult [symmetric])
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   480
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   481
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   482
lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<subseteq> O(f)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   483
  apply (rule bigo_elt_subset)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   484
  apply (rule bigo_const_mult1)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   485
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   486
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   487
lemma bigo_const_mult3:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   488
  fixes c :: "'a::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   489
  shows "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   490
  apply (simp add: bigo_def)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   491
  apply (rule_tac x = "abs (inverse c)" in exI)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   492
  apply (simp add: abs_mult [symmetric] mult.assoc [symmetric])
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   493
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   494
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   495
lemma bigo_const_mult4:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   496
  fixes c :: "'a::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   497
  shows "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   498
  apply (rule bigo_elt_subset)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   499
  apply (rule bigo_const_mult3)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   500
  apply assumption
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   501
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   502
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   503
lemma bigo_const_mult [simp]:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   504
  fixes c :: "'a::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   505
  shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   506
  apply (rule equalityI)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   507
  apply (rule bigo_const_mult2)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   508
  apply (erule bigo_const_mult4)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   509
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   510
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   511
lemma bigo_const_mult5 [simp]:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   512
  fixes c :: "'a::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   513
  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   514
  apply (auto del: subsetI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   515
  apply (rule order_trans)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   516
  apply (rule bigo_mult2)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   517
  apply (simp add: func_times)
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   518
  apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   519
  apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   520
  apply (simp add: mult.assoc [symmetric] abs_mult)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   521
  apply (rule_tac x = "abs (inverse c) * ca" in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   522
  apply (rule allI)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   523
  apply (subst mult.assoc)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   524
  apply (rule mult_left_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   525
  apply (erule spec)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   526
  apply force
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   527
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   528
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   529
lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) \<subseteq> O(f)"
41528
276078f01ada eliminated global prems;
wenzelm
parents: 38622
diff changeset
   530
  apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   531
  apply (rule_tac x = "ca * abs c" in exI)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   532
  apply (rule allI)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   533
  apply (subgoal_tac "ca * abs c * abs (f x) = abs c * (ca * abs (f x))")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   534
  apply (erule ssubst)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   535
  apply (subst abs_mult)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   536
  apply (rule mult_left_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   537
  apply (erule spec)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   538
  apply simp
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   539
  apply(simp add: ac_simps)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   540
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   541
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   542
lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   543
proof -
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   544
  assume "f =o O(g)"
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   545
  then have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   546
    by auto
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   547
  also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   548
    by (simp add: func_times)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   549
  also have "(\<lambda>x. c) *o O(g) \<subseteq> O(g)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   550
    by (auto del: subsetI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   551
  finally show ?thesis .
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   552
qed
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   553
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   554
lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f (k x)) =o O(\<lambda>x. g (k x))"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   555
  unfolding bigo_def by auto
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   556
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   557
lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   558
    (\<lambda>x. f (k x)) =o (\<lambda>x. g (k x)) +o O(\<lambda>x. h(k x))"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 47445
diff changeset
   559
  apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   560
  apply (drule bigo_compose1)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   561
  apply (simp add: fun_diff_def)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 47445
diff changeset
   562
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   563
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   564
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   565
subsection {* Setsum *}
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   566
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   567
lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 \<le> h x y \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   568
    \<exists>c. \<forall>x. \<forall>y \<in> A x. abs (f x y) \<le> c * (h x y) \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   569
      (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   570
  apply (auto simp add: bigo_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   571
  apply (rule_tac x = "abs c" in exI)
17199
59c1bfc81d91 moved lemmas that require the HOL-Complex logic image to Complex/ex/BigO_Complex.thy;
wenzelm
parents: 16961
diff changeset
   572
  apply (subst abs_of_nonneg) back back
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   573
  apply (rule setsum_nonneg)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   574
  apply force
19279
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 17199
diff changeset
   575
  apply (subst setsum_right_distrib)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   576
  apply (rule allI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   577
  apply (rule order_trans)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   578
  apply (rule setsum_abs)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   579
  apply (rule setsum_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   580
  apply (rule order_trans)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   581
  apply (drule spec)+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   582
  apply (drule bspec)+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   583
  apply assumption+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   584
  apply (drule bspec)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   585
  apply assumption+
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   586
  apply (rule mult_right_mono)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   587
  apply (rule abs_ge_self)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   588
  apply force
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   589
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   590
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   591
lemma bigo_setsum1: "\<forall>x y. 0 \<le> h x y \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   592
    \<exists>c. \<forall>x y. abs (f x y) \<le> c * h x y \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   593
      (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   594
  apply (rule bigo_setsum_main)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   595
  apply force
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   596
  apply clarsimp
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   597
  apply (rule_tac x = c in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   598
  apply force
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   599
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   600
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   601
lemma bigo_setsum2: "\<forall>y. 0 \<le> h y \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   602
    \<exists>c. \<forall>y. abs (f y) \<le> c * (h y) \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   603
      (\<lambda>x. \<Sum>y \<in> A x. f y) =o O(\<lambda>x. \<Sum>y \<in> A x. h y)"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   604
  by (rule bigo_setsum1) auto
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   605
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   606
lemma bigo_setsum3: "f =o O(h) \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   607
    (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o O(\<lambda>x. \<Sum>y \<in> A x. abs (l x y * h (k x y)))"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   608
  apply (rule bigo_setsum1)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   609
  apply (rule allI)+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   610
  apply (rule abs_ge_zero)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   611
  apply (unfold bigo_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   612
  apply auto
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   613
  apply (rule_tac x = c in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   614
  apply (rule allI)+
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   615
  apply (subst abs_mult)+
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   616
  apply (subst mult.left_commute)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   617
  apply (rule mult_left_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   618
  apply (erule spec)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   619
  apply (rule abs_ge_zero)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   620
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   621
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   622
lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   623
    (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   624
      (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   625
        O(\<lambda>x. \<Sum>y \<in> A x. abs (l x y * h (k x y)))"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   626
  apply (rule set_minus_imp_plus)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   627
  apply (subst fun_diff_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   628
  apply (subst setsum_subtractf [symmetric])
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   629
  apply (subst right_diff_distrib [symmetric])
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   630
  apply (rule bigo_setsum3)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   631
  apply (subst fun_diff_def [symmetric])
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   632
  apply (erule set_plus_imp_minus)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   633
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   634
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   635
lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   636
    \<forall>x. 0 \<le> h x \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   637
      (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   638
        O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   639
  apply (subgoal_tac "(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y)) =
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   640
      (\<lambda>x. \<Sum>y \<in> A x. abs (l x y * h (k x y)))")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   641
  apply (erule ssubst)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   642
  apply (erule bigo_setsum3)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   643
  apply (rule ext)
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56796
diff changeset
   644
  apply (rule setsum.cong)
6ab1c7cb0b8d fact consolidation
haftmann
parents: 56796
diff changeset
   645
  apply (rule refl)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   646
  apply (subst abs_of_nonneg)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   647
  apply auto
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   648
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   649
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   650
lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   651
    \<forall>x. 0 \<le> h x \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   652
      (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   653
        (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   654
          O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   655
  apply (rule set_minus_imp_plus)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   656
  apply (subst fun_diff_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   657
  apply (subst setsum_subtractf [symmetric])
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   658
  apply (subst right_diff_distrib [symmetric])
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   659
  apply (rule bigo_setsum5)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   660
  apply (subst fun_diff_def [symmetric])
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   661
  apply (drule set_plus_imp_minus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   662
  apply auto
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   663
  done
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   664
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   665
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   666
subsection {* Misc useful stuff *}
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   667
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   668
lemma bigo_useful_intro: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   669
  apply (subst bigo_plus_idemp [symmetric])
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   670
  apply (rule set_plus_mono2)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   671
  apply assumption+
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   672
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   673
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   674
lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   675
  apply (subst bigo_plus_idemp [symmetric])
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   676
  apply (rule set_plus_intro)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   677
  apply assumption+
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   678
  done
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   679
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   680
lemma bigo_useful_const_mult:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   681
  fixes c :: "'a::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   682
  shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   683
  apply (rule subsetD)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   684
  apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   685
  apply assumption
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   686
  apply (rule bigo_const_mult6)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   687
  apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   688
  apply (erule ssubst)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   689
  apply (erule set_times_intro2)
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23373
diff changeset
   690
  apply (simp add: func_times)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   691
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   692
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   693
lemma bigo_fix: "(\<lambda>x::nat. f (x + 1)) =o O(\<lambda>x. h (x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow> f =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   694
  apply (simp add: bigo_alt_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   695
  apply auto
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   696
  apply (rule_tac x = c in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   697
  apply auto
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   698
  apply (case_tac "x = 0")
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   699
  apply simp
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   700
  apply (subgoal_tac "x = Suc (x - 1)")
17199
59c1bfc81d91 moved lemmas that require the HOL-Complex logic image to Complex/ex/BigO_Complex.thy;
wenzelm
parents: 16961
diff changeset
   701
  apply (erule ssubst) back
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   702
  apply (erule spec)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   703
  apply simp
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   704
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   705
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   706
lemma bigo_fix2:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   707
    "(\<lambda>x. f ((x::nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   708
       f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   709
  apply (rule set_minus_imp_plus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   710
  apply (rule bigo_fix)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   711
  apply (subst fun_diff_def)
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   712
  apply (subst fun_diff_def [symmetric])
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   713
  apply (rule set_plus_imp_minus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   714
  apply simp
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   715
  apply (simp add: fun_diff_def)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   716
  done
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   717
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   718
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   719
subsection {* Less than or equal to *}
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   720
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   721
definition lesso :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"  (infixl "<o" 70)
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   722
  where "f <o g = (\<lambda>x. max (f x - g x) 0)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   723
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   724
lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) \<le> abs (f x) \<Longrightarrow> g =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   725
  apply (unfold bigo_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   726
  apply clarsimp
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   727
  apply (rule_tac x = c in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   728
  apply (rule allI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   729
  apply (rule order_trans)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   730
  apply (erule spec)+
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   731
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   732
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   733
lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) \<le> f x \<Longrightarrow> g =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   734
  apply (erule bigo_lesseq1)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   735
  apply (rule allI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   736
  apply (drule_tac x = x in spec)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   737
  apply (rule order_trans)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   738
  apply assumption
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   739
  apply (rule abs_ge_self)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   740
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   741
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   742
lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> f x \<Longrightarrow> g =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   743
  apply (erule bigo_lesseq2)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   744
  apply (rule allI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   745
  apply (subst abs_of_nonneg)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   746
  apply (erule spec)+
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   747
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   748
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   749
lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   750
    \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> abs (f x) \<Longrightarrow> g =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   751
  apply (erule bigo_lesseq1)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   752
  apply (rule allI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   753
  apply (subst abs_of_nonneg)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   754
  apply (erule spec)+
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   755
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   756
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   757
lemma bigo_lesso1: "\<forall>x. f x \<le> g x \<Longrightarrow> f <o g =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   758
  apply (unfold lesso_def)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   759
  apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   760
  apply (erule ssubst)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   761
  apply (rule bigo_zero)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   762
  apply (unfold func_zero)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   763
  apply (rule ext)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   764
  apply (simp split: split_max)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   765
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   766
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   767
lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   768
    \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. k x \<le> f x \<Longrightarrow> k <o g =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   769
  apply (unfold lesso_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   770
  apply (rule bigo_lesseq4)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   771
  apply (erule set_plus_imp_minus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   772
  apply (rule allI)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54230
diff changeset
   773
  apply (rule max.cobounded2)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   774
  apply (rule allI)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   775
  apply (subst fun_diff_def)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   776
  apply (case_tac "0 \<le> k x - g x")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   777
  apply simp
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   778
  apply (subst abs_of_nonneg)
17199
59c1bfc81d91 moved lemmas that require the HOL-Complex logic image to Complex/ex/BigO_Complex.thy;
wenzelm
parents: 16961
diff changeset
   779
  apply (drule_tac x = x in spec) back
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 27487
diff changeset
   780
  apply (simp add: algebra_simps)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 47445
diff changeset
   781
  apply (subst diff_conv_add_uminus)+
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   782
  apply (rule add_right_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   783
  apply (erule spec)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   784
  apply (rule order_trans)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   785
  prefer 2
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   786
  apply (rule abs_ge_zero)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 27487
diff changeset
   787
  apply (simp add: algebra_simps)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   788
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   789
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   790
lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow>
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   791
    \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. g x \<le> k x \<Longrightarrow> f <o k =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   792
  apply (unfold lesso_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   793
  apply (rule bigo_lesseq4)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   794
  apply (erule set_plus_imp_minus)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   795
  apply (rule allI)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54230
diff changeset
   796
  apply (rule max.cobounded2)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   797
  apply (rule allI)
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   798
  apply (subst fun_diff_def)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   799
  apply (case_tac "0 \<le> f x - k x")
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   800
  apply simp
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   801
  apply (subst abs_of_nonneg)
17199
59c1bfc81d91 moved lemmas that require the HOL-Complex logic image to Complex/ex/BigO_Complex.thy;
wenzelm
parents: 16961
diff changeset
   802
  apply (drule_tac x = x in spec) back
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 27487
diff changeset
   803
  apply (simp add: algebra_simps)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 47445
diff changeset
   804
  apply (subst diff_conv_add_uminus)+
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   805
  apply (rule add_left_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   806
  apply (rule le_imp_neg_le)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   807
  apply (erule spec)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   808
  apply (rule order_trans)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   809
  prefer 2
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   810
  apply (rule abs_ge_zero)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 27487
diff changeset
   811
  apply (simp add: algebra_simps)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   812
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   813
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   814
lemma bigo_lesso4:
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   815
  fixes k :: "'a \<Rightarrow> 'b::linordered_field"
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   816
  shows "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   817
  apply (unfold lesso_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   818
  apply (drule set_plus_imp_minus)
17199
59c1bfc81d91 moved lemmas that require the HOL-Complex logic image to Complex/ex/BigO_Complex.thy;
wenzelm
parents: 16961
diff changeset
   819
  apply (drule bigo_abs5) back
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25592
diff changeset
   820
  apply (simp add: fun_diff_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   821
  apply (drule bigo_useful_add)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   822
  apply assumption
17199
59c1bfc81d91 moved lemmas that require the HOL-Complex logic image to Complex/ex/BigO_Complex.thy;
wenzelm
parents: 16961
diff changeset
   823
  apply (erule bigo_lesseq2) back
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   824
  apply (rule allI)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   825
  apply (auto simp add: func_plus fun_diff_def algebra_simps split: split_max abs_split)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   826
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   827
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   828
lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x \<le> g x + C * abs (h x)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   829
  apply (simp only: lesso_def bigo_alt_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   830
  apply clarsimp
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   831
  apply (rule_tac x = c in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   832
  apply (rule allI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   833
  apply (drule_tac x = x in spec)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   834
  apply (subgoal_tac "abs (max (f x - g x) 0) = max (f x - g x) 0")
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   835
  apply (clarsimp simp add: algebra_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   836
  apply (rule abs_of_nonneg)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54230
diff changeset
   837
  apply (rule max.cobounded2)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   838
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   839
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   840
lemma lesso_add: "f <o g =o O(h) \<Longrightarrow> k <o l =o O(h) \<Longrightarrow> (f + k) <o (g + l) =o O(h)"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   841
  apply (unfold lesso_def)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   842
  apply (rule bigo_lesseq3)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   843
  apply (erule bigo_useful_add)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   844
  apply assumption
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   845
  apply (force split: split_max)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   846
  apply (auto split: split_max simp add: func_plus)
22665
cf152ff55d16 tuned document (headers, sections, spacing);
wenzelm
parents: 21404
diff changeset
   847
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   848
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   849
lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g ----> 0 \<Longrightarrow> f ----> (0::real)"
31337
a9ed5fcc5e39 LIMSEQ_def -> LIMSEQ_iff
huffman
parents: 29786
diff changeset
   850
  apply (simp add: LIMSEQ_iff bigo_alt_def)
29786
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   851
  apply clarify
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   852
  apply (drule_tac x = "r / c" in spec)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   853
  apply (drule mp)
56541
0e3abadbef39 made divide_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   854
  apply simp
29786
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   855
  apply clarify
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   856
  apply (rule_tac x = no in exI)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   857
  apply (rule allI)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   858
  apply (drule_tac x = n in spec)+
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   859
  apply (rule impI)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   860
  apply (drule mp)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   861
  apply assumption
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   862
  apply (rule order_le_less_trans)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   863
  apply assumption
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   864
  apply (rule order_less_le_trans)
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   865
  apply (subgoal_tac "c * abs (g n) < c * (r / c)")
29786
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   866
  apply assumption
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   867
  apply (erule mult_strict_left_mono)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   868
  apply assumption
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   869
  apply simp
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   870
  done
29786
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   871
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   872
lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h ----> 0 \<Longrightarrow> f ----> a \<Longrightarrow> g ----> (a::real)"
29786
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   873
  apply (drule set_plus_imp_minus)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   874
  apply (drule bigo_LIMSEQ1)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   875
  apply assumption
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   876
  apply (simp only: fun_diff_def)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   877
  apply (erule LIMSEQ_diff_approach_zero2)
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   878
  apply assumption
55821
44055f07cbd8 more symbols, less parentheses;
wenzelm
parents: 54863
diff changeset
   879
  done
29786
84a3f86441eb merged Big0
haftmann
parents: 29667
diff changeset
   880
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   881
end