author | wenzelm |
Wed, 13 Jul 2016 14:28:15 +0200 | |
changeset 63473 | 151bb79536a7 |
parent 63462 | c1fe30f2bc32 |
child 63485 | ea8dfb0ed10e |
permissions | -rw-r--r-- |
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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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section \<open>Big O notation\<close> |
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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 \<open> |
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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 \<open>f = O(g)\<close> and \<open>f = g + O(h)\<close>. |
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An earlier version of this library is described in detail in @{cite |
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"Avigad-Donnelly"}. |
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||
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The main changes in this version are as follows: |
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\<^item> We have eliminated the \<open>O\<close> operator on sets. (Most uses of this seem |
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to be inessential.) |
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\<^item> We no longer use \<open>+\<close> as output syntax for \<open>+o\<close> |
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\<^item> Lemmas involving \<open>sumr\<close> have been replaced by more general lemmas |
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involving `\<open>setsum\<close>. |
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\<^item> The library has been expanded, with e.g.~support for expressions of |
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the form \<open>f < g + O(h)\<close>. |
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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 one |
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should redeclare the theorem \<open>subsetI\<close> as an intro rule, rather than as an |
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\<open>intro!\<close> rule, for example, using \<^theory_text>\<open>declare subsetI [del, intro]\<close>. |
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\<close> |
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subsection \<open>Definitions\<close> |
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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. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>}" |
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lemma bigo_pos_const: |
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"(\<exists>c::'a::linordered_idom. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>) \<longleftrightarrow> |
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(\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))" |
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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 = "\<bar>c\<bar>" in exI) |
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apply auto |
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apply (subgoal_tac "c * \<bar>f x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>") |
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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. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)}" |
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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 * \<bar>f xa\<bar> \<le> ca * (c * \<bar>g xa\<bar>)") |
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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 \<bar>g n\<bar> \<le> \<bar>f n\<bar> 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 * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>f xa\<bar>") |
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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 * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)") |
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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 \<bar>f n\<bar> < \<bar>g n\<bar> 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 * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>g xa\<bar>") |
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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 * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)") |
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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 (rule abs_triangle_ineq) |
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apply (simp add: order_less_le) |
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done |
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lemma bigo_plus_subset2 [intro]: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)" |
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apply (subgoal_tac "A + B \<subseteq> O(f) + O(f)") |
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apply (erule order_trans) |
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apply simp |
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apply (auto del: subsetI simp del: bigo_plus_idemp) |
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done |
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lemma bigo_plus_eq: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> O(f + g) = O(f) + O(g)" |
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apply (rule equalityI) |
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apply (rule bigo_plus_subset) |
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apply (simp add: bigo_alt_def set_plus_def func_plus) |
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apply clarify |
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apply (rule_tac x = "max c ca" in exI) |
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apply (rule conjI) |
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apply (subgoal_tac "c \<le> max c ca") |
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apply (erule order_less_le_trans) |
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apply assumption |
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apply (rule max.cobounded1) |
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apply clarify |
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apply (drule_tac x = "xa" in spec)+ |
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apply (subgoal_tac "0 \<le> f xa + g xa") |
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apply (simp add: ring_distribs) |
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apply (subgoal_tac "\<bar>a xa + b xa\<bar> \<le> \<bar>a xa\<bar> + \<bar>b xa\<bar>") |
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apply (subgoal_tac "\<bar>a xa\<bar> + \<bar>b xa\<bar> \<le> max c ca * f xa + max c ca * g xa") |
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apply force |
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apply (rule add_mono) |
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apply (subgoal_tac "c * f xa \<le> max c ca * f xa") |
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apply force |
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apply (rule mult_right_mono) |
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apply (rule max.cobounded1) |
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apply assumption |
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apply (subgoal_tac "ca * g xa \<le> max c ca * g xa") |
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apply force |
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apply (rule mult_right_mono) |
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apply (rule max.cobounded2) |
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apply assumption |
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apply (rule abs_triangle_ineq) |
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apply (rule add_nonneg_nonneg) |
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apply assumption+ |
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done |
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lemma bigo_bounded_alt: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> c * g x \<Longrightarrow> f \<in> O(g)" |
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apply (auto simp add: bigo_def) |
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apply (rule_tac x = "\<bar>c\<bar>" in exI) |
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apply auto |
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apply (drule_tac x = x in spec)+ |
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apply (simp add: abs_mult [symmetric]) |
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done |
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lemma bigo_bounded: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> g x \<Longrightarrow> f \<in> O(g)" |
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apply (erule bigo_bounded_alt [of f 1 g]) |
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apply simp |
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done |
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|
55821 | 196 |
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
|
197 |
apply (rule set_minus_imp_plus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
198 |
apply (rule bigo_bounded) |
63473 | 199 |
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
|
200 |
apply (drule_tac x = x in spec)+ |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
201 |
apply force |
22665 | 202 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
203 |
|
61945 | 204 |
lemma bigo_abs: "(\<lambda>x. \<bar>f x\<bar>) =o O(f)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
205 |
apply (unfold bigo_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
206 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
207 |
apply (rule_tac x = 1 in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
208 |
apply auto |
22665 | 209 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
210 |
|
61945 | 211 |
lemma bigo_abs2: "f =o O(\<lambda>x. \<bar>f x\<bar>)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
212 |
apply (unfold bigo_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
213 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
214 |
apply (rule_tac x = 1 in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
215 |
apply auto |
22665 | 216 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
217 |
|
61945 | 218 |
lemma bigo_abs3: "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
219 |
apply (rule equalityI) |
63473 | 220 |
apply (rule bigo_elt_subset) |
221 |
apply (rule bigo_abs2) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
222 |
apply (rule bigo_elt_subset) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
223 |
apply (rule bigo_abs) |
22665 | 224 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
225 |
|
61945 | 226 |
lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o (\<lambda>x. \<bar>g x\<bar>) +o O(h)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
227 |
apply (drule set_plus_imp_minus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
228 |
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
|
229 |
apply (subst fun_diff_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
230 |
proof - |
63473 | 231 |
assume *: "f - g \<in> O(h)" |
61945 | 232 |
have "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) =o O(\<lambda>x. \<bar>\<bar>f x\<bar> - \<bar>g x\<bar>\<bar>)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
233 |
by (rule bigo_abs2) |
61945 | 234 |
also have "\<dots> \<subseteq> O(\<lambda>x. \<bar>f x - g x\<bar>)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
235 |
apply (rule bigo_elt_subset) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
236 |
apply (rule bigo_bounded) |
63473 | 237 |
apply force |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
238 |
apply (rule allI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
239 |
apply (rule abs_triangle_ineq3) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
240 |
done |
55821 | 241 |
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
|
242 |
apply (rule bigo_elt_subset) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25592
diff
changeset
|
243 |
apply (subst fun_diff_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
244 |
apply (rule bigo_abs) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
245 |
done |
63473 | 246 |
also from * have "\<dots> \<subseteq> O(h)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
247 |
by (rule bigo_elt_subset) |
61945 | 248 |
finally show "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) \<in> O(h)". |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
249 |
qed |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
250 |
|
61945 | 251 |
lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o O(g)" |
63473 | 252 |
by (auto simp: bigo_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
253 |
|
63473 | 254 |
lemma bigo_elt_subset2 [intro]: |
255 |
assumes *: "f \<in> g +o O(h)" |
|
256 |
shows "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
|
257 |
proof - |
63473 | 258 |
note * |
259 |
also have "g +o O(h) \<subseteq> O(g) + O(h)" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
260 |
by (auto del: subsetI) |
61945 | 261 |
also have "\<dots> = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)" |
63473 | 262 |
by (subst bigo_abs3 [symmetric])+ (rule refl) |
61945 | 263 |
also have "\<dots> = O((\<lambda>x. \<bar>g x\<bar>) + (\<lambda>x. \<bar>h x\<bar>))" |
55821 | 264 |
by (rule bigo_plus_eq [symmetric]) auto |
265 |
finally have "f \<in> \<dots>" . |
|
266 |
then have "O(f) \<subseteq> \<dots>" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
267 |
by (elim bigo_elt_subset) |
61945 | 268 |
also have "\<dots> = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
269 |
by (rule bigo_plus_eq, auto) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
270 |
finally show ?thesis |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
271 |
by (simp add: bigo_abs3 [symmetric]) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
272 |
qed |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
273 |
|
55821 | 274 |
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
|
275 |
apply (rule subsetI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
276 |
apply (subst bigo_def) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25592
diff
changeset
|
277 |
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
|
278 |
apply (rule_tac x = "c * ca" in exI) |
55821 | 279 |
apply (rule allI) |
280 |
apply (erule_tac x = x in allE)+ |
|
61945 | 281 |
apply (subgoal_tac "c * ca * \<bar>f x * g x\<bar> = (c * \<bar>f x\<bar>) * (ca * \<bar>g x\<bar>)") |
63473 | 282 |
apply (erule ssubst) |
283 |
apply (subst abs_mult) |
|
284 |
apply (rule mult_mono) |
|
285 |
apply assumption+ |
|
286 |
apply auto |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
287 |
apply (simp add: ac_simps abs_mult) |
22665 | 288 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
289 |
|
55821 | 290 |
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
|
291 |
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
|
292 |
apply (rule_tac x = c in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
293 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
294 |
apply (drule_tac x = x in spec) |
61945 | 295 |
apply (subgoal_tac "\<bar>f x\<bar> * \<bar>b x\<bar> \<le> \<bar>f x\<bar> * (c * \<bar>g x\<bar>)") |
63473 | 296 |
apply (force simp add: ac_simps) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
297 |
apply (rule mult_left_mono, assumption) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
298 |
apply (rule abs_ge_zero) |
22665 | 299 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
300 |
|
55821 | 301 |
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
|
302 |
apply (rule subsetD) |
63473 | 303 |
apply (rule bigo_mult) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
304 |
apply (erule set_times_intro, assumption) |
22665 | 305 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
306 |
|
55821 | 307 |
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
|
308 |
apply (drule set_plus_imp_minus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
309 |
apply (rule set_minus_imp_plus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
310 |
apply (drule bigo_mult3 [where g = g and j = g]) |
63473 | 311 |
apply (auto simp add: algebra_simps) |
22665 | 312 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
313 |
|
41528 | 314 |
lemma bigo_mult5: |
55821 | 315 |
fixes f :: "'a \<Rightarrow> 'b::linordered_field" |
316 |
assumes "\<forall>x. f x \<noteq> 0" |
|
317 |
shows "O(f * g) \<subseteq> f *o O(g)" |
|
41528 | 318 |
proof |
319 |
fix h |
|
55821 | 320 |
assume "h \<in> O(f * g)" |
321 |
then have "(\<lambda>x. 1 / (f x)) * h \<in> (\<lambda>x. 1 / f x) *o O(f * g)" |
|
41528 | 322 |
by auto |
55821 | 323 |
also have "\<dots> \<subseteq> O((\<lambda>x. 1 / f x) * (f * g))" |
41528 | 324 |
by (rule bigo_mult2) |
55821 | 325 |
also have "(\<lambda>x. 1 / f x) * (f * g) = g" |
326 |
apply (simp add: func_times) |
|
41528 | 327 |
apply (rule ext) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
328 |
apply (simp add: assms nonzero_divide_eq_eq ac_simps) |
41528 | 329 |
done |
55821 | 330 |
finally have "(\<lambda>x. (1::'b) / f x) * h \<in> O(g)" . |
331 |
then have "f * ((\<lambda>x. (1::'b) / f x) * h) \<in> f *o O(g)" |
|
41528 | 332 |
by auto |
55821 | 333 |
also have "f * ((\<lambda>x. (1::'b) / f x) * h) = h" |
334 |
apply (simp add: func_times) |
|
41528 | 335 |
apply (rule ext) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
336 |
apply (simp add: assms nonzero_divide_eq_eq ac_simps) |
41528 | 337 |
done |
55821 | 338 |
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
|
339 |
qed |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
340 |
|
63473 | 341 |
lemma bigo_mult6: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)" |
342 |
for f :: "'a \<Rightarrow> 'b::linordered_field" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
343 |
apply (rule equalityI) |
63473 | 344 |
apply (erule bigo_mult5) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
345 |
apply (rule bigo_mult2) |
22665 | 346 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
347 |
|
63473 | 348 |
lemma bigo_mult7: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)" |
349 |
for f :: "'a \<Rightarrow> 'b::linordered_field" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
350 |
apply (subst bigo_mult6) |
63473 | 351 |
apply assumption |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
352 |
apply (rule set_times_mono3) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
353 |
apply (rule bigo_refl) |
22665 | 354 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
355 |
|
63473 | 356 |
lemma bigo_mult8: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)" |
357 |
for f :: "'a \<Rightarrow> 'b::linordered_field" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
358 |
apply (rule equalityI) |
63473 | 359 |
apply (erule bigo_mult7) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
360 |
apply (rule bigo_mult) |
22665 | 361 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
362 |
|
55821 | 363 |
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
|
364 |
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
|
365 |
|
55821 | 366 |
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
|
367 |
apply (rule set_minus_imp_plus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
368 |
apply (drule set_plus_imp_minus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
369 |
apply (drule bigo_minus) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
47445
diff
changeset
|
370 |
apply simp |
22665 | 371 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
372 |
|
55821 | 373 |
lemma bigo_minus3: "O(- f) = O(f)" |
41528 | 374 |
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
|
375 |
|
63473 | 376 |
lemma bigo_plus_absorb_lemma1: |
377 |
assumes *: "f \<in> O(g)" |
|
378 |
shows "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
|
379 |
proof - |
63473 | 380 |
have "f \<in> O(f)" by auto |
381 |
then have "f +o O(g) \<subseteq> O(f) + O(g)" |
|
382 |
by (auto del: subsetI) |
|
383 |
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
|
384 |
proof - |
63473 | 385 |
from * have "O(f) \<subseteq> O(g)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
386 |
by (auto del: subsetI) |
63473 | 387 |
then show ?thesis |
388 |
by (auto del: subsetI) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
389 |
qed |
63473 | 390 |
also have "\<dots> \<subseteq> O(g)" by simp |
391 |
finally show ?thesis . |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
392 |
qed |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
393 |
|
63473 | 394 |
lemma bigo_plus_absorb_lemma2: |
395 |
assumes *: "f \<in> O(g)" |
|
396 |
shows "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
|
397 |
proof - |
63473 | 398 |
from * have "- f \<in> O(g)" |
399 |
by auto |
|
400 |
then have "- f +o O(g) \<subseteq> O(g)" |
|
401 |
by (elim bigo_plus_absorb_lemma1) |
|
402 |
then have "f +o (- f +o O(g)) \<subseteq> f +o O(g)" |
|
403 |
by auto |
|
404 |
also have "f +o (- f +o O(g)) = O(g)" |
|
405 |
by (simp add: set_plus_rearranges) |
|
406 |
finally show ?thesis . |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
407 |
qed |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
408 |
|
55821 | 409 |
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
|
410 |
apply (rule equalityI) |
63473 | 411 |
apply (erule bigo_plus_absorb_lemma1) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
412 |
apply (erule bigo_plus_absorb_lemma2) |
22665 | 413 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
414 |
|
55821 | 415 |
lemma bigo_plus_absorb2 [intro]: "f \<in> O(g) \<Longrightarrow> A \<subseteq> O(g) \<Longrightarrow> f +o A \<subseteq> O(g)" |
416 |
apply (subgoal_tac "f +o A \<subseteq> f +o O(g)") |
|
63473 | 417 |
apply force+ |
22665 | 418 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
419 |
|
55821 | 420 |
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
|
421 |
apply (subst set_minus_plus [symmetric]) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
422 |
apply (subgoal_tac "g - f = - (f - g)") |
63473 | 423 |
apply (erule ssubst) |
424 |
apply (rule bigo_minus) |
|
425 |
apply (subst set_minus_plus) |
|
426 |
apply assumption |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
427 |
apply (simp add: ac_simps) |
22665 | 428 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
429 |
|
55821 | 430 |
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
|
431 |
apply (rule iffI) |
63473 | 432 |
apply (erule bigo_add_commute_imp)+ |
22665 | 433 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
434 |
|
55821 | 435 |
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
|
436 |
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
|
437 |
|
55821 | 438 |
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
|
439 |
apply (rule bigo_elt_subset) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
440 |
apply (rule bigo_const1) |
22665 | 441 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
442 |
|
63473 | 443 |
lemma bigo_const3: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)" |
444 |
for c :: "'a::linordered_field" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
445 |
apply (simp add: bigo_def) |
61945 | 446 |
apply (rule_tac x = "\<bar>inverse c\<bar>" in exI) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
447 |
apply (simp add: abs_mult [symmetric]) |
22665 | 448 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
449 |
|
63473 | 450 |
lemma bigo_const4: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)" |
451 |
for c :: "'a::linordered_field" |
|
55821 | 452 |
apply (rule bigo_elt_subset) |
453 |
apply (rule bigo_const3) |
|
454 |
apply assumption |
|
455 |
done |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
456 |
|
63473 | 457 |
lemma bigo_const [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)" |
458 |
for c :: "'a::linordered_field" |
|
55821 | 459 |
apply (rule equalityI) |
63473 | 460 |
apply (rule bigo_const2) |
55821 | 461 |
apply (rule bigo_const4) |
462 |
apply assumption |
|
463 |
done |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
464 |
|
55821 | 465 |
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
|
466 |
apply (simp add: bigo_def) |
61945 | 467 |
apply (rule_tac x = "\<bar>c\<bar>" in exI) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
468 |
apply (auto simp add: abs_mult [symmetric]) |
22665 | 469 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
470 |
|
55821 | 471 |
lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<subseteq> O(f)" |
472 |
apply (rule bigo_elt_subset) |
|
473 |
apply (rule bigo_const_mult1) |
|
474 |
done |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
475 |
|
63473 | 476 |
lemma bigo_const_mult3: "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)" |
477 |
for c :: "'a::linordered_field" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
478 |
apply (simp add: bigo_def) |
61945 | 479 |
apply (rule_tac x = "\<bar>inverse c\<bar>" in exI) |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58881
diff
changeset
|
480 |
apply (simp add: abs_mult mult.assoc [symmetric]) |
22665 | 481 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
482 |
|
63473 | 483 |
lemma bigo_const_mult4: "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)" |
484 |
for c :: "'a::linordered_field" |
|
55821 | 485 |
apply (rule bigo_elt_subset) |
486 |
apply (rule bigo_const_mult3) |
|
487 |
apply assumption |
|
488 |
done |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
489 |
|
63473 | 490 |
lemma bigo_const_mult [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)" |
491 |
for c :: "'a::linordered_field" |
|
55821 | 492 |
apply (rule equalityI) |
63473 | 493 |
apply (rule bigo_const_mult2) |
55821 | 494 |
apply (erule bigo_const_mult4) |
495 |
done |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
496 |
|
63473 | 497 |
lemma bigo_const_mult5 [simp]: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)" |
498 |
for c :: "'a::linordered_field" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
499 |
apply (auto del: subsetI) |
63473 | 500 |
apply (rule order_trans) |
501 |
apply (rule bigo_mult2) |
|
502 |
apply (simp add: func_times) |
|
41528 | 503 |
apply (auto intro!: simp add: bigo_def elt_set_times_def func_times) |
55821 | 504 |
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
|
505 |
apply (simp add: mult.assoc [symmetric] abs_mult) |
61945 | 506 |
apply (rule_tac x = "\<bar>inverse c\<bar> * ca" in exI) |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58881
diff
changeset
|
507 |
apply auto |
22665 | 508 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
509 |
|
55821 | 510 |
lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) \<subseteq> O(f)" |
41528 | 511 |
apply (auto intro!: simp add: bigo_def elt_set_times_def func_times) |
61945 | 512 |
apply (rule_tac x = "ca * \<bar>c\<bar>" in exI) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
513 |
apply (rule allI) |
61945 | 514 |
apply (subgoal_tac "ca * \<bar>c\<bar> * \<bar>f x\<bar> = \<bar>c\<bar> * (ca * \<bar>f x\<bar>)") |
63473 | 515 |
apply (erule ssubst) |
516 |
apply (subst abs_mult) |
|
517 |
apply (rule mult_left_mono) |
|
518 |
apply (erule spec) |
|
519 |
apply simp |
|
520 |
apply (simp add: ac_simps) |
|
22665 | 521 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
522 |
|
63473 | 523 |
lemma bigo_const_mult7 [intro]: |
524 |
assumes *: "f =o O(g)" |
|
525 |
shows "(\<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
|
526 |
proof - |
63473 | 527 |
from * 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
|
528 |
by auto |
55821 | 529 |
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
|
530 |
by (simp add: func_times) |
55821 | 531 |
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
|
532 |
by (auto del: subsetI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
533 |
finally show ?thesis . |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
534 |
qed |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
535 |
|
55821 | 536 |
lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f (k x)) =o O(\<lambda>x. g (k x))" |
63473 | 537 |
by (auto simp: bigo_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
538 |
|
63473 | 539 |
lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow> (\<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
|
540 |
apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus) |
55821 | 541 |
apply (drule bigo_compose1) |
542 |
apply (simp add: fun_diff_def) |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
47445
diff
changeset
|
543 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
544 |
|
22665 | 545 |
|
60500 | 546 |
subsection \<open>Setsum\<close> |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
547 |
|
55821 | 548 |
lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 \<le> h x y \<Longrightarrow> |
61945 | 549 |
\<exists>c. \<forall>x. \<forall>y \<in> A x. \<bar>f x y\<bar> \<le> c * h x y \<Longrightarrow> |
55821 | 550 |
(\<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
|
551 |
apply (auto simp add: bigo_def) |
61945 | 552 |
apply (rule_tac x = "\<bar>c\<bar>" in exI) |
17199
59c1bfc81d91
moved lemmas that require the HOL-Complex logic image to Complex/ex/BigO_Complex.thy;
wenzelm
parents:
16961
diff
changeset
|
553 |
apply (subst abs_of_nonneg) back back |
63473 | 554 |
apply (rule setsum_nonneg) |
555 |
apply force |
|
19279 | 556 |
apply (subst setsum_right_distrib) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
557 |
apply (rule allI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
558 |
apply (rule order_trans) |
63473 | 559 |
apply (rule setsum_abs) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
560 |
apply (rule setsum_mono) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
561 |
apply (rule order_trans) |
63473 | 562 |
apply (drule spec)+ |
563 |
apply (drule bspec)+ |
|
564 |
apply assumption+ |
|
565 |
apply (drule bspec) |
|
566 |
apply assumption+ |
|
55821 | 567 |
apply (rule mult_right_mono) |
63473 | 568 |
apply (rule abs_ge_self) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
569 |
apply force |
22665 | 570 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
571 |
|
55821 | 572 |
lemma bigo_setsum1: "\<forall>x y. 0 \<le> h x y \<Longrightarrow> |
61945 | 573 |
\<exists>c. \<forall>x y. \<bar>f x y\<bar> \<le> c * h x y \<Longrightarrow> |
55821 | 574 |
(\<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
|
575 |
apply (rule bigo_setsum_main) |
63473 | 576 |
apply force |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
577 |
apply clarsimp |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
578 |
apply (rule_tac x = c in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
579 |
apply force |
22665 | 580 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
581 |
|
55821 | 582 |
lemma bigo_setsum2: "\<forall>y. 0 \<le> h y \<Longrightarrow> |
61945 | 583 |
\<exists>c. \<forall>y. \<bar>f y\<bar> \<le> c * (h y) \<Longrightarrow> |
55821 | 584 |
(\<lambda>x. \<Sum>y \<in> A x. f y) =o O(\<lambda>x. \<Sum>y \<in> A x. h y)" |
585 |
by (rule bigo_setsum1) auto |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
586 |
|
55821 | 587 |
lemma bigo_setsum3: "f =o O(h) \<Longrightarrow> |
61945 | 588 |
(\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
589 |
apply (rule bigo_setsum1) |
63473 | 590 |
apply (rule allI)+ |
591 |
apply (rule abs_ge_zero) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
592 |
apply (unfold bigo_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
593 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
594 |
apply (rule_tac x = c in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
595 |
apply (rule allI)+ |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
596 |
apply (subst abs_mult)+ |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
597 |
apply (subst mult.left_commute) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
598 |
apply (rule mult_left_mono) |
63473 | 599 |
apply (erule spec) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
600 |
apply (rule abs_ge_zero) |
22665 | 601 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
602 |
|
55821 | 603 |
lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow> |
604 |
(\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o |
|
605 |
(\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o |
|
61945 | 606 |
O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
607 |
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
|
608 |
apply (subst fun_diff_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
609 |
apply (subst setsum_subtractf [symmetric]) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
610 |
apply (subst right_diff_distrib [symmetric]) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
611 |
apply (rule bigo_setsum3) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25592
diff
changeset
|
612 |
apply (subst fun_diff_def [symmetric]) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
613 |
apply (erule set_plus_imp_minus) |
22665 | 614 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
615 |
|
55821 | 616 |
lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow> |
617 |
\<forall>x. 0 \<le> h x \<Longrightarrow> |
|
618 |
(\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o |
|
619 |
O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))" |
|
620 |
apply (subgoal_tac "(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y)) = |
|
61945 | 621 |
(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)") |
63473 | 622 |
apply (erule ssubst) |
623 |
apply (erule bigo_setsum3) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
624 |
apply (rule ext) |
57418 | 625 |
apply (rule setsum.cong) |
63473 | 626 |
apply (rule refl) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
627 |
apply (subst abs_of_nonneg) |
63473 | 628 |
apply auto |
22665 | 629 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
630 |
|
55821 | 631 |
lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow> |
632 |
\<forall>x. 0 \<le> h x \<Longrightarrow> |
|
633 |
(\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o |
|
634 |
(\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o |
|
635 |
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
|
636 |
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
|
637 |
apply (subst fun_diff_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
638 |
apply (subst setsum_subtractf [symmetric]) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
639 |
apply (subst right_diff_distrib [symmetric]) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
640 |
apply (rule bigo_setsum5) |
63473 | 641 |
apply (subst fun_diff_def [symmetric]) |
642 |
apply (drule set_plus_imp_minus) |
|
643 |
apply auto |
|
22665 | 644 |
done |
645 |
||
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
646 |
|
60500 | 647 |
subsection \<open>Misc useful stuff\<close> |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
648 |
|
55821 | 649 |
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
|
650 |
apply (subst bigo_plus_idemp [symmetric]) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
651 |
apply (rule set_plus_mono2) |
63473 | 652 |
apply assumption+ |
22665 | 653 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
654 |
|
55821 | 655 |
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
|
656 |
apply (subst bigo_plus_idemp [symmetric]) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
657 |
apply (rule set_plus_intro) |
63473 | 658 |
apply assumption+ |
22665 | 659 |
done |
55821 | 660 |
|
63473 | 661 |
lemma bigo_useful_const_mult: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)" |
662 |
for c :: "'a::linordered_field" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
663 |
apply (rule subsetD) |
63473 | 664 |
apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)") |
665 |
apply assumption |
|
666 |
apply (rule bigo_const_mult6) |
|
55821 | 667 |
apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)") |
63473 | 668 |
apply (erule ssubst) |
669 |
apply (erule set_times_intro2) |
|
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23373
diff
changeset
|
670 |
apply (simp add: func_times) |
22665 | 671 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
672 |
|
55821 | 673 |
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
|
674 |
apply (simp add: bigo_alt_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
675 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
676 |
apply (rule_tac x = c in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
677 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
678 |
apply (case_tac "x = 0") |
63473 | 679 |
apply simp |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
680 |
apply (subgoal_tac "x = Suc (x - 1)") |
63473 | 681 |
apply (erule ssubst) back |
682 |
apply (erule spec) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
683 |
apply simp |
22665 | 684 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
685 |
|
55821 | 686 |
lemma bigo_fix2: |
687 |
"(\<lambda>x. f ((x::nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow> |
|
688 |
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
|
689 |
apply (rule set_minus_imp_plus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
690 |
apply (rule bigo_fix) |
63473 | 691 |
apply (subst fun_diff_def) |
692 |
apply (subst fun_diff_def [symmetric]) |
|
693 |
apply (rule set_plus_imp_minus) |
|
694 |
apply simp |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25592
diff
changeset
|
695 |
apply (simp add: fun_diff_def) |
22665 | 696 |
done |
697 |
||
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
698 |
|
60500 | 699 |
subsection \<open>Less than or equal to\<close> |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
700 |
|
55821 | 701 |
definition lesso :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" (infixl "<o" 70) |
702 |
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
|
703 |
|
61945 | 704 |
lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> \<le> \<bar>f x\<bar> \<Longrightarrow> g =o O(h)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
705 |
apply (unfold bigo_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
706 |
apply clarsimp |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
707 |
apply (rule_tac x = c in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
708 |
apply (rule allI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
709 |
apply (rule order_trans) |
63473 | 710 |
apply (erule spec)+ |
22665 | 711 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
712 |
|
61945 | 713 |
lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> \<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
|
714 |
apply (erule bigo_lesseq1) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
715 |
apply (rule allI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
716 |
apply (drule_tac x = x in spec) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
717 |
apply (rule order_trans) |
63473 | 718 |
apply assumption |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
719 |
apply (rule abs_ge_self) |
22665 | 720 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
721 |
|
55821 | 722 |
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
|
723 |
apply (erule bigo_lesseq2) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
724 |
apply (rule allI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
725 |
apply (subst abs_of_nonneg) |
63473 | 726 |
apply (erule spec)+ |
22665 | 727 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
728 |
|
55821 | 729 |
lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow> |
61945 | 730 |
\<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> \<bar>f x\<bar> \<Longrightarrow> g =o O(h)" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
731 |
apply (erule bigo_lesseq1) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
732 |
apply (rule allI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
733 |
apply (subst abs_of_nonneg) |
63473 | 734 |
apply (erule spec)+ |
22665 | 735 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
736 |
|
55821 | 737 |
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
|
738 |
apply (unfold lesso_def) |
55821 | 739 |
apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0") |
63473 | 740 |
apply (erule ssubst) |
741 |
apply (rule bigo_zero) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
742 |
apply (unfold func_zero) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
743 |
apply (rule ext) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
744 |
apply (simp split: split_max) |
22665 | 745 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
746 |
|
63473 | 747 |
lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow> \<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
|
748 |
apply (unfold lesso_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
749 |
apply (rule bigo_lesseq4) |
63473 | 750 |
apply (erule set_plus_imp_minus) |
751 |
apply (rule allI) |
|
752 |
apply (rule max.cobounded2) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
753 |
apply (rule allI) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25592
diff
changeset
|
754 |
apply (subst fun_diff_def) |
55821 | 755 |
apply (case_tac "0 \<le> k x - g x") |
63473 | 756 |
apply simp |
757 |
apply (subst abs_of_nonneg) |
|
758 |
apply (drule_tac x = x in spec) back |
|
759 |
apply (simp add: algebra_simps) |
|
760 |
apply (subst diff_conv_add_uminus)+ |
|
761 |
apply (rule add_right_mono) |
|
762 |
apply (erule spec) |
|
55821 | 763 |
apply (rule order_trans) |
63473 | 764 |
prefer 2 |
765 |
apply (rule abs_ge_zero) |
|
29667 | 766 |
apply (simp add: algebra_simps) |
22665 | 767 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
768 |
|
63473 | 769 |
lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow> \<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
|
770 |
apply (unfold lesso_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
771 |
apply (rule bigo_lesseq4) |
63473 | 772 |
apply (erule set_plus_imp_minus) |
773 |
apply (rule allI) |
|
774 |
apply (rule max.cobounded2) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
775 |
apply (rule allI) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25592
diff
changeset
|
776 |
apply (subst fun_diff_def) |
55821 | 777 |
apply (case_tac "0 \<le> f x - k x") |
63473 | 778 |
apply simp |
779 |
apply (subst abs_of_nonneg) |
|
780 |
apply (drule_tac x = x in spec) back |
|
781 |
apply (simp add: algebra_simps) |
|
782 |
apply (subst diff_conv_add_uminus)+ |
|
783 |
apply (rule add_left_mono) |
|
784 |
apply (rule le_imp_neg_le) |
|
785 |
apply (erule spec) |
|
55821 | 786 |
apply (rule order_trans) |
63473 | 787 |
prefer 2 |
788 |
apply (rule abs_ge_zero) |
|
29667 | 789 |
apply (simp add: algebra_simps) |
22665 | 790 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
791 |
|
63473 | 792 |
lemma bigo_lesso4: "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)" |
793 |
for k :: "'a \<Rightarrow> 'b::linordered_field" |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
794 |
apply (unfold lesso_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
795 |
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
|
796 |
apply (drule bigo_abs5) back |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
25592
diff
changeset
|
797 |
apply (simp add: fun_diff_def) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
798 |
apply (drule bigo_useful_add) |
63473 | 799 |
apply assumption |
17199
59c1bfc81d91
moved lemmas that require the HOL-Complex logic image to Complex/ex/BigO_Complex.thy;
wenzelm
parents:
16961
diff
changeset
|
800 |
apply (erule bigo_lesseq2) back |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
801 |
apply (rule allI) |
55821 | 802 |
apply (auto simp add: func_plus fun_diff_def algebra_simps split: split_max abs_split) |
22665 | 803 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
804 |
|
61945 | 805 |
lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x \<le> g x + C * \<bar>h x\<bar>" |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
806 |
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
|
807 |
apply clarsimp |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
808 |
apply (rule_tac x = c in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
809 |
apply (rule allI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
810 |
apply (drule_tac x = x in spec) |
61945 | 811 |
apply (subgoal_tac "\<bar>max (f x - g x) 0\<bar> = max (f x - g x) 0") |
63473 | 812 |
apply (clarsimp simp add: algebra_simps) |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
813 |
apply (rule abs_of_nonneg) |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54230
diff
changeset
|
814 |
apply (rule max.cobounded2) |
22665 | 815 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
816 |
|
55821 | 817 |
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
|
818 |
apply (unfold lesso_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
819 |
apply (rule bigo_lesseq3) |
63473 | 820 |
apply (erule bigo_useful_add) |
821 |
apply assumption |
|
822 |
apply (force split: split_max) |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
823 |
apply (auto split: split_max simp add: func_plus) |
22665 | 824 |
done |
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
825 |
|
63473 | 826 |
lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> 0" |
827 |
for f g :: "nat \<Rightarrow> real" |
|
31337 | 828 |
apply (simp add: LIMSEQ_iff bigo_alt_def) |
29786 | 829 |
apply clarify |
830 |
apply (drule_tac x = "r / c" in spec) |
|
831 |
apply (drule mp) |
|
63473 | 832 |
apply simp |
29786 | 833 |
apply clarify |
834 |
apply (rule_tac x = no in exI) |
|
835 |
apply (rule allI) |
|
836 |
apply (drule_tac x = n in spec)+ |
|
837 |
apply (rule impI) |
|
838 |
apply (drule mp) |
|
63473 | 839 |
apply assumption |
29786 | 840 |
apply (rule order_le_less_trans) |
63473 | 841 |
apply assumption |
29786 | 842 |
apply (rule order_less_le_trans) |
63473 | 843 |
apply (subgoal_tac "c * \<bar>g n\<bar> < c * (r / c)") |
844 |
apply assumption |
|
845 |
apply (erule mult_strict_left_mono) |
|
846 |
apply assumption |
|
29786 | 847 |
apply simp |
55821 | 848 |
done |
29786 | 849 |
|
63473 | 850 |
lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> g \<longlonglongrightarrow> a" |
851 |
for f g h :: "nat \<Rightarrow> real" |
|
29786 | 852 |
apply (drule set_plus_imp_minus) |
853 |
apply (drule bigo_LIMSEQ1) |
|
63473 | 854 |
apply assumption |
29786 | 855 |
apply (simp only: fun_diff_def) |
60142 | 856 |
apply (erule Lim_transform2) |
29786 | 857 |
apply assumption |
55821 | 858 |
done |
29786 | 859 |
|
16908
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
860 |
end |