src/HOL/Metis_Examples/BigO.thy
author paulson
Mon Feb 28 15:06:36 2011 +0000 (2011-02-28)
changeset 41865 4e8483cc2cc5
parent 41541 1fa4725c4656
child 42103 6066a35f6678
permissions -rw-r--r--
declare ext [intro]: Extensionality now available by default
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(*  Title:      HOL/Metis_Examples/BigO.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Jasmin Blanchette, TU Muenchen
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Testing Metis.
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*)
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header {* Big O notation *}
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theory BigO
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imports
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  "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
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  Main
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  "~~/src/HOL/Library/Function_Algebras"
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  "~~/src/HOL/Library/Set_Algebras"
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begin
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subsection {* Definitions *}
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definition bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set"    ("(1O'(_'))") where
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  "O(f::('a => 'b)) ==   {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_pos_const" ]]
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lemma bigo_pos_const: "(EX (c::'a::linordered_idom). 
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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  apply (metis abs_ge_zero abs_of_nonneg Orderings.xt1(6) abs_mult)
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  done
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(*** Now various verions with an increasing shrink factor ***)
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sledgehammer_params [isar_proof, isar_shrink_factor = 1]
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lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom). 
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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proof -
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  fix c :: 'a and x :: 'b
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  assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
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  have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> \<bar>x\<^isub>1\<bar>" by (metis abs_ge_zero)
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  have F2: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
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  have F3: "\<forall>x\<^isub>1 x\<^isub>3. x\<^isub>3 \<le> \<bar>h x\<^isub>1\<bar> \<longrightarrow> x\<^isub>3 \<le> c * \<bar>f x\<^isub>1\<bar>" by (metis A1 order_trans)
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  have F4: "\<forall>x\<^isub>2 x\<^isub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^isub>3\<bar> * \<bar>x\<^isub>2\<bar> = \<bar>x\<^isub>3 * x\<^isub>2\<bar>"
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    by (metis abs_mult)
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  have F5: "\<forall>x\<^isub>3 x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> x\<^isub>1 \<longrightarrow> \<bar>x\<^isub>3 * x\<^isub>1\<bar> = \<bar>x\<^isub>3\<bar> * x\<^isub>1"
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    by (metis abs_mult_pos)
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  hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = \<bar>1\<bar> * x\<^isub>1" by (metis F2)
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  hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F2 abs_one)
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  hence "\<forall>x\<^isub>3. 0 \<le> \<bar>h x\<^isub>3\<bar> \<longrightarrow> \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>" by (metis F3)
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  hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>" by (metis F1)
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  hence "\<forall>x\<^isub>3. (0\<Colon>'a) \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^isub>3\<bar>" by (metis F5)
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  hence "\<forall>x\<^isub>3. (0\<Colon>'a) \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c * f x\<^isub>3\<bar>" by (metis F4)
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  hence "\<forall>x\<^isub>3. c * \<bar>f x\<^isub>3\<bar> = \<bar>c * f x\<^isub>3\<bar>" by (metis F1)
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  hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1)
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  thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F4)
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qed
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sledgehammer_params [isar_proof, isar_shrink_factor = 2]
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lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom). 
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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proof -
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  fix c :: 'a and x :: 'b
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  assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
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  have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
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  have F2: "\<forall>x\<^isub>2 x\<^isub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^isub>3\<bar> * \<bar>x\<^isub>2\<bar> = \<bar>x\<^isub>3 * x\<^isub>2\<bar>"
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    by (metis abs_mult)
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  have "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F1 abs_mult_pos abs_one)
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  hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>" by (metis A1 abs_ge_zero order_trans)
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  hence "\<forall>x\<^isub>3. 0 \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c * f x\<^isub>3\<bar>" by (metis F2 abs_mult_pos)
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  hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero)
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  thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F2)
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qed
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sledgehammer_params [isar_proof, isar_shrink_factor = 3]
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lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom). 
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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proof -
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  fix c :: 'a and x :: 'b
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  assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
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  have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
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  have F2: "\<forall>x\<^isub>3 x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> x\<^isub>1 \<longrightarrow> \<bar>x\<^isub>3 * x\<^isub>1\<bar> = \<bar>x\<^isub>3\<bar> * x\<^isub>1" by (metis abs_mult_pos)
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  hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F1 abs_one)
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  hence "\<forall>x\<^isub>3. 0 \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^isub>3\<bar>" by (metis F2 A1 abs_ge_zero order_trans)
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  thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis A1 abs_mult abs_ge_zero)
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qed
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sledgehammer_params [isar_proof, isar_shrink_factor = 4]
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lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom). 
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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proof -
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  fix c :: 'a and x :: 'b
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  assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
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  have "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
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  hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>"
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    by (metis A1 abs_ge_zero order_trans abs_mult_pos abs_one)
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  hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero abs_mult_pos abs_mult)
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  thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis abs_mult)
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qed
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sledgehammer_params [isar_proof, isar_shrink_factor = 1]
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lemma bigo_alt_def: "O(f) = 
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    {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
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by (auto simp add: bigo_def bigo_pos_const)
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_elt_subset" ]]
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lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
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  apply (auto simp add: bigo_alt_def)
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  apply (rule_tac x = "ca * c" in exI)
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  apply (rule conjI)
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  apply (rule mult_pos_pos)
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  apply (assumption)+ 
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(*sledgehammer*)
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  apply (rule allI)
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  apply (drule_tac x = "xa" in spec)+
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  apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))")
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  apply (erule order_trans)
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  apply (simp add: mult_ac)
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  apply (rule mult_left_mono, assumption)
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  apply (rule order_less_imp_le, assumption)
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done
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_refl" ]]
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lemma bigo_refl [intro]: "f : O(f)"
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apply (auto simp add: bigo_def)
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by (metis mult_1 order_refl)
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_zero" ]]
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lemma bigo_zero: "0 : O(g)"
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apply (auto simp add: bigo_def func_zero)
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by (metis mult_zero_left order_refl)
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lemma bigo_zero2: "O(%x.0) = {%x.0}"
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  by (auto simp add: bigo_def) 
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lemma bigo_plus_self_subset [intro]: 
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  "O(f) \<oplus> O(f) <= O(f)"
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  apply (auto simp add: bigo_alt_def set_plus_def)
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  apply (rule_tac x = "c + ca" in exI)
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  apply auto
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  apply (simp add: ring_distribs func_plus)
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  apply (blast intro:order_trans abs_triangle_ineq add_mono elim:) 
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done
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lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
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  apply (rule equalityI)
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  apply (rule bigo_plus_self_subset)
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  apply (rule set_zero_plus2) 
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  apply (rule bigo_zero)
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done
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lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) \<oplus> O(g)"
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  apply (rule subsetI)
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  apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
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  apply (subst bigo_pos_const [symmetric])+
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  apply (rule_tac x = 
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    "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
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  apply (rule conjI)
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  apply (rule_tac x = "c + c" in exI)
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  apply (clarsimp)
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  apply (auto)
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  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
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  apply (erule_tac x = xa in allE)
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  apply (erule order_trans)
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  apply (simp)
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  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
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  apply (erule order_trans)
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  apply (simp add: ring_distribs)
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  apply (rule mult_left_mono)
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  apply assumption
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  apply (simp add: order_less_le)
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  apply (rule mult_left_mono)
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  apply (simp add: abs_triangle_ineq)
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  apply (simp add: order_less_le)
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  apply (rule mult_nonneg_nonneg)
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  apply (rule add_nonneg_nonneg)
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  apply auto
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  apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0" 
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     in exI)
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  apply (rule conjI)
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  apply (rule_tac x = "c + c" in exI)
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  apply auto
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  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
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  apply (erule_tac x = xa in allE)
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  apply (erule order_trans)
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  apply (simp)
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  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
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  apply (erule order_trans)
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  apply (simp add: ring_distribs)
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  apply (rule mult_left_mono)
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  apply (simp add: order_less_le)
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  apply (simp add: order_less_le)
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  apply (rule mult_left_mono)
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  apply (rule abs_triangle_ineq)
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  apply (simp add: order_less_le)
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apply (metis abs_not_less_zero double_less_0_iff less_not_permute linorder_not_less mult_less_0_iff)
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  apply (rule ext)
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  apply (auto simp add: if_splits linorder_not_le)
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done
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lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A \<oplus> B <= O(f)"
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  apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> O(f)")
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  apply (erule order_trans)
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  apply simp
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  apply (auto del: subsetI simp del: bigo_plus_idemp)
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done
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq" ]]
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lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> 
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  O(f + g) = O(f) \<oplus> O(g)"
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  apply (rule equalityI)
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  apply (rule bigo_plus_subset)
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  apply (simp add: bigo_alt_def set_plus_def func_plus)
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  apply clarify 
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   242
(*sledgehammer*) 
paulson@23449
   243
  apply (rule_tac x = "max c ca" in exI)
paulson@23449
   244
  apply (rule conjI)
paulson@25087
   245
   apply (metis Orderings.less_max_iff_disj)
paulson@23449
   246
  apply clarify
paulson@23449
   247
  apply (drule_tac x = "xa" in spec)+
paulson@23449
   248
  apply (subgoal_tac "0 <= f xa + g xa")
nipkow@23477
   249
  apply (simp add: ring_distribs)
paulson@23449
   250
  apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
paulson@23449
   251
  apply (subgoal_tac "abs(a xa) + abs(b xa) <= 
paulson@23449
   252
      max c ca * f xa + max c ca * g xa")
paulson@23449
   253
  apply (blast intro: order_trans)
paulson@23449
   254
  defer 1
paulson@23449
   255
  apply (rule abs_triangle_ineq)
paulson@25087
   256
  apply (metis add_nonneg_nonneg)
paulson@23449
   257
  apply (rule add_mono)
blanchet@39259
   258
using [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq_simpler" ]]
blanchet@39259
   259
  apply (metis le_maxI2 linorder_linear min_max.sup_absorb1 mult_right_mono xt1(6))
blanchet@39259
   260
  apply (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans)
paulson@23449
   261
done
paulson@23449
   262
blanchet@38991
   263
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt" ]]
paulson@23449
   264
lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
paulson@23449
   265
    f : O(g)" 
paulson@23449
   266
  apply (auto simp add: bigo_def)
blanchet@36561
   267
(* Version 1: one-line proof *)
haftmann@35050
   268
  apply (metis abs_le_D1 linorder_class.not_less  order_less_le  Orderings.xt1(12)  abs_mult)
paulson@23449
   269
  done
paulson@23449
   270
wenzelm@26312
   271
lemma (*bigo_bounded_alt:*) "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
blanchet@36561
   272
    f : O(g)"
blanchet@36561
   273
apply (auto simp add: bigo_def)
blanchet@36561
   274
(* Version 2: structured proof *)
blanchet@36561
   275
proof -
blanchet@36561
   276
  assume "\<forall>x. f x \<le> c * g x"
blanchet@36561
   277
  thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans)
paulson@23449
   278
qed
paulson@23449
   279
blanchet@36561
   280
text{*So here is the easier (and more natural) problem using transitivity*}
blanchet@38991
   281
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]]
blanchet@36561
   282
lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" 
blanchet@36561
   283
apply (auto simp add: bigo_def)
blanchet@36561
   284
(* Version 1: one-line proof *)
blanchet@36561
   285
by (metis abs_ge_self abs_mult order_trans)
paulson@23449
   286
paulson@23449
   287
text{*So here is the easier (and more natural) problem using transitivity*}
blanchet@38991
   288
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]]
paulson@23449
   289
lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" 
paulson@23449
   290
  apply (auto simp add: bigo_def)
blanchet@36561
   291
(* Version 2: structured proof *)
blanchet@36561
   292
proof -
blanchet@36561
   293
  assume "\<forall>x. f x \<le> c * g x"
blanchet@36561
   294
  thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans)
paulson@23449
   295
qed
paulson@23449
   296
paulson@23449
   297
lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> 
paulson@23449
   298
    f : O(g)" 
paulson@23449
   299
  apply (erule bigo_bounded_alt [of f 1 g])
paulson@23449
   300
  apply simp
paulson@23449
   301
done
paulson@23449
   302
blanchet@38991
   303
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded2" ]]
paulson@23449
   304
lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
paulson@23449
   305
    f : lb +o O(g)"
blanchet@36561
   306
apply (rule set_minus_imp_plus)
blanchet@36561
   307
apply (rule bigo_bounded)
blanchet@36561
   308
 apply (auto simp add: diff_minus fun_Compl_def func_plus)
blanchet@36561
   309
 prefer 2
blanchet@36561
   310
 apply (drule_tac x = x in spec)+
hoelzl@36844
   311
 apply (metis add_right_mono add_commute diff_add_cancel diff_minus_eq_add le_less order_trans)
blanchet@36561
   312
proof -
blanchet@36561
   313
  fix x :: 'a
blanchet@36561
   314
  assume "\<forall>x. lb x \<le> f x"
blanchet@36561
   315
  thus "(0\<Colon>'b) \<le> f x + - lb x" by (metis not_leE diff_minus less_iff_diff_less_0 less_le_not_le)
paulson@23449
   316
qed
paulson@23449
   317
blanchet@38991
   318
declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs" ]]
paulson@23449
   319
lemma bigo_abs: "(%x. abs(f x)) =o O(f)" 
blanchet@36561
   320
apply (unfold bigo_def)
blanchet@36561
   321
apply auto
hoelzl@36844
   322
by (metis mult_1 order_refl)
paulson@23449
   323
blanchet@38991
   324
declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs2" ]]
paulson@23449
   325
lemma bigo_abs2: "f =o O(%x. abs(f x))"
blanchet@36561
   326
apply (unfold bigo_def)
blanchet@36561
   327
apply auto
hoelzl@36844
   328
by (metis mult_1 order_refl)
paulson@23449
   329
 
paulson@23449
   330
lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
blanchet@36561
   331
proof -
blanchet@36561
   332
  have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset)
blanchet@36561
   333
  have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs)
blanchet@36561
   334
  have "\<forall>u. u \<in> O(\<lambda>R. \<bar>u R\<bar>)" by (metis bigo_abs2)
blanchet@36561
   335
  thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" using F1 F2 by auto
blanchet@36561
   336
qed 
paulson@23449
   337
paulson@23449
   338
lemma bigo_abs4: "f =o g +o O(h) ==> 
paulson@23449
   339
    (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
paulson@23449
   340
  apply (drule set_plus_imp_minus)
paulson@23449
   341
  apply (rule set_minus_imp_plus)
berghofe@26814
   342
  apply (subst fun_diff_def)
paulson@23449
   343
proof -
paulson@23449
   344
  assume a: "f - g : O(h)"
paulson@23449
   345
  have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
paulson@23449
   346
    by (rule bigo_abs2)
paulson@23449
   347
  also have "... <= O(%x. abs (f x - g x))"
paulson@23449
   348
    apply (rule bigo_elt_subset)
paulson@23449
   349
    apply (rule bigo_bounded)
paulson@23449
   350
    apply force
paulson@23449
   351
    apply (rule allI)
paulson@23449
   352
    apply (rule abs_triangle_ineq3)
paulson@23449
   353
    done
paulson@23449
   354
  also have "... <= O(f - g)"
paulson@23449
   355
    apply (rule bigo_elt_subset)
berghofe@26814
   356
    apply (subst fun_diff_def)
paulson@23449
   357
    apply (rule bigo_abs)
paulson@23449
   358
    done
paulson@23449
   359
  also have "... <= O(h)"
wenzelm@23464
   360
    using a by (rule bigo_elt_subset)
paulson@23449
   361
  finally show "(%x. abs (f x) - abs (g x)) : O(h)".
paulson@23449
   362
qed
paulson@23449
   363
paulson@23449
   364
lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" 
paulson@23449
   365
by (unfold bigo_def, auto)
paulson@23449
   366
berghofe@26814
   367
lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) \<oplus> O(h)"
paulson@23449
   368
proof -
paulson@23449
   369
  assume "f : g +o O(h)"
berghofe@26814
   370
  also have "... <= O(g) \<oplus> O(h)"
paulson@23449
   371
    by (auto del: subsetI)
berghofe@26814
   372
  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
paulson@23449
   373
    apply (subst bigo_abs3 [symmetric])+
paulson@23449
   374
    apply (rule refl)
paulson@23449
   375
    done
paulson@23449
   376
  also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
paulson@23449
   377
    by (rule bigo_plus_eq [symmetric], auto)
paulson@23449
   378
  finally have "f : ...".
paulson@23449
   379
  then have "O(f) <= ..."
paulson@23449
   380
    by (elim bigo_elt_subset)
berghofe@26814
   381
  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
paulson@23449
   382
    by (rule bigo_plus_eq, auto)
paulson@23449
   383
  finally show ?thesis
paulson@23449
   384
    by (simp add: bigo_abs3 [symmetric])
paulson@23449
   385
qed
paulson@23449
   386
blanchet@38991
   387
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult" ]]
berghofe@26814
   388
lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)"
paulson@23449
   389
  apply (rule subsetI)
paulson@23449
   390
  apply (subst bigo_def)
paulson@23449
   391
  apply (auto simp del: abs_mult mult_ac
berghofe@26814
   392
              simp add: bigo_alt_def set_times_def func_times)
paulson@41865
   393
(*sledgehammer*)
paulson@23449
   394
  apply (rule_tac x = "c * ca" in exI)
paulson@23449
   395
  apply(rule allI)
paulson@23449
   396
  apply(erule_tac x = x in allE)+
paulson@23449
   397
  apply(subgoal_tac "c * ca * abs(f x * g x) = 
paulson@23449
   398
      (c * abs(f x)) * (ca * abs(g x))")
blanchet@38991
   399
using [[ sledgehammer_problem_prefix = "BigO__bigo_mult_simpler" ]]
paulson@23449
   400
prefer 2 
haftmann@26041
   401
apply (metis mult_assoc mult_left_commute
haftmann@35050
   402
  abs_of_pos mult_left_commute
haftmann@35050
   403
  abs_mult mult_pos_pos)
haftmann@26041
   404
  apply (erule ssubst) 
paulson@23449
   405
  apply (subst abs_mult)
blanchet@36561
   406
(* not quite as hard as BigO__bigo_mult_simpler_1 (a hard problem!) since
blanchet@36561
   407
   abs_mult has just been done *)
blanchet@36561
   408
by (metis abs_ge_zero mult_mono')
paulson@23449
   409
blanchet@38991
   410
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult2" ]]
paulson@23449
   411
lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
paulson@23449
   412
  apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
paulson@41865
   413
(*sledgehammer*)
paulson@23449
   414
  apply (rule_tac x = c in exI)
paulson@23449
   415
  apply clarify
paulson@23449
   416
  apply (drule_tac x = x in spec)
blanchet@38991
   417
using [[ sledgehammer_problem_prefix = "BigO__bigo_mult2_simpler" ]]
paulson@41865
   418
(*sledgehammer [no luck]*)
paulson@23449
   419
  apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
paulson@23449
   420
  apply (simp add: mult_ac)
paulson@23449
   421
  apply (rule mult_left_mono, assumption)
paulson@23449
   422
  apply (rule abs_ge_zero)
paulson@23449
   423
done
paulson@23449
   424
blanchet@38991
   425
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult3" ]]
paulson@23449
   426
lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
blanchet@36561
   427
by (metis bigo_mult set_rev_mp set_times_intro)
paulson@23449
   428
blanchet@38991
   429
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult4" ]]
paulson@23449
   430
lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
paulson@23449
   431
by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib)
paulson@23449
   432
paulson@23449
   433
paulson@23449
   434
lemma bigo_mult5: "ALL x. f x ~= 0 ==>
haftmann@35028
   435
    O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
paulson@23449
   436
proof -
wenzelm@41541
   437
  assume a: "ALL x. f x ~= 0"
paulson@23449
   438
  show "O(f * g) <= f *o O(g)"
paulson@23449
   439
  proof
paulson@23449
   440
    fix h
wenzelm@41541
   441
    assume h: "h : O(f * g)"
paulson@23449
   442
    then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
paulson@23449
   443
      by auto
paulson@23449
   444
    also have "... <= O((%x. 1 / f x) * (f * g))"
paulson@23449
   445
      by (rule bigo_mult2)
paulson@23449
   446
    also have "(%x. 1 / f x) * (f * g) = g"
paulson@23449
   447
      apply (simp add: func_times) 
paulson@23449
   448
      apply (rule ext)
wenzelm@41541
   449
      apply (simp add: a h nonzero_divide_eq_eq mult_ac)
paulson@23449
   450
      done
paulson@23449
   451
    finally have "(%x. (1::'b) / f x) * h : O(g)".
paulson@23449
   452
    then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
paulson@23449
   453
      by auto
paulson@23449
   454
    also have "f * ((%x. (1::'b) / f x) * h) = h"
paulson@23449
   455
      apply (simp add: func_times) 
paulson@23449
   456
      apply (rule ext)
wenzelm@41541
   457
      apply (simp add: a h nonzero_divide_eq_eq mult_ac)
paulson@23449
   458
      done
paulson@23449
   459
    finally show "h : f *o O(g)".
paulson@23449
   460
  qed
paulson@23449
   461
qed
paulson@23449
   462
blanchet@38991
   463
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult6" ]]
paulson@23449
   464
lemma bigo_mult6: "ALL x. f x ~= 0 ==>
haftmann@35028
   465
    O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)"
paulson@23449
   466
by (metis bigo_mult2 bigo_mult5 order_antisym)
paulson@23449
   467
paulson@23449
   468
(*proof requires relaxing relevance: 2007-01-25*)
blanchet@38991
   469
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult7" ]]
paulson@23449
   470
  declare bigo_mult6 [simp]
paulson@23449
   471
lemma bigo_mult7: "ALL x. f x ~= 0 ==>
haftmann@35028
   472
    O(f * g) <= O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
paulson@23449
   473
(*sledgehammer*)
paulson@23449
   474
  apply (subst bigo_mult6)
paulson@23449
   475
  apply assumption
paulson@23449
   476
  apply (rule set_times_mono3) 
paulson@23449
   477
  apply (rule bigo_refl)
paulson@23449
   478
done
paulson@23449
   479
  declare bigo_mult6 [simp del]
paulson@23449
   480
blanchet@38991
   481
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult8" ]]
paulson@23449
   482
  declare bigo_mult7[intro!]
paulson@23449
   483
lemma bigo_mult8: "ALL x. f x ~= 0 ==>
haftmann@35028
   484
    O(f * g) = O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
paulson@23449
   485
by (metis bigo_mult bigo_mult7 order_antisym_conv)
paulson@23449
   486
paulson@23449
   487
lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
berghofe@26814
   488
  by (auto simp add: bigo_def fun_Compl_def)
paulson@23449
   489
paulson@23449
   490
lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
paulson@23449
   491
  apply (rule set_minus_imp_plus)
paulson@23449
   492
  apply (drule set_plus_imp_minus)
paulson@23449
   493
  apply (drule bigo_minus)
paulson@23449
   494
  apply (simp add: diff_minus)
paulson@23449
   495
done
paulson@23449
   496
paulson@23449
   497
lemma bigo_minus3: "O(-f) = O(f)"
berghofe@26814
   498
  by (auto simp add: bigo_def fun_Compl_def abs_minus_cancel)
paulson@23449
   499
paulson@23449
   500
lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
paulson@23449
   501
proof -
paulson@23449
   502
  assume a: "f : O(g)"
paulson@23449
   503
  show "f +o O(g) <= O(g)"
paulson@23449
   504
  proof -
paulson@23449
   505
    have "f : O(f)" by auto
berghofe@26814
   506
    then have "f +o O(g) <= O(f) \<oplus> O(g)"
paulson@23449
   507
      by (auto del: subsetI)
berghofe@26814
   508
    also have "... <= O(g) \<oplus> O(g)"
paulson@23449
   509
    proof -
paulson@23449
   510
      from a have "O(f) <= O(g)" by (auto del: subsetI)
paulson@23449
   511
      thus ?thesis by (auto del: subsetI)
paulson@23449
   512
    qed
paulson@23449
   513
    also have "... <= O(g)" by (simp add: bigo_plus_idemp)
paulson@23449
   514
    finally show ?thesis .
paulson@23449
   515
  qed
paulson@23449
   516
qed
paulson@23449
   517
paulson@23449
   518
lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
paulson@23449
   519
proof -
paulson@23449
   520
  assume a: "f : O(g)"
paulson@23449
   521
  show "O(g) <= f +o O(g)"
paulson@23449
   522
  proof -
paulson@23449
   523
    from a have "-f : O(g)" by auto
paulson@23449
   524
    then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
paulson@23449
   525
    then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
paulson@23449
   526
    also have "f +o (-f +o O(g)) = O(g)"
paulson@23449
   527
      by (simp add: set_plus_rearranges)
paulson@23449
   528
    finally show ?thesis .
paulson@23449
   529
  qed
paulson@23449
   530
qed
paulson@23449
   531
blanchet@38991
   532
declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_absorb" ]]
paulson@23449
   533
lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
paulson@41865
   534
by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff)
paulson@23449
   535
paulson@23449
   536
lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
paulson@23449
   537
  apply (subgoal_tac "f +o A <= f +o O(g)")
paulson@23449
   538
  apply force+
paulson@23449
   539
done
paulson@23449
   540
paulson@23449
   541
lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
paulson@23449
   542
  apply (subst set_minus_plus [symmetric])
paulson@23449
   543
  apply (subgoal_tac "g - f = - (f - g)")
paulson@23449
   544
  apply (erule ssubst)
paulson@23449
   545
  apply (rule bigo_minus)
paulson@23449
   546
  apply (subst set_minus_plus)
paulson@23449
   547
  apply assumption
paulson@23449
   548
  apply  (simp add: diff_minus add_ac)
paulson@23449
   549
done
paulson@23449
   550
paulson@23449
   551
lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
paulson@23449
   552
  apply (rule iffI)
paulson@23449
   553
  apply (erule bigo_add_commute_imp)+
paulson@23449
   554
done
paulson@23449
   555
paulson@23449
   556
lemma bigo_const1: "(%x. c) : O(%x. 1)"
paulson@23449
   557
by (auto simp add: bigo_def mult_ac)
paulson@23449
   558
blanchet@38991
   559
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const2" ]]
paulson@23449
   560
lemma (*bigo_const2 [intro]:*) "O(%x. c) <= O(%x. 1)"
paulson@41865
   561
by (metis bigo_const1 bigo_elt_subset)
paulson@23449
   562
paulson@41865
   563
lemma bigo_const2 [intro]: "O(%x. c::'b::linordered_idom) <= O(%x. 1)"
blanchet@36561
   564
(* "thus" had to be replaced by "show" with an explicit reference to "F1" *)
blanchet@36561
   565
proof -
blanchet@36561
   566
  have F1: "\<forall>u. (\<lambda>Q. u) \<in> O(\<lambda>Q. 1)" by (metis bigo_const1)
blanchet@36561
   567
  show "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)" by (metis F1 bigo_elt_subset)
paulson@23449
   568
qed
paulson@23449
   569
blanchet@38991
   570
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const3" ]]
haftmann@35028
   571
lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
paulson@23449
   572
apply (simp add: bigo_def)
blanchet@36561
   573
by (metis abs_eq_0 left_inverse order_refl)
paulson@23449
   574
haftmann@35028
   575
lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
paulson@23449
   576
by (rule bigo_elt_subset, rule bigo_const3, assumption)
paulson@23449
   577
haftmann@35028
   578
lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==> 
paulson@23449
   579
    O(%x. c) = O(%x. 1)"
paulson@23449
   580
by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
paulson@23449
   581
blanchet@38991
   582
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult1" ]]
paulson@23449
   583
lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
paulson@24937
   584
  apply (simp add: bigo_def abs_mult)
blanchet@36561
   585
by (metis le_less)
paulson@23449
   586
paulson@23449
   587
lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
paulson@23449
   588
by (rule bigo_elt_subset, rule bigo_const_mult1)
paulson@23449
   589
blanchet@38991
   590
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult3" ]]
haftmann@35028
   591
lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%x. c * f x)"
paulson@23449
   592
  apply (simp add: bigo_def)
blanchet@36561
   593
(*sledgehammer [no luck]*)
paulson@23449
   594
  apply (rule_tac x = "abs(inverse c)" in exI)
paulson@23449
   595
  apply (simp only: abs_mult [symmetric] mult_assoc [symmetric])
paulson@23449
   596
apply (subst left_inverse) 
paulson@41865
   597
apply (auto )
paulson@23449
   598
done
paulson@23449
   599
haftmann@35028
   600
lemma bigo_const_mult4: "(c::'a::linordered_field) ~= 0 ==> 
paulson@23449
   601
    O(f) <= O(%x. c * f x)"
paulson@23449
   602
by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
paulson@23449
   603
haftmann@35028
   604
lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==> 
paulson@23449
   605
    O(%x. c * f x) = O(f)"
paulson@23449
   606
by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
paulson@23449
   607
blanchet@38991
   608
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult5" ]]
haftmann@35028
   609
lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==> 
paulson@23449
   610
    (%x. c) *o O(f) = O(f)"
paulson@23449
   611
  apply (auto del: subsetI)
paulson@23449
   612
  apply (rule order_trans)
paulson@23449
   613
  apply (rule bigo_mult2)
paulson@23449
   614
  apply (simp add: func_times)
paulson@23449
   615
  apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times)
paulson@23449
   616
  apply (rule_tac x = "%y. inverse c * x y" in exI)
paulson@24942
   617
  apply (rename_tac g d) 
paulson@24942
   618
  apply safe
paulson@24942
   619
  apply (rule_tac [2] ext) 
paulson@24942
   620
   prefer 2 
haftmann@26041
   621
   apply simp
paulson@24942
   622
  apply (simp add: mult_assoc [symmetric] abs_mult)
blanchet@39259
   623
  (* couldn't get this proof without the step above *)
blanchet@39259
   624
proof -
blanchet@39259
   625
  fix g :: "'b \<Rightarrow> 'a" and d :: 'a
blanchet@39259
   626
  assume A1: "c \<noteq> (0\<Colon>'a)"
blanchet@39259
   627
  assume A2: "\<forall>x\<Colon>'b. \<bar>g x\<bar> \<le> d * \<bar>f x\<bar>"
blanchet@39259
   628
  have F1: "inverse \<bar>c\<bar> = \<bar>inverse c\<bar>" using A1 by (metis nonzero_abs_inverse)
blanchet@39259
   629
  have F2: "(0\<Colon>'a) < \<bar>c\<bar>" using A1 by (metis zero_less_abs_iff)
blanchet@39259
   630
  have "(0\<Colon>'a) < \<bar>c\<bar> \<longrightarrow> (0\<Colon>'a) < \<bar>inverse c\<bar>" using F1 by (metis positive_imp_inverse_positive)
blanchet@39259
   631
  hence "(0\<Colon>'a) < \<bar>inverse c\<bar>" using F2 by metis
blanchet@39259
   632
  hence F3: "(0\<Colon>'a) \<le> \<bar>inverse c\<bar>" by (metis order_le_less)
blanchet@39259
   633
  have "\<exists>(u\<Colon>'a) SKF\<^isub>7\<Colon>'a \<Rightarrow> 'b. \<bar>g (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar>"
blanchet@39259
   634
    using A2 by metis
blanchet@39259
   635
  hence F4: "\<exists>(u\<Colon>'a) SKF\<^isub>7\<Colon>'a \<Rightarrow> 'b. \<bar>g (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar> \<and> (0\<Colon>'a) \<le> \<bar>inverse c\<bar>"
blanchet@39259
   636
    using F3 by metis
blanchet@39259
   637
  hence "\<exists>(v\<Colon>'a) (u\<Colon>'a) SKF\<^isub>7\<Colon>'a \<Rightarrow> 'b. \<bar>inverse c\<bar> * \<bar>g (SKF\<^isub>7 (u * v))\<bar> \<le> u * (v * \<bar>f (SKF\<^isub>7 (u * v))\<bar>)"
blanchet@39259
   638
    by (metis comm_mult_left_mono)
blanchet@39259
   639
  thus "\<exists>ca\<Colon>'a. \<forall>x\<Colon>'b. \<bar>inverse c\<bar> * \<bar>g x\<bar> \<le> ca * \<bar>f x\<bar>"
blanchet@39259
   640
    using A2 F4 by (metis ab_semigroup_mult_class.mult_ac(1) comm_mult_left_mono)
blanchet@39259
   641
qed
paulson@23449
   642
paulson@23449
   643
blanchet@38991
   644
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult6" ]]
paulson@23449
   645
lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
paulson@23449
   646
  apply (auto intro!: subsetI
paulson@23449
   647
    simp add: bigo_def elt_set_times_def func_times
paulson@23449
   648
    simp del: abs_mult mult_ac)
paulson@41865
   649
(*sledgehammer*)
paulson@23449
   650
  apply (rule_tac x = "ca * (abs c)" in exI)
paulson@23449
   651
  apply (rule allI)
paulson@23449
   652
  apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
paulson@23449
   653
  apply (erule ssubst)
paulson@23449
   654
  apply (subst abs_mult)
paulson@23449
   655
  apply (rule mult_left_mono)
paulson@23449
   656
  apply (erule spec)
paulson@23449
   657
  apply simp
paulson@23449
   658
  apply(simp add: mult_ac)
paulson@23449
   659
done
paulson@23449
   660
paulson@23449
   661
lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
paulson@23449
   662
proof -
paulson@23449
   663
  assume "f =o O(g)"
paulson@23449
   664
  then have "(%x. c) * f =o (%x. c) *o O(g)"
paulson@23449
   665
    by auto
paulson@23449
   666
  also have "(%x. c) * f = (%x. c * f x)"
paulson@23449
   667
    by (simp add: func_times)
paulson@23449
   668
  also have "(%x. c) *o O(g) <= O(g)"
paulson@23449
   669
    by (auto del: subsetI)
paulson@23449
   670
  finally show ?thesis .
paulson@23449
   671
qed
paulson@23449
   672
paulson@23449
   673
lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
paulson@23449
   674
by (unfold bigo_def, auto)
paulson@23449
   675
paulson@23449
   676
lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o 
paulson@23449
   677
    O(%x. h(k x))"
berghofe@26814
   678
  apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def
paulson@23449
   679
      func_plus)
paulson@23449
   680
  apply (erule bigo_compose1)
paulson@23449
   681
done
paulson@23449
   682
paulson@23449
   683
subsection {* Setsum *}
paulson@23449
   684
paulson@23449
   685
lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==> 
paulson@23449
   686
    EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
paulson@23449
   687
      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"  
paulson@23449
   688
  apply (auto simp add: bigo_def)
paulson@23449
   689
  apply (rule_tac x = "abs c" in exI)
paulson@23449
   690
  apply (subst abs_of_nonneg) back back
paulson@23449
   691
  apply (rule setsum_nonneg)
paulson@23449
   692
  apply force
paulson@23449
   693
  apply (subst setsum_right_distrib)
paulson@23449
   694
  apply (rule allI)
paulson@23449
   695
  apply (rule order_trans)
paulson@23449
   696
  apply (rule setsum_abs)
paulson@23449
   697
  apply (rule setsum_mono)
paulson@23449
   698
apply (blast intro: order_trans mult_right_mono abs_ge_self) 
paulson@23449
   699
done
paulson@23449
   700
blanchet@38991
   701
declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum1" ]]
paulson@23449
   702
lemma bigo_setsum1: "ALL x y. 0 <= h x y ==> 
paulson@23449
   703
    EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
paulson@23449
   704
      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
paulson@23449
   705
  apply (rule bigo_setsum_main)
paulson@41865
   706
(*sledgehammer*)
paulson@23449
   707
  apply force
paulson@23449
   708
  apply clarsimp
paulson@23449
   709
  apply (rule_tac x = c in exI)
paulson@23449
   710
  apply force
paulson@23449
   711
done
paulson@23449
   712
paulson@23449
   713
lemma bigo_setsum2: "ALL y. 0 <= h y ==> 
paulson@23449
   714
    EX c. ALL y. abs(f y) <= c * (h y) ==>
paulson@23449
   715
      (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
paulson@23449
   716
by (rule bigo_setsum1, auto)  
paulson@23449
   717
blanchet@38991
   718
declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum3" ]]
paulson@23449
   719
lemma bigo_setsum3: "f =o O(h) ==>
paulson@23449
   720
    (%x. SUM y : A x. (l x y) * f(k x y)) =o
paulson@23449
   721
      O(%x. SUM y : A x. abs(l x y * h(k x y)))"
paulson@23449
   722
  apply (rule bigo_setsum1)
paulson@23449
   723
  apply (rule allI)+
paulson@23449
   724
  apply (rule abs_ge_zero)
paulson@23449
   725
  apply (unfold bigo_def)
paulson@41865
   726
  apply (auto simp add: abs_mult)
paulson@41865
   727
(*sledgehammer*)
paulson@23449
   728
  apply (rule_tac x = c in exI)
paulson@23449
   729
  apply (rule allI)+
paulson@23449
   730
  apply (subst mult_left_commute)
paulson@23449
   731
  apply (rule mult_left_mono)
paulson@23449
   732
  apply (erule spec)
paulson@23449
   733
  apply (rule abs_ge_zero)
paulson@23449
   734
done
paulson@23449
   735
paulson@23449
   736
lemma bigo_setsum4: "f =o g +o O(h) ==>
paulson@23449
   737
    (%x. SUM y : A x. l x y * f(k x y)) =o
paulson@23449
   738
      (%x. SUM y : A x. l x y * g(k x y)) +o
paulson@23449
   739
        O(%x. SUM y : A x. abs(l x y * h(k x y)))"
paulson@23449
   740
  apply (rule set_minus_imp_plus)
berghofe@26814
   741
  apply (subst fun_diff_def)
paulson@23449
   742
  apply (subst setsum_subtractf [symmetric])
paulson@23449
   743
  apply (subst right_diff_distrib [symmetric])
paulson@23449
   744
  apply (rule bigo_setsum3)
berghofe@26814
   745
  apply (subst fun_diff_def [symmetric])
paulson@23449
   746
  apply (erule set_plus_imp_minus)
paulson@23449
   747
done
paulson@23449
   748
blanchet@38991
   749
declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum5" ]]
paulson@23449
   750
lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==> 
paulson@23449
   751
    ALL x. 0 <= h x ==>
paulson@23449
   752
      (%x. SUM y : A x. (l x y) * f(k x y)) =o
paulson@23449
   753
        O(%x. SUM y : A x. (l x y) * h(k x y))" 
paulson@23449
   754
  apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) = 
paulson@23449
   755
      (%x. SUM y : A x. abs((l x y) * h(k x y)))")
paulson@23449
   756
  apply (erule ssubst)
paulson@23449
   757
  apply (erule bigo_setsum3)
paulson@23449
   758
  apply (rule ext)
paulson@23449
   759
  apply (rule setsum_cong2)
paulson@23449
   760
  apply (thin_tac "f \<in> O(h)") 
paulson@24942
   761
apply (metis abs_of_nonneg zero_le_mult_iff)
paulson@23449
   762
done
paulson@23449
   763
paulson@23449
   764
lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
paulson@23449
   765
    ALL x. 0 <= h x ==>
paulson@23449
   766
      (%x. SUM y : A x. (l x y) * f(k x y)) =o
paulson@23449
   767
        (%x. SUM y : A x. (l x y) * g(k x y)) +o
paulson@23449
   768
          O(%x. SUM y : A x. (l x y) * h(k x y))" 
paulson@23449
   769
  apply (rule set_minus_imp_plus)
berghofe@26814
   770
  apply (subst fun_diff_def)
paulson@23449
   771
  apply (subst setsum_subtractf [symmetric])
paulson@23449
   772
  apply (subst right_diff_distrib [symmetric])
paulson@23449
   773
  apply (rule bigo_setsum5)
berghofe@26814
   774
  apply (subst fun_diff_def [symmetric])
paulson@23449
   775
  apply (drule set_plus_imp_minus)
paulson@23449
   776
  apply auto
paulson@23449
   777
done
paulson@23449
   778
paulson@23449
   779
subsection {* Misc useful stuff *}
paulson@23449
   780
paulson@23449
   781
lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
berghofe@26814
   782
  A \<oplus> B <= O(f)"
paulson@23449
   783
  apply (subst bigo_plus_idemp [symmetric])
paulson@23449
   784
  apply (rule set_plus_mono2)
paulson@23449
   785
  apply assumption+
paulson@23449
   786
done
paulson@23449
   787
paulson@23449
   788
lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
paulson@23449
   789
  apply (subst bigo_plus_idemp [symmetric])
paulson@23449
   790
  apply (rule set_plus_intro)
paulson@23449
   791
  apply assumption+
paulson@23449
   792
done
paulson@23449
   793
  
haftmann@35028
   794
lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 ==> 
paulson@23449
   795
    (%x. c) * f =o O(h) ==> f =o O(h)"
paulson@23449
   796
  apply (rule subsetD)
paulson@23449
   797
  apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
paulson@23449
   798
  apply assumption
paulson@23449
   799
  apply (rule bigo_const_mult6)
paulson@23449
   800
  apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
paulson@23449
   801
  apply (erule ssubst)
paulson@23449
   802
  apply (erule set_times_intro2)
paulson@23449
   803
  apply (simp add: func_times) 
paulson@23449
   804
done
paulson@23449
   805
blanchet@38991
   806
declare [[ sledgehammer_problem_prefix = "BigO__bigo_fix" ]]
paulson@23449
   807
lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
paulson@23449
   808
    f =o O(h)"
paulson@23449
   809
  apply (simp add: bigo_alt_def)
paulson@41865
   810
(*sledgehammer*)
paulson@23449
   811
  apply clarify
paulson@23449
   812
  apply (rule_tac x = c in exI)
paulson@23449
   813
  apply safe
paulson@23449
   814
  apply (case_tac "x = 0")
haftmann@35050
   815
apply (metis abs_ge_zero  abs_zero  order_less_le  split_mult_pos_le) 
paulson@23449
   816
  apply (subgoal_tac "x = Suc (x - 1)")
paulson@23816
   817
  apply metis
paulson@23449
   818
  apply simp
paulson@23449
   819
  done
paulson@23449
   820
paulson@23449
   821
paulson@23449
   822
lemma bigo_fix2: 
paulson@23449
   823
    "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==> 
paulson@23449
   824
       f 0 = g 0 ==> f =o g +o O(h)"
paulson@23449
   825
  apply (rule set_minus_imp_plus)
paulson@23449
   826
  apply (rule bigo_fix)
berghofe@26814
   827
  apply (subst fun_diff_def)
berghofe@26814
   828
  apply (subst fun_diff_def [symmetric])
paulson@23449
   829
  apply (rule set_plus_imp_minus)
paulson@23449
   830
  apply simp
berghofe@26814
   831
  apply (simp add: fun_diff_def)
paulson@23449
   832
done
paulson@23449
   833
paulson@23449
   834
subsection {* Less than or equal to *}
paulson@23449
   835
haftmann@35416
   836
definition lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where
paulson@23449
   837
  "f <o g == (%x. max (f x - g x) 0)"
paulson@23449
   838
paulson@23449
   839
lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
paulson@23449
   840
    g =o O(h)"
paulson@23449
   841
  apply (unfold bigo_def)
paulson@23449
   842
  apply clarsimp
paulson@23449
   843
apply (blast intro: order_trans) 
paulson@23449
   844
done
paulson@23449
   845
paulson@23449
   846
lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
paulson@23449
   847
      g =o O(h)"
paulson@23449
   848
  apply (erule bigo_lesseq1)
paulson@23449
   849
apply (blast intro: abs_ge_self order_trans) 
paulson@23449
   850
done
paulson@23449
   851
paulson@23449
   852
lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
paulson@23449
   853
      g =o O(h)"
paulson@23449
   854
  apply (erule bigo_lesseq2)
paulson@23449
   855
  apply (rule allI)
paulson@23449
   856
  apply (subst abs_of_nonneg)
paulson@23449
   857
  apply (erule spec)+
paulson@23449
   858
done
paulson@23449
   859
paulson@23449
   860
lemma bigo_lesseq4: "f =o O(h) ==>
paulson@23449
   861
    ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
paulson@23449
   862
      g =o O(h)"
paulson@23449
   863
  apply (erule bigo_lesseq1)
paulson@23449
   864
  apply (rule allI)
paulson@23449
   865
  apply (subst abs_of_nonneg)
paulson@23449
   866
  apply (erule spec)+
paulson@23449
   867
done
paulson@23449
   868
blanchet@38991
   869
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso1" ]]
paulson@23449
   870
lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
blanchet@36561
   871
apply (unfold lesso_def)
blanchet@36561
   872
apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
blanchet@36561
   873
proof -
blanchet@36561
   874
  assume "(\<lambda>x. max (f x - g x) 0) = 0"
blanchet@36561
   875
  thus "(\<lambda>x. max (f x - g x) 0) \<in> O(h)" by (metis bigo_zero)
blanchet@36561
   876
next
blanchet@36561
   877
  show "\<forall>x\<Colon>'a. f x \<le> g x \<Longrightarrow> (\<lambda>x\<Colon>'a. max (f x - g x) (0\<Colon>'b)) = (0\<Colon>'a \<Rightarrow> 'b)"
paulson@23449
   878
  apply (unfold func_zero)
paulson@23449
   879
  apply (rule ext)
blanchet@36561
   880
  by (simp split: split_max)
blanchet@36561
   881
qed
paulson@23449
   882
blanchet@38991
   883
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso2" ]]
paulson@23449
   884
lemma bigo_lesso2: "f =o g +o O(h) ==>
paulson@23449
   885
    ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
paulson@23449
   886
      k <o g =o O(h)"
paulson@23449
   887
  apply (unfold lesso_def)
paulson@23449
   888
  apply (rule bigo_lesseq4)
paulson@23449
   889
  apply (erule set_plus_imp_minus)
paulson@23449
   890
  apply (rule allI)
paulson@23449
   891
  apply (rule le_maxI2)
paulson@23449
   892
  apply (rule allI)
berghofe@26814
   893
  apply (subst fun_diff_def)
paulson@23449
   894
apply (erule thin_rl)
paulson@41865
   895
(*sledgehammer*)
paulson@23449
   896
  apply (case_tac "0 <= k x - g x")
blanchet@36561
   897
(* apply (metis abs_le_iff add_le_imp_le_right diff_minus le_less
blanchet@36561
   898
                le_max_iff_disj min_max.le_supE min_max.sup_absorb2
blanchet@36561
   899
                min_max.sup_commute) *)
blanchet@37320
   900
proof -
blanchet@37320
   901
  fix x :: 'a
blanchet@37320
   902
  assume "\<forall>x\<Colon>'a. k x \<le> f x"
blanchet@37320
   903
  hence F1: "\<forall>x\<^isub>1\<Colon>'a. max (k x\<^isub>1) (f x\<^isub>1) = f x\<^isub>1" by (metis min_max.sup_absorb2)
blanchet@37320
   904
  assume "(0\<Colon>'b) \<le> k x - g x"
blanchet@37320
   905
  hence F2: "max (0\<Colon>'b) (k x - g x) = k x - g x" by (metis min_max.sup_absorb2)
blanchet@37320
   906
  have F3: "\<forall>x\<^isub>1\<Colon>'b. x\<^isub>1 \<le> \<bar>x\<^isub>1\<bar>" by (metis abs_le_iff le_less)
blanchet@37320
   907
  have "\<forall>(x\<^isub>2\<Colon>'b) x\<^isub>1\<Colon>'b. max x\<^isub>1 x\<^isub>2 \<le> x\<^isub>2 \<or> max x\<^isub>1 x\<^isub>2 \<le> x\<^isub>1" by (metis le_less le_max_iff_disj)
blanchet@37320
   908
  hence "\<forall>(x\<^isub>3\<Colon>'b) (x\<^isub>2\<Colon>'b) x\<^isub>1\<Colon>'b. x\<^isub>1 - x\<^isub>2 \<le> x\<^isub>3 - x\<^isub>2 \<or> x\<^isub>3 \<le> x\<^isub>1" by (metis add_le_imp_le_right diff_minus min_max.le_supE)
blanchet@37320
   909
  hence "k x - g x \<le> f x - g x" by (metis F1 le_less min_max.sup_absorb2 min_max.sup_commute)
blanchet@37320
   910
  hence "k x - g x \<le> \<bar>f x - g x\<bar>" by (metis F3 le_max_iff_disj min_max.sup_absorb2)
blanchet@37320
   911
  thus "max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>" by (metis F2 min_max.sup_commute)
blanchet@36561
   912
next
blanchet@36561
   913
  show "\<And>x\<Colon>'a.
blanchet@36561
   914
       \<lbrakk>\<forall>x\<Colon>'a. (0\<Colon>'b) \<le> k x; \<forall>x\<Colon>'a. k x \<le> f x; \<not> (0\<Colon>'b) \<le> k x - g x\<rbrakk>
blanchet@36561
   915
       \<Longrightarrow> max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>"
blanchet@36561
   916
    by (metis abs_ge_zero le_cases min_max.sup_absorb2)
paulson@24545
   917
qed
paulson@23449
   918
blanchet@38991
   919
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3" ]]
paulson@23449
   920
lemma bigo_lesso3: "f =o g +o O(h) ==>
paulson@23449
   921
    ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
paulson@23449
   922
      f <o k =o O(h)"
paulson@23449
   923
  apply (unfold lesso_def)
paulson@23449
   924
  apply (rule bigo_lesseq4)
paulson@23449
   925
  apply (erule set_plus_imp_minus)
paulson@23449
   926
  apply (rule allI)
paulson@23449
   927
  apply (rule le_maxI2)
paulson@23449
   928
  apply (rule allI)
berghofe@26814
   929
  apply (subst fun_diff_def)
paulson@23449
   930
apply (erule thin_rl) 
paulson@41865
   931
(*sledgehammer*)
paulson@23449
   932
  apply (case_tac "0 <= f x - k x")
nipkow@29667
   933
  apply (simp)
paulson@23449
   934
  apply (subst abs_of_nonneg)
paulson@23449
   935
  apply (drule_tac x = x in spec) back
blanchet@38991
   936
using [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3_simpler" ]]
paulson@24545
   937
apply (metis diff_less_0_iff_less linorder_not_le not_leE uminus_add_conv_diff xt1(12) xt1(6))
paulson@24545
   938
apply (metis add_minus_cancel diff_le_eq le_diff_eq uminus_add_conv_diff)
haftmann@29511
   939
apply (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute)
paulson@23449
   940
done
paulson@23449
   941
haftmann@35028
   942
lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::linordered_field) ==>
paulson@23449
   943
    g =o h +o O(k) ==> f <o h =o O(k)"
paulson@23449
   944
  apply (unfold lesso_def)
paulson@23449
   945
  apply (drule set_plus_imp_minus)
paulson@23449
   946
  apply (drule bigo_abs5) back
berghofe@26814
   947
  apply (simp add: fun_diff_def)
paulson@23449
   948
  apply (drule bigo_useful_add)
paulson@23449
   949
  apply assumption
paulson@23449
   950
  apply (erule bigo_lesseq2) back
paulson@23449
   951
  apply (rule allI)
nipkow@29667
   952
  apply (auto simp add: func_plus fun_diff_def algebra_simps
paulson@23449
   953
    split: split_max abs_split)
paulson@23449
   954
done
paulson@23449
   955
blanchet@38991
   956
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso5" ]]
paulson@23449
   957
lemma bigo_lesso5: "f <o g =o O(h) ==>
paulson@23449
   958
    EX C. ALL x. f x <= g x + C * abs(h x)"
paulson@23449
   959
  apply (simp only: lesso_def bigo_alt_def)
paulson@23449
   960
  apply clarsimp
paulson@24855
   961
  apply (metis abs_if abs_mult add_commute diff_le_eq less_not_permute)  
paulson@23449
   962
done
paulson@23449
   963
paulson@23449
   964
end