author | wenzelm |
Thu, 13 Jan 2011 23:50:16 +0100 | |
changeset 41541 | 1fa4725c4656 |
parent 41413 | 64cd30d6b0b8 |
child 41865 | 4e8483cc2cc5 |
permissions | -rw-r--r-- |
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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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|
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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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||
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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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|
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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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apply (auto simp add: bigo_def) |
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apply (rule ext) |
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apply auto |
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done |
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lemma bigo_plus_self_subset [intro]: |
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"O(f) \<oplus> O(f) <= O(f)" |
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apply (auto simp add: bigo_alt_def set_plus_def) |
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apply (rule_tac x = "c + ca" in exI) |
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apply auto |
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apply (simp add: ring_distribs func_plus) |
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apply (blast intro:order_trans abs_triangle_ineq add_mono elim:) |
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done |
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||
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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) |
200 |
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) |
221 |
apply (simp add: order_less_le) |
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222 |
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) |
23449 | 227 |
apply (rule ext) |
228 |
apply (auto simp add: if_splits linorder_not_le) |
|
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done |
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||
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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) |
234 |
apply simp |
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apply (auto del: subsetI simp del: bigo_plus_idemp) |
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236 |
done |
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||
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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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Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
240 |
O(f + g) = O(f) \<oplus> O(g)" |
23449 | 241 |
apply (rule equalityI) |
242 |
apply (rule bigo_plus_subset) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
243 |
apply (simp add: bigo_alt_def set_plus_def func_plus) |
23449 | 244 |
apply clarify |
36844 | 245 |
(*sledgehammer*) |
23449 | 246 |
apply (rule_tac x = "max c ca" in exI) |
247 |
apply (rule conjI) |
|
25087 | 248 |
apply (metis Orderings.less_max_iff_disj) |
23449 | 249 |
apply clarify |
250 |
apply (drule_tac x = "xa" in spec)+ |
|
251 |
apply (subgoal_tac "0 <= f xa + g xa") |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23464
diff
changeset
|
252 |
apply (simp add: ring_distribs) |
23449 | 253 |
apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)") |
254 |
apply (subgoal_tac "abs(a xa) + abs(b xa) <= |
|
255 |
max c ca * f xa + max c ca * g xa") |
|
256 |
apply (blast intro: order_trans) |
|
257 |
defer 1 |
|
258 |
apply (rule abs_triangle_ineq) |
|
25087 | 259 |
apply (metis add_nonneg_nonneg) |
23449 | 260 |
apply (rule add_mono) |
39259
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
261 |
using [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq_simpler" ]] |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
262 |
apply (metis le_maxI2 linorder_linear min_max.sup_absorb1 mult_right_mono xt1(6)) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
263 |
apply (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans) |
23449 | 264 |
done |
265 |
||
38991 | 266 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt" ]] |
23449 | 267 |
lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> |
268 |
f : O(g)" |
|
269 |
apply (auto simp add: bigo_def) |
|
36561 | 270 |
(* Version 1: one-line proof *) |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35028
diff
changeset
|
271 |
apply (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult) |
23449 | 272 |
done |
273 |
||
26312 | 274 |
lemma (*bigo_bounded_alt:*) "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> |
36561 | 275 |
f : O(g)" |
276 |
apply (auto simp add: bigo_def) |
|
277 |
(* Version 2: structured proof *) |
|
278 |
proof - |
|
279 |
assume "\<forall>x. f x \<le> c * g x" |
|
280 |
thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans) |
|
23449 | 281 |
qed |
282 |
||
36561 | 283 |
text{*So here is the easier (and more natural) problem using transitivity*} |
38991 | 284 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]] |
36561 | 285 |
lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" |
286 |
apply (auto simp add: bigo_def) |
|
287 |
(* Version 1: one-line proof *) |
|
288 |
by (metis abs_ge_self abs_mult order_trans) |
|
23449 | 289 |
|
290 |
text{*So here is the easier (and more natural) problem using transitivity*} |
|
38991 | 291 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]] |
23449 | 292 |
lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" |
293 |
apply (auto simp add: bigo_def) |
|
36561 | 294 |
(* Version 2: structured proof *) |
295 |
proof - |
|
296 |
assume "\<forall>x. f x \<le> c * g x" |
|
297 |
thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans) |
|
23449 | 298 |
qed |
299 |
||
300 |
lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> |
|
301 |
f : O(g)" |
|
302 |
apply (erule bigo_bounded_alt [of f 1 g]) |
|
303 |
apply simp |
|
304 |
done |
|
305 |
||
38991 | 306 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded2" ]] |
23449 | 307 |
lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==> |
308 |
f : lb +o O(g)" |
|
36561 | 309 |
apply (rule set_minus_imp_plus) |
310 |
apply (rule bigo_bounded) |
|
311 |
apply (auto simp add: diff_minus fun_Compl_def func_plus) |
|
312 |
prefer 2 |
|
313 |
apply (drule_tac x = x in spec)+ |
|
36844 | 314 |
apply (metis add_right_mono add_commute diff_add_cancel diff_minus_eq_add le_less order_trans) |
36561 | 315 |
proof - |
316 |
fix x :: 'a |
|
317 |
assume "\<forall>x. lb x \<le> f x" |
|
318 |
thus "(0\<Colon>'b) \<le> f x + - lb x" by (metis not_leE diff_minus less_iff_diff_less_0 less_le_not_le) |
|
23449 | 319 |
qed |
320 |
||
38991 | 321 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs" ]] |
23449 | 322 |
lemma bigo_abs: "(%x. abs(f x)) =o O(f)" |
36561 | 323 |
apply (unfold bigo_def) |
324 |
apply auto |
|
36844 | 325 |
by (metis mult_1 order_refl) |
23449 | 326 |
|
38991 | 327 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs2" ]] |
23449 | 328 |
lemma bigo_abs2: "f =o O(%x. abs(f x))" |
36561 | 329 |
apply (unfold bigo_def) |
330 |
apply auto |
|
36844 | 331 |
by (metis mult_1 order_refl) |
23449 | 332 |
|
333 |
lemma bigo_abs3: "O(f) = O(%x. abs(f x))" |
|
36561 | 334 |
proof - |
335 |
have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset) |
|
336 |
have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs) |
|
337 |
have "\<forall>u. u \<in> O(\<lambda>R. \<bar>u R\<bar>)" by (metis bigo_abs2) |
|
338 |
thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" using F1 F2 by auto |
|
339 |
qed |
|
23449 | 340 |
|
341 |
lemma bigo_abs4: "f =o g +o O(h) ==> |
|
342 |
(%x. abs (f x)) =o (%x. abs (g x)) +o O(h)" |
|
343 |
apply (drule set_plus_imp_minus) |
|
344 |
apply (rule set_minus_imp_plus) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
345 |
apply (subst fun_diff_def) |
23449 | 346 |
proof - |
347 |
assume a: "f - g : O(h)" |
|
348 |
have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))" |
|
349 |
by (rule bigo_abs2) |
|
350 |
also have "... <= O(%x. abs (f x - g x))" |
|
351 |
apply (rule bigo_elt_subset) |
|
352 |
apply (rule bigo_bounded) |
|
353 |
apply force |
|
354 |
apply (rule allI) |
|
355 |
apply (rule abs_triangle_ineq3) |
|
356 |
done |
|
357 |
also have "... <= O(f - g)" |
|
358 |
apply (rule bigo_elt_subset) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
359 |
apply (subst fun_diff_def) |
23449 | 360 |
apply (rule bigo_abs) |
361 |
done |
|
362 |
also have "... <= O(h)" |
|
23464 | 363 |
using a by (rule bigo_elt_subset) |
23449 | 364 |
finally show "(%x. abs (f x) - abs (g x)) : O(h)". |
365 |
qed |
|
366 |
||
367 |
lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" |
|
368 |
by (unfold bigo_def, auto) |
|
369 |
||
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
370 |
lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) \<oplus> O(h)" |
23449 | 371 |
proof - |
372 |
assume "f : g +o O(h)" |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
373 |
also have "... <= O(g) \<oplus> O(h)" |
23449 | 374 |
by (auto del: subsetI) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
375 |
also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))" |
23449 | 376 |
apply (subst bigo_abs3 [symmetric])+ |
377 |
apply (rule refl) |
|
378 |
done |
|
379 |
also have "... = O((%x. abs(g x)) + (%x. abs(h x)))" |
|
380 |
by (rule bigo_plus_eq [symmetric], auto) |
|
381 |
finally have "f : ...". |
|
382 |
then have "O(f) <= ..." |
|
383 |
by (elim bigo_elt_subset) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
384 |
also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))" |
23449 | 385 |
by (rule bigo_plus_eq, auto) |
386 |
finally show ?thesis |
|
387 |
by (simp add: bigo_abs3 [symmetric]) |
|
388 |
qed |
|
389 |
||
38991 | 390 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult" ]] |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
391 |
lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)" |
23449 | 392 |
apply (rule subsetI) |
393 |
apply (subst bigo_def) |
|
394 |
apply (auto simp del: abs_mult mult_ac |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
395 |
simp add: bigo_alt_def set_times_def func_times) |
23449 | 396 |
(*sledgehammer*); |
397 |
apply (rule_tac x = "c * ca" in exI) |
|
398 |
apply(rule allI) |
|
399 |
apply(erule_tac x = x in allE)+ |
|
400 |
apply(subgoal_tac "c * ca * abs(f x * g x) = |
|
401 |
(c * abs(f x)) * (ca * abs(g x))") |
|
38991 | 402 |
using [[ sledgehammer_problem_prefix = "BigO__bigo_mult_simpler" ]] |
23449 | 403 |
prefer 2 |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25710
diff
changeset
|
404 |
apply (metis mult_assoc mult_left_commute |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35028
diff
changeset
|
405 |
abs_of_pos mult_left_commute |
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35028
diff
changeset
|
406 |
abs_mult mult_pos_pos) |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25710
diff
changeset
|
407 |
apply (erule ssubst) |
23449 | 408 |
apply (subst abs_mult) |
36561 | 409 |
(* not quite as hard as BigO__bigo_mult_simpler_1 (a hard problem!) since |
410 |
abs_mult has just been done *) |
|
411 |
by (metis abs_ge_zero mult_mono') |
|
23449 | 412 |
|
38991 | 413 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult2" ]] |
23449 | 414 |
lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)" |
415 |
apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult) |
|
416 |
(*sledgehammer*); |
|
417 |
apply (rule_tac x = c in exI) |
|
418 |
apply clarify |
|
419 |
apply (drule_tac x = x in spec) |
|
38991 | 420 |
using [[ sledgehammer_problem_prefix = "BigO__bigo_mult2_simpler" ]] |
24942 | 421 |
(*sledgehammer [no luck]*); |
23449 | 422 |
apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))") |
423 |
apply (simp add: mult_ac) |
|
424 |
apply (rule mult_left_mono, assumption) |
|
425 |
apply (rule abs_ge_zero) |
|
426 |
done |
|
427 |
||
38991 | 428 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult3" ]] |
23449 | 429 |
lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)" |
36561 | 430 |
by (metis bigo_mult set_rev_mp set_times_intro) |
23449 | 431 |
|
38991 | 432 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult4" ]] |
23449 | 433 |
lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)" |
434 |
by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib) |
|
435 |
||
436 |
||
437 |
lemma bigo_mult5: "ALL x. f x ~= 0 ==> |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
438 |
O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)" |
23449 | 439 |
proof - |
41541 | 440 |
assume a: "ALL x. f x ~= 0" |
23449 | 441 |
show "O(f * g) <= f *o O(g)" |
442 |
proof |
|
443 |
fix h |
|
41541 | 444 |
assume h: "h : O(f * g)" |
23449 | 445 |
then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)" |
446 |
by auto |
|
447 |
also have "... <= O((%x. 1 / f x) * (f * g))" |
|
448 |
by (rule bigo_mult2) |
|
449 |
also have "(%x. 1 / f x) * (f * g) = g" |
|
450 |
apply (simp add: func_times) |
|
451 |
apply (rule ext) |
|
41541 | 452 |
apply (simp add: a h nonzero_divide_eq_eq mult_ac) |
23449 | 453 |
done |
454 |
finally have "(%x. (1::'b) / f x) * h : O(g)". |
|
455 |
then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)" |
|
456 |
by auto |
|
457 |
also have "f * ((%x. (1::'b) / f x) * h) = h" |
|
458 |
apply (simp add: func_times) |
|
459 |
apply (rule ext) |
|
41541 | 460 |
apply (simp add: a h nonzero_divide_eq_eq mult_ac) |
23449 | 461 |
done |
462 |
finally show "h : f *o O(g)". |
|
463 |
qed |
|
464 |
qed |
|
465 |
||
38991 | 466 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult6" ]] |
23449 | 467 |
lemma bigo_mult6: "ALL x. f x ~= 0 ==> |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
468 |
O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)" |
23449 | 469 |
by (metis bigo_mult2 bigo_mult5 order_antisym) |
470 |
||
471 |
(*proof requires relaxing relevance: 2007-01-25*) |
|
38991 | 472 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult7" ]] |
23449 | 473 |
declare bigo_mult6 [simp] |
474 |
lemma bigo_mult7: "ALL x. f x ~= 0 ==> |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
475 |
O(f * g) <= O(f::'a => ('b::linordered_field)) \<otimes> O(g)" |
23449 | 476 |
(*sledgehammer*) |
477 |
apply (subst bigo_mult6) |
|
478 |
apply assumption |
|
479 |
apply (rule set_times_mono3) |
|
480 |
apply (rule bigo_refl) |
|
481 |
done |
|
482 |
declare bigo_mult6 [simp del] |
|
483 |
||
38991 | 484 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult8" ]] |
23449 | 485 |
declare bigo_mult7[intro!] |
486 |
lemma bigo_mult8: "ALL x. f x ~= 0 ==> |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
487 |
O(f * g) = O(f::'a => ('b::linordered_field)) \<otimes> O(g)" |
23449 | 488 |
by (metis bigo_mult bigo_mult7 order_antisym_conv) |
489 |
||
490 |
lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)" |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
491 |
by (auto simp add: bigo_def fun_Compl_def) |
23449 | 492 |
|
493 |
lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)" |
|
494 |
apply (rule set_minus_imp_plus) |
|
495 |
apply (drule set_plus_imp_minus) |
|
496 |
apply (drule bigo_minus) |
|
497 |
apply (simp add: diff_minus) |
|
498 |
done |
|
499 |
||
500 |
lemma bigo_minus3: "O(-f) = O(f)" |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
501 |
by (auto simp add: bigo_def fun_Compl_def abs_minus_cancel) |
23449 | 502 |
|
503 |
lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)" |
|
504 |
proof - |
|
505 |
assume a: "f : O(g)" |
|
506 |
show "f +o O(g) <= O(g)" |
|
507 |
proof - |
|
508 |
have "f : O(f)" by auto |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
509 |
then have "f +o O(g) <= O(f) \<oplus> O(g)" |
23449 | 510 |
by (auto del: subsetI) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
511 |
also have "... <= O(g) \<oplus> O(g)" |
23449 | 512 |
proof - |
513 |
from a have "O(f) <= O(g)" by (auto del: subsetI) |
|
514 |
thus ?thesis by (auto del: subsetI) |
|
515 |
qed |
|
516 |
also have "... <= O(g)" by (simp add: bigo_plus_idemp) |
|
517 |
finally show ?thesis . |
|
518 |
qed |
|
519 |
qed |
|
520 |
||
521 |
lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)" |
|
522 |
proof - |
|
523 |
assume a: "f : O(g)" |
|
524 |
show "O(g) <= f +o O(g)" |
|
525 |
proof - |
|
526 |
from a have "-f : O(g)" by auto |
|
527 |
then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1) |
|
528 |
then have "f +o (-f +o O(g)) <= f +o O(g)" by auto |
|
529 |
also have "f +o (-f +o O(g)) = O(g)" |
|
530 |
by (simp add: set_plus_rearranges) |
|
531 |
finally show ?thesis . |
|
532 |
qed |
|
533 |
qed |
|
534 |
||
38991 | 535 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_absorb" ]] |
23449 | 536 |
lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)" |
537 |
by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff); |
|
538 |
||
539 |
lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)" |
|
540 |
apply (subgoal_tac "f +o A <= f +o O(g)") |
|
541 |
apply force+ |
|
542 |
done |
|
543 |
||
544 |
lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)" |
|
545 |
apply (subst set_minus_plus [symmetric]) |
|
546 |
apply (subgoal_tac "g - f = - (f - g)") |
|
547 |
apply (erule ssubst) |
|
548 |
apply (rule bigo_minus) |
|
549 |
apply (subst set_minus_plus) |
|
550 |
apply assumption |
|
551 |
apply (simp add: diff_minus add_ac) |
|
552 |
done |
|
553 |
||
554 |
lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))" |
|
555 |
apply (rule iffI) |
|
556 |
apply (erule bigo_add_commute_imp)+ |
|
557 |
done |
|
558 |
||
559 |
lemma bigo_const1: "(%x. c) : O(%x. 1)" |
|
560 |
by (auto simp add: bigo_def mult_ac) |
|
561 |
||
38991 | 562 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const2" ]] |
23449 | 563 |
lemma (*bigo_const2 [intro]:*) "O(%x. c) <= O(%x. 1)" |
564 |
by (metis bigo_const1 bigo_elt_subset); |
|
565 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
566 |
lemma bigo_const2 [intro]: "O(%x. c::'b::linordered_idom) <= O(%x. 1)"; |
36561 | 567 |
(* "thus" had to be replaced by "show" with an explicit reference to "F1" *) |
568 |
proof - |
|
569 |
have F1: "\<forall>u. (\<lambda>Q. u) \<in> O(\<lambda>Q. 1)" by (metis bigo_const1) |
|
570 |
show "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)" by (metis F1 bigo_elt_subset) |
|
23449 | 571 |
qed |
572 |
||
38991 | 573 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const3" ]] |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
574 |
lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)" |
23449 | 575 |
apply (simp add: bigo_def) |
36561 | 576 |
by (metis abs_eq_0 left_inverse order_refl) |
23449 | 577 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
578 |
lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)" |
23449 | 579 |
by (rule bigo_elt_subset, rule bigo_const3, assumption) |
580 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
581 |
lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==> |
23449 | 582 |
O(%x. c) = O(%x. 1)" |
583 |
by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption) |
|
584 |
||
38991 | 585 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult1" ]] |
23449 | 586 |
lemma bigo_const_mult1: "(%x. c * f x) : O(f)" |
24937
340523598914
context-based treatment of generalization; also handling TFrees in axiom clauses
paulson
parents:
24855
diff
changeset
|
587 |
apply (simp add: bigo_def abs_mult) |
36561 | 588 |
by (metis le_less) |
23449 | 589 |
|
590 |
lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)" |
|
591 |
by (rule bigo_elt_subset, rule bigo_const_mult1) |
|
592 |
||
38991 | 593 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult3" ]] |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
594 |
lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%x. c * f x)" |
23449 | 595 |
apply (simp add: bigo_def) |
36561 | 596 |
(*sledgehammer [no luck]*) |
23449 | 597 |
apply (rule_tac x = "abs(inverse c)" in exI) |
598 |
apply (simp only: abs_mult [symmetric] mult_assoc [symmetric]) |
|
599 |
apply (subst left_inverse) |
|
600 |
apply (auto ); |
|
601 |
done |
|
602 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
603 |
lemma bigo_const_mult4: "(c::'a::linordered_field) ~= 0 ==> |
23449 | 604 |
O(f) <= O(%x. c * f x)" |
605 |
by (rule bigo_elt_subset, rule bigo_const_mult3, assumption) |
|
606 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
607 |
lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==> |
23449 | 608 |
O(%x. c * f x) = O(f)" |
609 |
by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4) |
|
610 |
||
38991 | 611 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult5" ]] |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
612 |
lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==> |
23449 | 613 |
(%x. c) *o O(f) = O(f)" |
614 |
apply (auto del: subsetI) |
|
615 |
apply (rule order_trans) |
|
616 |
apply (rule bigo_mult2) |
|
617 |
apply (simp add: func_times) |
|
618 |
apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times) |
|
619 |
apply (rule_tac x = "%y. inverse c * x y" in exI) |
|
24942 | 620 |
apply (rename_tac g d) |
621 |
apply safe |
|
622 |
apply (rule_tac [2] ext) |
|
623 |
prefer 2 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25710
diff
changeset
|
624 |
apply simp |
24942 | 625 |
apply (simp add: mult_assoc [symmetric] abs_mult) |
39259
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
626 |
(* couldn't get this proof without the step above *) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
627 |
proof - |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
628 |
fix g :: "'b \<Rightarrow> 'a" and d :: 'a |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
629 |
assume A1: "c \<noteq> (0\<Colon>'a)" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
630 |
assume A2: "\<forall>x\<Colon>'b. \<bar>g x\<bar> \<le> d * \<bar>f x\<bar>" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
631 |
have F1: "inverse \<bar>c\<bar> = \<bar>inverse c\<bar>" using A1 by (metis nonzero_abs_inverse) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
632 |
have F2: "(0\<Colon>'a) < \<bar>c\<bar>" using A1 by (metis zero_less_abs_iff) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
633 |
have "(0\<Colon>'a) < \<bar>c\<bar> \<longrightarrow> (0\<Colon>'a) < \<bar>inverse c\<bar>" using F1 by (metis positive_imp_inverse_positive) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
634 |
hence "(0\<Colon>'a) < \<bar>inverse c\<bar>" using F2 by metis |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
635 |
hence F3: "(0\<Colon>'a) \<le> \<bar>inverse c\<bar>" by (metis order_le_less) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
636 |
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>" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
637 |
using A2 by metis |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
638 |
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>" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
639 |
using F3 by metis |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
640 |
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>)" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
641 |
by (metis comm_mult_left_mono) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
642 |
thus "\<exists>ca\<Colon>'a. \<forall>x\<Colon>'b. \<bar>inverse c\<bar> * \<bar>g x\<bar> \<le> ca * \<bar>f x\<bar>" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
643 |
using A2 F4 by (metis ab_semigroup_mult_class.mult_ac(1) comm_mult_left_mono) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
644 |
qed |
23449 | 645 |
|
646 |
||
38991 | 647 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult6" ]] |
23449 | 648 |
lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)" |
649 |
apply (auto intro!: subsetI |
|
650 |
simp add: bigo_def elt_set_times_def func_times |
|
651 |
simp del: abs_mult mult_ac) |
|
652 |
(*sledgehammer*); |
|
653 |
apply (rule_tac x = "ca * (abs c)" in exI) |
|
654 |
apply (rule allI) |
|
655 |
apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))") |
|
656 |
apply (erule ssubst) |
|
657 |
apply (subst abs_mult) |
|
658 |
apply (rule mult_left_mono) |
|
659 |
apply (erule spec) |
|
660 |
apply simp |
|
661 |
apply(simp add: mult_ac) |
|
662 |
done |
|
663 |
||
664 |
lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)" |
|
665 |
proof - |
|
666 |
assume "f =o O(g)" |
|
667 |
then have "(%x. c) * f =o (%x. c) *o O(g)" |
|
668 |
by auto |
|
669 |
also have "(%x. c) * f = (%x. c * f x)" |
|
670 |
by (simp add: func_times) |
|
671 |
also have "(%x. c) *o O(g) <= O(g)" |
|
672 |
by (auto del: subsetI) |
|
673 |
finally show ?thesis . |
|
674 |
qed |
|
675 |
||
676 |
lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))" |
|
677 |
by (unfold bigo_def, auto) |
|
678 |
||
679 |
lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o |
|
680 |
O(%x. h(k x))" |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
681 |
apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def |
23449 | 682 |
func_plus) |
683 |
apply (erule bigo_compose1) |
|
684 |
done |
|
685 |
||
686 |
subsection {* Setsum *} |
|
687 |
||
688 |
lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==> |
|
689 |
EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==> |
|
690 |
(%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)" |
|
691 |
apply (auto simp add: bigo_def) |
|
692 |
apply (rule_tac x = "abs c" in exI) |
|
693 |
apply (subst abs_of_nonneg) back back |
|
694 |
apply (rule setsum_nonneg) |
|
695 |
apply force |
|
696 |
apply (subst setsum_right_distrib) |
|
697 |
apply (rule allI) |
|
698 |
apply (rule order_trans) |
|
699 |
apply (rule setsum_abs) |
|
700 |
apply (rule setsum_mono) |
|
701 |
apply (blast intro: order_trans mult_right_mono abs_ge_self) |
|
702 |
done |
|
703 |
||
38991 | 704 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum1" ]] |
23449 | 705 |
lemma bigo_setsum1: "ALL x y. 0 <= h x y ==> |
706 |
EX c. ALL x y. abs(f x y) <= c * (h x y) ==> |
|
707 |
(%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)" |
|
708 |
apply (rule bigo_setsum_main) |
|
709 |
(*sledgehammer*); |
|
710 |
apply force |
|
711 |
apply clarsimp |
|
712 |
apply (rule_tac x = c in exI) |
|
713 |
apply force |
|
714 |
done |
|
715 |
||
716 |
lemma bigo_setsum2: "ALL y. 0 <= h y ==> |
|
717 |
EX c. ALL y. abs(f y) <= c * (h y) ==> |
|
718 |
(%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)" |
|
719 |
by (rule bigo_setsum1, auto) |
|
720 |
||
38991 | 721 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum3" ]] |
23449 | 722 |
lemma bigo_setsum3: "f =o O(h) ==> |
723 |
(%x. SUM y : A x. (l x y) * f(k x y)) =o |
|
724 |
O(%x. SUM y : A x. abs(l x y * h(k x y)))" |
|
725 |
apply (rule bigo_setsum1) |
|
726 |
apply (rule allI)+ |
|
727 |
apply (rule abs_ge_zero) |
|
728 |
apply (unfold bigo_def) |
|
729 |
apply (auto simp add: abs_mult); |
|
730 |
(*sledgehammer*); |
|
731 |
apply (rule_tac x = c in exI) |
|
732 |
apply (rule allI)+ |
|
733 |
apply (subst mult_left_commute) |
|
734 |
apply (rule mult_left_mono) |
|
735 |
apply (erule spec) |
|
736 |
apply (rule abs_ge_zero) |
|
737 |
done |
|
738 |
||
739 |
lemma bigo_setsum4: "f =o g +o O(h) ==> |
|
740 |
(%x. SUM y : A x. l x y * f(k x y)) =o |
|
741 |
(%x. SUM y : A x. l x y * g(k x y)) +o |
|
742 |
O(%x. SUM y : A x. abs(l x y * h(k x y)))" |
|
743 |
apply (rule set_minus_imp_plus) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
744 |
apply (subst fun_diff_def) |
23449 | 745 |
apply (subst setsum_subtractf [symmetric]) |
746 |
apply (subst right_diff_distrib [symmetric]) |
|
747 |
apply (rule bigo_setsum3) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
748 |
apply (subst fun_diff_def [symmetric]) |
23449 | 749 |
apply (erule set_plus_imp_minus) |
750 |
done |
|
751 |
||
38991 | 752 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum5" ]] |
23449 | 753 |
lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==> |
754 |
ALL x. 0 <= h x ==> |
|
755 |
(%x. SUM y : A x. (l x y) * f(k x y)) =o |
|
756 |
O(%x. SUM y : A x. (l x y) * h(k x y))" |
|
757 |
apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) = |
|
758 |
(%x. SUM y : A x. abs((l x y) * h(k x y)))") |
|
759 |
apply (erule ssubst) |
|
760 |
apply (erule bigo_setsum3) |
|
761 |
apply (rule ext) |
|
762 |
apply (rule setsum_cong2) |
|
763 |
apply (thin_tac "f \<in> O(h)") |
|
24942 | 764 |
apply (metis abs_of_nonneg zero_le_mult_iff) |
23449 | 765 |
done |
766 |
||
767 |
lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==> |
|
768 |
ALL x. 0 <= h x ==> |
|
769 |
(%x. SUM y : A x. (l x y) * f(k x y)) =o |
|
770 |
(%x. SUM y : A x. (l x y) * g(k x y)) +o |
|
771 |
O(%x. SUM y : A x. (l x y) * h(k x y))" |
|
772 |
apply (rule set_minus_imp_plus) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
773 |
apply (subst fun_diff_def) |
23449 | 774 |
apply (subst setsum_subtractf [symmetric]) |
775 |
apply (subst right_diff_distrib [symmetric]) |
|
776 |
apply (rule bigo_setsum5) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
777 |
apply (subst fun_diff_def [symmetric]) |
23449 | 778 |
apply (drule set_plus_imp_minus) |
779 |
apply auto |
|
780 |
done |
|
781 |
||
782 |
subsection {* Misc useful stuff *} |
|
783 |
||
784 |
lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==> |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
785 |
A \<oplus> B <= O(f)" |
23449 | 786 |
apply (subst bigo_plus_idemp [symmetric]) |
787 |
apply (rule set_plus_mono2) |
|
788 |
apply assumption+ |
|
789 |
done |
|
790 |
||
791 |
lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)" |
|
792 |
apply (subst bigo_plus_idemp [symmetric]) |
|
793 |
apply (rule set_plus_intro) |
|
794 |
apply assumption+ |
|
795 |
done |
|
796 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
797 |
lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 ==> |
23449 | 798 |
(%x. c) * f =o O(h) ==> f =o O(h)" |
799 |
apply (rule subsetD) |
|
800 |
apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)") |
|
801 |
apply assumption |
|
802 |
apply (rule bigo_const_mult6) |
|
803 |
apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)") |
|
804 |
apply (erule ssubst) |
|
805 |
apply (erule set_times_intro2) |
|
806 |
apply (simp add: func_times) |
|
807 |
done |
|
808 |
||
38991 | 809 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_fix" ]] |
23449 | 810 |
lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==> |
811 |
f =o O(h)" |
|
812 |
apply (simp add: bigo_alt_def) |
|
813 |
(*sledgehammer*); |
|
814 |
apply clarify |
|
815 |
apply (rule_tac x = c in exI) |
|
816 |
apply safe |
|
817 |
apply (case_tac "x = 0") |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35028
diff
changeset
|
818 |
apply (metis abs_ge_zero abs_zero order_less_le split_mult_pos_le) |
23449 | 819 |
apply (subgoal_tac "x = Suc (x - 1)") |
23816 | 820 |
apply metis |
23449 | 821 |
apply simp |
822 |
done |
|
823 |
||
824 |
||
825 |
lemma bigo_fix2: |
|
826 |
"(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==> |
|
827 |
f 0 = g 0 ==> f =o g +o O(h)" |
|
828 |
apply (rule set_minus_imp_plus) |
|
829 |
apply (rule bigo_fix) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
830 |
apply (subst fun_diff_def) |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
831 |
apply (subst fun_diff_def [symmetric]) |
23449 | 832 |
apply (rule set_plus_imp_minus) |
833 |
apply simp |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
834 |
apply (simp add: fun_diff_def) |
23449 | 835 |
done |
836 |
||
837 |
subsection {* Less than or equal to *} |
|
838 |
||
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35050
diff
changeset
|
839 |
definition lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where |
23449 | 840 |
"f <o g == (%x. max (f x - g x) 0)" |
841 |
||
842 |
lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==> |
|
843 |
g =o O(h)" |
|
844 |
apply (unfold bigo_def) |
|
845 |
apply clarsimp |
|
846 |
apply (blast intro: order_trans) |
|
847 |
done |
|
848 |
||
849 |
lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==> |
|
850 |
g =o O(h)" |
|
851 |
apply (erule bigo_lesseq1) |
|
852 |
apply (blast intro: abs_ge_self order_trans) |
|
853 |
done |
|
854 |
||
855 |
lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==> |
|
856 |
g =o O(h)" |
|
857 |
apply (erule bigo_lesseq2) |
|
858 |
apply (rule allI) |
|
859 |
apply (subst abs_of_nonneg) |
|
860 |
apply (erule spec)+ |
|
861 |
done |
|
862 |
||
863 |
lemma bigo_lesseq4: "f =o O(h) ==> |
|
864 |
ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==> |
|
865 |
g =o O(h)" |
|
866 |
apply (erule bigo_lesseq1) |
|
867 |
apply (rule allI) |
|
868 |
apply (subst abs_of_nonneg) |
|
869 |
apply (erule spec)+ |
|
870 |
done |
|
871 |
||
38991 | 872 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso1" ]] |
23449 | 873 |
lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)" |
36561 | 874 |
apply (unfold lesso_def) |
875 |
apply (subgoal_tac "(%x. max (f x - g x) 0) = 0") |
|
876 |
proof - |
|
877 |
assume "(\<lambda>x. max (f x - g x) 0) = 0" |
|
878 |
thus "(\<lambda>x. max (f x - g x) 0) \<in> O(h)" by (metis bigo_zero) |
|
879 |
next |
|
880 |
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)" |
|
23449 | 881 |
apply (unfold func_zero) |
882 |
apply (rule ext) |
|
36561 | 883 |
by (simp split: split_max) |
884 |
qed |
|
23449 | 885 |
|
38991 | 886 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso2" ]] |
23449 | 887 |
lemma bigo_lesso2: "f =o g +o O(h) ==> |
888 |
ALL x. 0 <= k x ==> ALL x. k x <= f x ==> |
|
889 |
k <o g =o O(h)" |
|
890 |
apply (unfold lesso_def) |
|
891 |
apply (rule bigo_lesseq4) |
|
892 |
apply (erule set_plus_imp_minus) |
|
893 |
apply (rule allI) |
|
894 |
apply (rule le_maxI2) |
|
895 |
apply (rule allI) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
896 |
apply (subst fun_diff_def) |
23449 | 897 |
apply (erule thin_rl) |
898 |
(*sledgehammer*); |
|
899 |
apply (case_tac "0 <= k x - g x") |
|
36561 | 900 |
(* apply (metis abs_le_iff add_le_imp_le_right diff_minus le_less |
901 |
le_max_iff_disj min_max.le_supE min_max.sup_absorb2 |
|
902 |
min_max.sup_commute) *) |
|
37320 | 903 |
proof - |
904 |
fix x :: 'a |
|
905 |
assume "\<forall>x\<Colon>'a. k x \<le> f x" |
|
906 |
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) |
|
907 |
assume "(0\<Colon>'b) \<le> k x - g x" |
|
908 |
hence F2: "max (0\<Colon>'b) (k x - g x) = k x - g x" by (metis min_max.sup_absorb2) |
|
909 |
have F3: "\<forall>x\<^isub>1\<Colon>'b. x\<^isub>1 \<le> \<bar>x\<^isub>1\<bar>" by (metis abs_le_iff le_less) |
|
910 |
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) |
|
911 |
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) |
|
912 |
hence "k x - g x \<le> f x - g x" by (metis F1 le_less min_max.sup_absorb2 min_max.sup_commute) |
|
913 |
hence "k x - g x \<le> \<bar>f x - g x\<bar>" by (metis F3 le_max_iff_disj min_max.sup_absorb2) |
|
914 |
thus "max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>" by (metis F2 min_max.sup_commute) |
|
36561 | 915 |
next |
916 |
show "\<And>x\<Colon>'a. |
|
917 |
\<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> |
|
918 |
\<Longrightarrow> max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>" |
|
919 |
by (metis abs_ge_zero le_cases min_max.sup_absorb2) |
|
24545 | 920 |
qed |
23449 | 921 |
|
38991 | 922 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3" ]] |
23449 | 923 |
lemma bigo_lesso3: "f =o g +o O(h) ==> |
924 |
ALL x. 0 <= k x ==> ALL x. g x <= k x ==> |
|
925 |
f <o k =o O(h)" |
|
926 |
apply (unfold lesso_def) |
|
927 |
apply (rule bigo_lesseq4) |
|
928 |
apply (erule set_plus_imp_minus) |
|
929 |
apply (rule allI) |
|
930 |
apply (rule le_maxI2) |
|
931 |
apply (rule allI) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
932 |
apply (subst fun_diff_def) |
23449 | 933 |
apply (erule thin_rl) |
934 |
(*sledgehammer*); |
|
935 |
apply (case_tac "0 <= f x - k x") |
|
29667 | 936 |
apply (simp) |
23449 | 937 |
apply (subst abs_of_nonneg) |
938 |
apply (drule_tac x = x in spec) back |
|
38991 | 939 |
using [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3_simpler" ]] |
24545 | 940 |
apply (metis diff_less_0_iff_less linorder_not_le not_leE uminus_add_conv_diff xt1(12) xt1(6)) |
941 |
apply (metis add_minus_cancel diff_le_eq le_diff_eq uminus_add_conv_diff) |
|
29511
7071b017cb35
migrated class package to new locale implementation
haftmann
parents:
28592
diff
changeset
|
942 |
apply (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute) |
23449 | 943 |
done |
944 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
33027
diff
changeset
|
945 |
lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::linordered_field) ==> |
23449 | 946 |
g =o h +o O(k) ==> f <o h =o O(k)" |
947 |
apply (unfold lesso_def) |
|
948 |
apply (drule set_plus_imp_minus) |
|
949 |
apply (drule bigo_abs5) back |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
950 |
apply (simp add: fun_diff_def) |
23449 | 951 |
apply (drule bigo_useful_add) |
952 |
apply assumption |
|
953 |
apply (erule bigo_lesseq2) back |
|
954 |
apply (rule allI) |
|
29667 | 955 |
apply (auto simp add: func_plus fun_diff_def algebra_simps |
23449 | 956 |
split: split_max abs_split) |
957 |
done |
|
958 |
||
38991 | 959 |
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso5" ]] |
23449 | 960 |
lemma bigo_lesso5: "f <o g =o O(h) ==> |
961 |
EX C. ALL x. f x <= g x + C * abs(h x)" |
|
962 |
apply (simp only: lesso_def bigo_alt_def) |
|
963 |
apply clarsimp |
|
24855 | 964 |
apply (metis abs_if abs_mult add_commute diff_le_eq less_not_permute) |
23449 | 965 |
done |
966 |
||
967 |
end |