author | blanchet |
Thu, 01 Dec 2011 13:34:12 +0100 | |
changeset 45705 | a25ff4283352 |
parent 45575 | 3a865fc42bbf |
child 46364 | abab10d1f4a3 |
permissions | -rw-r--r-- |
43197 | 1 |
(* Title: HOL/Metis_Examples/Big_O.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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Metis example featuring the Big O notation. |
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*) |
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header {* Metis Example Featuring the Big O Notation *} |
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theory Big_O |
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imports |
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"~~/src/HOL/Decision_Procs/Dense_Linear_Order" |
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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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||
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definition bigo :: "('a => 'b\<Colon>{linordered_idom,number_ring}) => ('a => 'b) set" ("(1O'(_'))") where |
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"O(f\<Colon>('a => 'b)) == {h. \<exists>c. \<forall>x. abs (h x) <= c * abs (f x)}" |
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lemma bigo_pos_const: |
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"(\<exists>(c\<Colon>'a\<Colon>linordered_idom). |
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\<forall>x. (abs (h x)) <= (c * (abs (f x)))) |
|
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= (\<exists>c. 0 < c & (\<forall>x. (abs(h x)) <= (c * (abs (f x)))))" |
|
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by (metis (hide_lams, no_types) abs_ge_zero |
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comm_semiring_1_class.normalizing_semiring_rules(7) mult.comm_neutral |
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mult_nonpos_nonneg not_leE order_trans zero_less_one) |
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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 |
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"(\<exists>(c\<Colon>'a\<Colon>linordered_idom). |
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\<forall>x. (abs (h x)) <= (c * (abs (f x)))) |
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= (\<exists>c. 0 < c & (\<forall>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 |
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"(\<exists>(c\<Colon>'a\<Colon>linordered_idom). |
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\<forall>x. (abs (h x)) <= (c * (abs (f x)))) |
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= (\<exists>c. 0 < c & (\<forall>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 |
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"(\<exists>(c\<Colon>'a\<Colon>linordered_idom). |
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\<forall>x. (abs (h x)) <= (c * (abs (f x)))) |
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= (\<exists>c. 0 < c & (\<forall>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 |
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"(\<exists>(c\<Colon>'a\<Colon>linordered_idom). |
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\<forall>x. (abs (h x)) <= (c * (abs (f x)))) |
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= (\<exists>c. 0 < c & (\<forall>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) = {h. \<exists>c. (0 < c & (\<forall>x. abs (h x) <= c * abs (f x)))}" |
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by (auto simp add: bigo_def bigo_pos_const) |
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lemma bigo_elt_subset [intro]: "f : O(g) \<Longrightarrow> 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 (metis comm_semiring_1_class.normalizing_semiring_rules(19) |
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comm_semiring_1_class.normalizing_semiring_rules(7) order_trans) |
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by (metis mult_le_cancel_left_pos) |
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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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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(\<lambda>x. 0) = {\<lambda>x. 0}" |
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by (auto simp add: bigo_def) |
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lemma bigo_plus_self_subset [intro]: |
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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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by (metis order_trans abs_triangle_ineq add_mono) |
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|
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lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)" |
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by (metis bigo_plus_self_subset bigo_zero set_eq_subset set_zero_plus2) |
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|
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parents:
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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 = "\<lambda>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 (metis mult_2 order_trans) |
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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) |
180 |
apply (simp add: ring_distribs) |
|
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apply (rule mult_left_mono) |
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apply (simp add: abs_triangle_ineq) |
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apply (simp add: order_less_le) |
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apply (rule mult_nonneg_nonneg) |
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apply auto |
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apply (rule_tac x = "\<lambda>n. if (abs (f n)) < abs (g n) then x n else 0" in exI) |
187 |
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 (metis order_trans semiring_mult_2) |
|
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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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194 |
apply (simp add: ring_distribs) |
45575 | 195 |
apply (metis abs_triangle_ineq mult_le_cancel_left_pos) |
196 |
by (metis abs_ge_zero abs_of_pos zero_le_mult_iff) |
|
23449 | 197 |
|
45575 | 198 |
lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A \<oplus> B <= O(f)" |
199 |
by (metis bigo_plus_idemp set_plus_mono2) |
|
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|
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lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) \<oplus> O(g)" |
202 |
apply (rule equalityI) |
|
203 |
apply (rule bigo_plus_subset) |
|
204 |
apply (simp add: bigo_alt_def set_plus_def func_plus) |
|
205 |
apply clarify |
|
206 |
(* sledgehammer *) |
|
207 |
apply (rule_tac x = "max c ca" in exI) |
|
208 |
apply (rule conjI) |
|
209 |
apply (metis less_max_iff_disj) |
|
210 |
apply clarify |
|
211 |
apply (drule_tac x = "xa" in spec)+ |
|
212 |
apply (subgoal_tac "0 <= f xa + g xa") |
|
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apply (simp add: ring_distribs) |
|
214 |
apply (subgoal_tac "abs (a xa + b xa) <= abs (a xa) + abs (b xa)") |
|
215 |
apply (subgoal_tac "abs (a xa) + abs (b xa) <= |
|
216 |
max c ca * f xa + max c ca * g xa") |
|
217 |
apply (metis order_trans) |
|
23449 | 218 |
defer 1 |
45575 | 219 |
apply (metis abs_triangle_ineq) |
220 |
apply (metis add_nonneg_nonneg) |
|
221 |
apply (rule add_mono) |
|
222 |
apply (metis le_maxI2 linorder_linear min_max.sup_absorb1 mult_right_mono xt1(6)) |
|
223 |
by (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans) |
|
23449 | 224 |
|
45575 | 225 |
lemma bigo_bounded_alt: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)" |
226 |
apply (auto simp add: bigo_def) |
|
36561 | 227 |
(* Version 1: one-line proof *) |
45575 | 228 |
by (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult) |
23449 | 229 |
|
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lemma "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)" |
36561 | 231 |
apply (auto simp add: bigo_def) |
232 |
(* Version 2: structured proof *) |
|
233 |
proof - |
|
234 |
assume "\<forall>x. f x \<le> c * g x" |
|
235 |
thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans) |
|
23449 | 236 |
qed |
237 |
||
45575 | 238 |
lemma bigo_bounded: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= g x \<Longrightarrow> f : O(g)" |
239 |
apply (erule bigo_bounded_alt [of f 1 g]) |
|
240 |
by (metis mult_1) |
|
23449 | 241 |
|
45575 | 242 |
lemma bigo_bounded2: "\<forall>x. lb x <= f x \<Longrightarrow> \<forall>x. f x <= lb x + g x \<Longrightarrow> f : lb +o O(g)" |
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apply (rule set_minus_imp_plus) |
244 |
apply (rule bigo_bounded) |
|
245 |
apply (auto simp add: diff_minus fun_Compl_def func_plus) |
|
246 |
prefer 2 |
|
247 |
apply (drule_tac x = x in spec)+ |
|
36844 | 248 |
apply (metis add_right_mono add_commute diff_add_cancel diff_minus_eq_add le_less order_trans) |
36561 | 249 |
proof - |
250 |
fix x :: 'a |
|
251 |
assume "\<forall>x. lb x \<le> f x" |
|
252 |
thus "(0\<Colon>'b) \<le> f x + - lb x" by (metis not_leE diff_minus less_iff_diff_less_0 less_le_not_le) |
|
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qed |
254 |
||
45575 | 255 |
lemma bigo_abs: "(\<lambda>x. abs(f x)) =o O(f)" |
36561 | 256 |
apply (unfold bigo_def) |
257 |
apply auto |
|
36844 | 258 |
by (metis mult_1 order_refl) |
23449 | 259 |
|
45575 | 260 |
lemma bigo_abs2: "f =o O(\<lambda>x. abs(f x))" |
36561 | 261 |
apply (unfold bigo_def) |
262 |
apply auto |
|
36844 | 263 |
by (metis mult_1 order_refl) |
43197 | 264 |
|
45575 | 265 |
lemma bigo_abs3: "O(f) = O(\<lambda>x. abs(f x))" |
36561 | 266 |
proof - |
267 |
have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset) |
|
268 |
have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs) |
|
269 |
have "\<forall>u. u \<in> O(\<lambda>R. \<bar>u R\<bar>)" by (metis bigo_abs2) |
|
270 |
thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" using F1 F2 by auto |
|
43197 | 271 |
qed |
23449 | 272 |
|
45575 | 273 |
lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. abs (f x)) =o (\<lambda>x. abs (g x)) +o O(h)" |
23449 | 274 |
apply (drule set_plus_imp_minus) |
275 |
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
|
276 |
apply (subst fun_diff_def) |
23449 | 277 |
proof - |
278 |
assume a: "f - g : O(h)" |
|
45575 | 279 |
have "(\<lambda>x. abs (f x) - abs (g x)) =o O(\<lambda>x. abs(abs (f x) - abs (g x)))" |
23449 | 280 |
by (rule bigo_abs2) |
45575 | 281 |
also have "... <= O(\<lambda>x. abs (f x - g x))" |
23449 | 282 |
apply (rule bigo_elt_subset) |
283 |
apply (rule bigo_bounded) |
|
284 |
apply force |
|
285 |
apply (rule allI) |
|
286 |
apply (rule abs_triangle_ineq3) |
|
287 |
done |
|
288 |
also have "... <= O(f - g)" |
|
289 |
apply (rule bigo_elt_subset) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
290 |
apply (subst fun_diff_def) |
23449 | 291 |
apply (rule bigo_abs) |
292 |
done |
|
293 |
also have "... <= O(h)" |
|
23464 | 294 |
using a by (rule bigo_elt_subset) |
45575 | 295 |
finally show "(\<lambda>x. abs (f x) - abs (g x)) : O(h)". |
23449 | 296 |
qed |
297 |
||
45575 | 298 |
lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. abs(f x)) =o O(g)" |
23449 | 299 |
by (unfold bigo_def, auto) |
300 |
||
45575 | 301 |
lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) \<oplus> O(h)" |
23449 | 302 |
proof - |
303 |
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
|
304 |
also have "... <= O(g) \<oplus> O(h)" |
23449 | 305 |
by (auto del: subsetI) |
45575 | 306 |
also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))" |
23449 | 307 |
apply (subst bigo_abs3 [symmetric])+ |
308 |
apply (rule refl) |
|
309 |
done |
|
45575 | 310 |
also have "... = O((\<lambda>x. abs(g x)) + (\<lambda>x. abs(h x)))" |
23449 | 311 |
by (rule bigo_plus_eq [symmetric], auto) |
312 |
finally have "f : ...". |
|
313 |
then have "O(f) <= ..." |
|
314 |
by (elim bigo_elt_subset) |
|
45575 | 315 |
also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))" |
23449 | 316 |
by (rule bigo_plus_eq, auto) |
317 |
finally show ?thesis |
|
318 |
by (simp add: bigo_abs3 [symmetric]) |
|
319 |
qed |
|
320 |
||
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
321 |
lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)" |
23449 | 322 |
apply (rule subsetI) |
323 |
apply (subst bigo_def) |
|
324 |
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
|
325 |
simp add: bigo_alt_def set_times_def func_times) |
45575 | 326 |
(* sledgehammer *) |
23449 | 327 |
apply (rule_tac x = "c * ca" in exI) |
328 |
apply(rule allI) |
|
329 |
apply(erule_tac x = x in allE)+ |
|
43197 | 330 |
apply(subgoal_tac "c * ca * abs(f x * g x) = |
23449 | 331 |
(c * abs(f x)) * (ca * abs(g x))") |
43197 | 332 |
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
|
333 |
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
|
334 |
abs_of_pos mult_left_commute |
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35028
diff
changeset
|
335 |
abs_mult mult_pos_pos) |
43197 | 336 |
apply (erule ssubst) |
23449 | 337 |
apply (subst abs_mult) |
36561 | 338 |
(* not quite as hard as BigO__bigo_mult_simpler_1 (a hard problem!) since |
339 |
abs_mult has just been done *) |
|
340 |
by (metis abs_ge_zero mult_mono') |
|
23449 | 341 |
|
342 |
lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)" |
|
343 |
apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult) |
|
45575 | 344 |
(* sledgehammer *) |
23449 | 345 |
apply (rule_tac x = c in exI) |
346 |
apply clarify |
|
347 |
apply (drule_tac x = x in spec) |
|
41865
4e8483cc2cc5
declare ext [intro]: Extensionality now available by default
paulson
parents:
41541
diff
changeset
|
348 |
(*sledgehammer [no luck]*) |
23449 | 349 |
apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))") |
350 |
apply (simp add: mult_ac) |
|
351 |
apply (rule mult_left_mono, assumption) |
|
352 |
apply (rule abs_ge_zero) |
|
353 |
done |
|
354 |
||
45575 | 355 |
lemma bigo_mult3: "f : O(h) \<Longrightarrow> g : O(j) \<Longrightarrow> f * g : O(h * j)" |
36561 | 356 |
by (metis bigo_mult set_rev_mp set_times_intro) |
23449 | 357 |
|
45575 | 358 |
lemma bigo_mult4 [intro]:"f : k +o O(h) \<Longrightarrow> g * f : (g * k) +o O(g * h)" |
23449 | 359 |
by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib) |
360 |
||
45575 | 361 |
lemma bigo_mult5: "\<forall>x. f x ~= 0 \<Longrightarrow> |
362 |
O(f * g) <= (f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) *o O(g)" |
|
23449 | 363 |
proof - |
45575 | 364 |
assume a: "\<forall>x. f x ~= 0" |
23449 | 365 |
show "O(f * g) <= f *o O(g)" |
366 |
proof |
|
367 |
fix h |
|
41541 | 368 |
assume h: "h : O(f * g)" |
45575 | 369 |
then have "(\<lambda>x. 1 / (f x)) * h : (\<lambda>x. 1 / f x) *o O(f * g)" |
23449 | 370 |
by auto |
45575 | 371 |
also have "... <= O((\<lambda>x. 1 / f x) * (f * g))" |
23449 | 372 |
by (rule bigo_mult2) |
45575 | 373 |
also have "(\<lambda>x. 1 / f x) * (f * g) = g" |
43197 | 374 |
apply (simp add: func_times) |
23449 | 375 |
apply (rule ext) |
41541 | 376 |
apply (simp add: a h nonzero_divide_eq_eq mult_ac) |
23449 | 377 |
done |
45575 | 378 |
finally have "(\<lambda>x. (1\<Colon>'b) / f x) * h : O(g)". |
379 |
then have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) : f *o O(g)" |
|
23449 | 380 |
by auto |
45575 | 381 |
also have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) = h" |
43197 | 382 |
apply (simp add: func_times) |
23449 | 383 |
apply (rule ext) |
41541 | 384 |
apply (simp add: a h nonzero_divide_eq_eq mult_ac) |
23449 | 385 |
done |
386 |
finally show "h : f *o O(g)". |
|
387 |
qed |
|
388 |
qed |
|
389 |
||
45575 | 390 |
lemma bigo_mult6: "\<forall>x. f x ~= 0 \<Longrightarrow> |
391 |
O(f * g) = (f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) *o O(g)" |
|
23449 | 392 |
by (metis bigo_mult2 bigo_mult5 order_antisym) |
393 |
||
394 |
(*proof requires relaxing relevance: 2007-01-25*) |
|
45705 | 395 |
declare bigo_mult6 [simp] |
396 |
||
45575 | 397 |
lemma bigo_mult7: "\<forall>x. f x ~= 0 \<Longrightarrow> |
398 |
O(f * g) <= O(f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) \<otimes> O(g)" |
|
399 |
(* sledgehammer *) |
|
23449 | 400 |
apply (subst bigo_mult6) |
401 |
apply assumption |
|
43197 | 402 |
apply (rule set_times_mono3) |
23449 | 403 |
apply (rule bigo_refl) |
404 |
done |
|
405 |
||
45575 | 406 |
declare bigo_mult6 [simp del] |
407 |
declare bigo_mult7 [intro!] |
|
408 |
||
409 |
lemma bigo_mult8: "\<forall>x. f x ~= 0 \<Longrightarrow> |
|
410 |
O(f * g) = O(f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) \<otimes> O(g)" |
|
23449 | 411 |
by (metis bigo_mult bigo_mult7 order_antisym_conv) |
412 |
||
45575 | 413 |
lemma bigo_minus [intro]: "f : O(g) \<Longrightarrow> - f : O(g)" |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
414 |
by (auto simp add: bigo_def fun_Compl_def) |
23449 | 415 |
|
45575 | 416 |
lemma bigo_minus2: "f : g +o O(h) \<Longrightarrow> -f : -g +o O(h)" |
23449 | 417 |
apply (rule set_minus_imp_plus) |
418 |
apply (drule set_plus_imp_minus) |
|
419 |
apply (drule bigo_minus) |
|
420 |
apply (simp add: diff_minus) |
|
421 |
done |
|
422 |
||
423 |
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
|
424 |
by (auto simp add: bigo_def fun_Compl_def abs_minus_cancel) |
23449 | 425 |
|
45575 | 426 |
lemma bigo_plus_absorb_lemma1: "f : O(g) \<Longrightarrow> f +o O(g) <= O(g)" |
23449 | 427 |
proof - |
428 |
assume a: "f : O(g)" |
|
429 |
show "f +o O(g) <= O(g)" |
|
430 |
proof - |
|
431 |
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
|
432 |
then have "f +o O(g) <= O(f) \<oplus> O(g)" |
23449 | 433 |
by (auto del: subsetI) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
434 |
also have "... <= O(g) \<oplus> O(g)" |
23449 | 435 |
proof - |
436 |
from a have "O(f) <= O(g)" by (auto del: subsetI) |
|
437 |
thus ?thesis by (auto del: subsetI) |
|
438 |
qed |
|
439 |
also have "... <= O(g)" by (simp add: bigo_plus_idemp) |
|
440 |
finally show ?thesis . |
|
441 |
qed |
|
442 |
qed |
|
443 |
||
45575 | 444 |
lemma bigo_plus_absorb_lemma2: "f : O(g) \<Longrightarrow> O(g) <= f +o O(g)" |
23449 | 445 |
proof - |
446 |
assume a: "f : O(g)" |
|
447 |
show "O(g) <= f +o O(g)" |
|
448 |
proof - |
|
449 |
from a have "-f : O(g)" by auto |
|
450 |
then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1) |
|
451 |
then have "f +o (-f +o O(g)) <= f +o O(g)" by auto |
|
452 |
also have "f +o (-f +o O(g)) = O(g)" |
|
453 |
by (simp add: set_plus_rearranges) |
|
454 |
finally show ?thesis . |
|
455 |
qed |
|
456 |
qed |
|
457 |
||
45575 | 458 |
lemma bigo_plus_absorb [simp]: "f : O(g) \<Longrightarrow> f +o O(g) = O(g)" |
41865
4e8483cc2cc5
declare ext [intro]: Extensionality now available by default
paulson
parents:
41541
diff
changeset
|
459 |
by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff) |
23449 | 460 |
|
45575 | 461 |
lemma bigo_plus_absorb2 [intro]: "f : O(g) \<Longrightarrow> A <= O(g) \<Longrightarrow> f +o A <= O(g)" |
23449 | 462 |
apply (subgoal_tac "f +o A <= f +o O(g)") |
463 |
apply force+ |
|
464 |
done |
|
465 |
||
45575 | 466 |
lemma bigo_add_commute_imp: "f : g +o O(h) \<Longrightarrow> g : f +o O(h)" |
23449 | 467 |
apply (subst set_minus_plus [symmetric]) |
468 |
apply (subgoal_tac "g - f = - (f - g)") |
|
469 |
apply (erule ssubst) |
|
470 |
apply (rule bigo_minus) |
|
471 |
apply (subst set_minus_plus) |
|
472 |
apply assumption |
|
45575 | 473 |
apply (simp add: diff_minus add_ac) |
23449 | 474 |
done |
475 |
||
476 |
lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))" |
|
477 |
apply (rule iffI) |
|
478 |
apply (erule bigo_add_commute_imp)+ |
|
479 |
done |
|
480 |
||
45575 | 481 |
lemma bigo_const1: "(\<lambda>x. c) : O(\<lambda>x. 1)" |
23449 | 482 |
by (auto simp add: bigo_def mult_ac) |
483 |
||
45575 | 484 |
lemma (*bigo_const2 [intro]:*) "O(\<lambda>x. c) <= O(\<lambda>x. 1)" |
41865
4e8483cc2cc5
declare ext [intro]: Extensionality now available by default
paulson
parents:
41541
diff
changeset
|
485 |
by (metis bigo_const1 bigo_elt_subset) |
23449 | 486 |
|
45575 | 487 |
lemma bigo_const2 [intro]: "O(\<lambda>x. c\<Colon>'b\<Colon>{linordered_idom,number_ring}) <= O(\<lambda>x. 1)" |
36561 | 488 |
proof - |
45575 | 489 |
have "\<forall>u. (\<lambda>Q. u) \<in> O(\<lambda>Q. 1)" by (metis bigo_const1) |
490 |
thus "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)" by (metis bigo_elt_subset) |
|
23449 | 491 |
qed |
492 |
||
45575 | 493 |
lemma bigo_const3: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> (\<lambda>x. 1) : O(\<lambda>x. c)" |
23449 | 494 |
apply (simp add: bigo_def) |
36561 | 495 |
by (metis abs_eq_0 left_inverse order_refl) |
23449 | 496 |
|
45575 | 497 |
lemma bigo_const4: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> O(\<lambda>x. 1) <= O(\<lambda>x. c)" |
23449 | 498 |
by (rule bigo_elt_subset, rule bigo_const3, assumption) |
499 |
||
45575 | 500 |
lemma bigo_const [simp]: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> |
501 |
O(\<lambda>x. c) = O(\<lambda>x. 1)" |
|
23449 | 502 |
by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption) |
503 |
||
45575 | 504 |
lemma bigo_const_mult1: "(\<lambda>x. c * f x) : O(f)" |
24937
340523598914
context-based treatment of generalization; also handling TFrees in axiom clauses
paulson
parents:
24855
diff
changeset
|
505 |
apply (simp add: bigo_def abs_mult) |
36561 | 506 |
by (metis le_less) |
23449 | 507 |
|
45575 | 508 |
lemma bigo_const_mult2: "O(\<lambda>x. c * f x) <= O(f)" |
23449 | 509 |
by (rule bigo_elt_subset, rule bigo_const_mult1) |
510 |
||
45575 | 511 |
lemma bigo_const_mult3: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> f : O(\<lambda>x. c * f x)" |
512 |
apply (simp add: bigo_def) |
|
513 |
(* sledgehammer *) |
|
514 |
apply (rule_tac x = "abs(inverse c)" in exI) |
|
515 |
apply (simp only: abs_mult [symmetric] mult_assoc [symmetric]) |
|
43197 | 516 |
apply (subst left_inverse) |
45575 | 517 |
by auto |
23449 | 518 |
|
45575 | 519 |
lemma bigo_const_mult4: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> |
520 |
O(f) <= O(\<lambda>x. c * f x)" |
|
23449 | 521 |
by (rule bigo_elt_subset, rule bigo_const_mult3, assumption) |
522 |
||
45575 | 523 |
lemma bigo_const_mult [simp]: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> |
524 |
O(\<lambda>x. c * f x) = O(f)" |
|
23449 | 525 |
by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4) |
526 |
||
45575 | 527 |
lemma bigo_const_mult5 [simp]: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> |
528 |
(\<lambda>x. c) *o O(f) = O(f)" |
|
23449 | 529 |
apply (auto del: subsetI) |
530 |
apply (rule order_trans) |
|
531 |
apply (rule bigo_mult2) |
|
532 |
apply (simp add: func_times) |
|
533 |
apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times) |
|
45575 | 534 |
apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI) |
43197 | 535 |
apply (rename_tac g d) |
24942 | 536 |
apply safe |
43197 | 537 |
apply (rule_tac [2] ext) |
538 |
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
|
539 |
apply simp |
24942 | 540 |
apply (simp add: mult_assoc [symmetric] abs_mult) |
39259
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
541 |
(* couldn't get this proof without the step above *) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
542 |
proof - |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
543 |
fix g :: "'b \<Rightarrow> 'a" and d :: 'a |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
544 |
assume A1: "c \<noteq> (0\<Colon>'a)" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
545 |
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
|
546 |
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
|
547 |
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
|
548 |
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
|
549 |
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
|
550 |
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
|
551 |
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
|
552 |
using A2 by metis |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
553 |
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
|
554 |
using F3 by metis |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
555 |
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
|
556 |
by (metis comm_mult_left_mono) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
557 |
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
|
558 |
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
|
559 |
qed |
23449 | 560 |
|
45575 | 561 |
lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) <= O(f)" |
23449 | 562 |
apply (auto intro!: subsetI |
563 |
simp add: bigo_def elt_set_times_def func_times |
|
564 |
simp del: abs_mult mult_ac) |
|
45575 | 565 |
(* sledgehammer *) |
23449 | 566 |
apply (rule_tac x = "ca * (abs c)" in exI) |
567 |
apply (rule allI) |
|
568 |
apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))") |
|
569 |
apply (erule ssubst) |
|
570 |
apply (subst abs_mult) |
|
571 |
apply (rule mult_left_mono) |
|
572 |
apply (erule spec) |
|
573 |
apply simp |
|
574 |
apply(simp add: mult_ac) |
|
575 |
done |
|
576 |
||
45575 | 577 |
lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)" |
23449 | 578 |
proof - |
579 |
assume "f =o O(g)" |
|
45575 | 580 |
then have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)" |
23449 | 581 |
by auto |
45575 | 582 |
also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)" |
23449 | 583 |
by (simp add: func_times) |
45575 | 584 |
also have "(\<lambda>x. c) *o O(g) <= O(g)" |
23449 | 585 |
by (auto del: subsetI) |
586 |
finally show ?thesis . |
|
587 |
qed |
|
588 |
||
45575 | 589 |
lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f(k x)) =o O(\<lambda>x. g(k x))" |
23449 | 590 |
by (unfold bigo_def, auto) |
591 |
||
45575 | 592 |
lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. f(k x)) =o (\<lambda>x. g(k x)) +o |
593 |
O(\<lambda>x. h(k x))" |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
594 |
apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def |
23449 | 595 |
func_plus) |
596 |
apply (erule bigo_compose1) |
|
597 |
done |
|
598 |
||
599 |
subsection {* Setsum *} |
|
600 |
||
45575 | 601 |
lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 <= h x y \<Longrightarrow> |
602 |
\<exists>c. \<forall>x. \<forall>y \<in> A x. abs (f x y) <= c * (h x y) \<Longrightarrow> |
|
603 |
(\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)" |
|
23449 | 604 |
apply (auto simp add: bigo_def) |
605 |
apply (rule_tac x = "abs c" in exI) |
|
606 |
apply (subst abs_of_nonneg) back back |
|
607 |
apply (rule setsum_nonneg) |
|
608 |
apply force |
|
609 |
apply (subst setsum_right_distrib) |
|
610 |
apply (rule allI) |
|
611 |
apply (rule order_trans) |
|
612 |
apply (rule setsum_abs) |
|
613 |
apply (rule setsum_mono) |
|
43197 | 614 |
apply (blast intro: order_trans mult_right_mono abs_ge_self) |
23449 | 615 |
done |
616 |
||
45575 | 617 |
lemma bigo_setsum1: "\<forall>x y. 0 <= h x y \<Longrightarrow> |
618 |
\<exists>c. \<forall>x y. abs (f x y) <= c * (h x y) \<Longrightarrow> |
|
619 |
(\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)" |
|
620 |
by (metis (no_types) bigo_setsum_main) |
|
23449 | 621 |
|
45575 | 622 |
lemma bigo_setsum2: "\<forall>y. 0 <= h y \<Longrightarrow> |
623 |
\<exists>c. \<forall>y. abs(f y) <= c * (h y) \<Longrightarrow> |
|
624 |
(\<lambda>x. SUM y : A x. f y) =o O(\<lambda>x. SUM y : A x. h y)" |
|
43197 | 625 |
by (rule bigo_setsum1, auto) |
23449 | 626 |
|
45575 | 627 |
lemma bigo_setsum3: "f =o O(h) \<Longrightarrow> |
628 |
(\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o |
|
629 |
O(\<lambda>x. SUM y : A x. abs(l x y * h(k x y)))" |
|
630 |
apply (rule bigo_setsum1) |
|
631 |
apply (rule allI)+ |
|
632 |
apply (rule abs_ge_zero) |
|
633 |
apply (unfold bigo_def) |
|
634 |
apply (auto simp add: abs_mult) |
|
635 |
(* sledgehammer *) |
|
636 |
apply (rule_tac x = c in exI) |
|
637 |
apply (rule allI)+ |
|
638 |
apply (subst mult_left_commute) |
|
639 |
apply (rule mult_left_mono) |
|
640 |
apply (erule spec) |
|
641 |
by (rule abs_ge_zero) |
|
23449 | 642 |
|
45575 | 643 |
lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow> |
644 |
(\<lambda>x. SUM y : A x. l x y * f(k x y)) =o |
|
645 |
(\<lambda>x. SUM y : A x. l x y * g(k x y)) +o |
|
646 |
O(\<lambda>x. SUM y : A x. abs(l x y * h(k x y)))" |
|
647 |
apply (rule set_minus_imp_plus) |
|
648 |
apply (subst fun_diff_def) |
|
649 |
apply (subst setsum_subtractf [symmetric]) |
|
650 |
apply (subst right_diff_distrib [symmetric]) |
|
651 |
apply (rule bigo_setsum3) |
|
652 |
apply (subst fun_diff_def [symmetric]) |
|
653 |
by (erule set_plus_imp_minus) |
|
23449 | 654 |
|
45575 | 655 |
lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow> |
656 |
\<forall>x. 0 <= h x \<Longrightarrow> |
|
657 |
(\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o |
|
658 |
O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))" |
|
659 |
apply (subgoal_tac "(\<lambda>x. SUM y : A x. (l x y) * h(k x y)) = |
|
660 |
(\<lambda>x. SUM y : A x. abs((l x y) * h(k x y)))") |
|
23449 | 661 |
apply (erule ssubst) |
662 |
apply (erule bigo_setsum3) |
|
663 |
apply (rule ext) |
|
664 |
apply (rule setsum_cong2) |
|
43197 | 665 |
apply (thin_tac "f \<in> O(h)") |
24942 | 666 |
apply (metis abs_of_nonneg zero_le_mult_iff) |
23449 | 667 |
done |
668 |
||
45575 | 669 |
lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow> |
670 |
\<forall>x. 0 <= h x \<Longrightarrow> |
|
671 |
(\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o |
|
672 |
(\<lambda>x. SUM y : A x. (l x y) * g(k x y)) +o |
|
673 |
O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))" |
|
23449 | 674 |
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
|
675 |
apply (subst fun_diff_def) |
23449 | 676 |
apply (subst setsum_subtractf [symmetric]) |
677 |
apply (subst right_diff_distrib [symmetric]) |
|
678 |
apply (rule bigo_setsum5) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
679 |
apply (subst fun_diff_def [symmetric]) |
23449 | 680 |
apply (drule set_plus_imp_minus) |
681 |
apply auto |
|
682 |
done |
|
683 |
||
684 |
subsection {* Misc useful stuff *} |
|
685 |
||
45575 | 686 |
lemma bigo_useful_intro: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
687 |
A \<oplus> B <= O(f)" |
23449 | 688 |
apply (subst bigo_plus_idemp [symmetric]) |
689 |
apply (rule set_plus_mono2) |
|
690 |
apply assumption+ |
|
691 |
done |
|
692 |
||
45575 | 693 |
lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)" |
23449 | 694 |
apply (subst bigo_plus_idemp [symmetric]) |
695 |
apply (rule set_plus_intro) |
|
696 |
apply assumption+ |
|
697 |
done |
|
43197 | 698 |
|
45575 | 699 |
lemma bigo_useful_const_mult: "(c\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> |
700 |
(\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)" |
|
23449 | 701 |
apply (rule subsetD) |
45575 | 702 |
apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) <= O(h)") |
23449 | 703 |
apply assumption |
704 |
apply (rule bigo_const_mult6) |
|
45575 | 705 |
apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)") |
23449 | 706 |
apply (erule ssubst) |
707 |
apply (erule set_times_intro2) |
|
43197 | 708 |
apply (simp add: func_times) |
23449 | 709 |
done |
710 |
||
45575 | 711 |
lemma bigo_fix: "(\<lambda>x. f ((x\<Colon>nat) + 1)) =o O(\<lambda>x. h(x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow> |
23449 | 712 |
f =o O(h)" |
45575 | 713 |
apply (simp add: bigo_alt_def) |
714 |
by (metis abs_ge_zero abs_mult abs_of_pos abs_zero not0_implies_Suc) |
|
23449 | 715 |
|
43197 | 716 |
lemma bigo_fix2: |
45575 | 717 |
"(\<lambda>x. f ((x\<Colon>nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow> |
718 |
f 0 = g 0 \<Longrightarrow> f =o g +o O(h)" |
|
23449 | 719 |
apply (rule set_minus_imp_plus) |
720 |
apply (rule bigo_fix) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
721 |
apply (subst fun_diff_def) |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
722 |
apply (subst fun_diff_def [symmetric]) |
23449 | 723 |
apply (rule set_plus_imp_minus) |
724 |
apply simp |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
725 |
apply (simp add: fun_diff_def) |
23449 | 726 |
done |
727 |
||
728 |
subsection {* Less than or equal to *} |
|
729 |
||
45575 | 730 |
definition lesso :: "('a => 'b\<Colon>linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where |
731 |
"f <o g == (\<lambda>x. max (f x - g x) 0)" |
|
23449 | 732 |
|
45575 | 733 |
lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) <= abs (f x) \<Longrightarrow> |
23449 | 734 |
g =o O(h)" |
735 |
apply (unfold bigo_def) |
|
736 |
apply clarsimp |
|
43197 | 737 |
apply (blast intro: order_trans) |
23449 | 738 |
done |
739 |
||
45575 | 740 |
lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) <= f x \<Longrightarrow> |
23449 | 741 |
g =o O(h)" |
742 |
apply (erule bigo_lesseq1) |
|
43197 | 743 |
apply (blast intro: abs_ge_self order_trans) |
23449 | 744 |
done |
745 |
||
45575 | 746 |
lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= f x \<Longrightarrow> |
23449 | 747 |
g =o O(h)" |
748 |
apply (erule bigo_lesseq2) |
|
749 |
apply (rule allI) |
|
750 |
apply (subst abs_of_nonneg) |
|
751 |
apply (erule spec)+ |
|
752 |
done |
|
753 |
||
45575 | 754 |
lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow> |
755 |
\<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= abs (f x) \<Longrightarrow> |
|
23449 | 756 |
g =o O(h)" |
757 |
apply (erule bigo_lesseq1) |
|
758 |
apply (rule allI) |
|
759 |
apply (subst abs_of_nonneg) |
|
760 |
apply (erule spec)+ |
|
761 |
done |
|
762 |
||
45575 | 763 |
lemma bigo_lesso1: "\<forall>x. f x <= g x \<Longrightarrow> f <o g =o O(h)" |
36561 | 764 |
apply (unfold lesso_def) |
45575 | 765 |
apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0") |
766 |
apply (metis bigo_zero) |
|
767 |
by (metis (lam_lifting, no_types) func_zero le_fun_def le_iff_diff_le_0 |
|
768 |
min_max.sup_absorb2 order_eq_iff) |
|
23449 | 769 |
|
45575 | 770 |
lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow> |
771 |
\<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. k x <= f x \<Longrightarrow> |
|
23449 | 772 |
k <o g =o O(h)" |
773 |
apply (unfold lesso_def) |
|
774 |
apply (rule bigo_lesseq4) |
|
775 |
apply (erule set_plus_imp_minus) |
|
776 |
apply (rule allI) |
|
777 |
apply (rule le_maxI2) |
|
778 |
apply (rule allI) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
779 |
apply (subst fun_diff_def) |
23449 | 780 |
apply (erule thin_rl) |
45575 | 781 |
(* sledgehammer *) |
782 |
apply (case_tac "0 <= k x - g x") |
|
783 |
apply (metis (hide_lams, no_types) abs_le_iff add_le_imp_le_right diff_minus le_less |
|
784 |
le_max_iff_disj min_max.le_supE min_max.sup_absorb2 |
|
785 |
min_max.sup_commute) |
|
786 |
by (metis abs_ge_zero le_cases min_max.sup_absorb2) |
|
23449 | 787 |
|
45575 | 788 |
lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow> |
789 |
\<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. g x <= k x \<Longrightarrow> |
|
23449 | 790 |
f <o k =o O(h)" |
791 |
apply (unfold lesso_def) |
|
792 |
apply (rule bigo_lesseq4) |
|
793 |
apply (erule set_plus_imp_minus) |
|
794 |
apply (rule allI) |
|
795 |
apply (rule le_maxI2) |
|
796 |
apply (rule allI) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
797 |
apply (subst fun_diff_def) |
45575 | 798 |
apply (erule thin_rl) |
799 |
(* sledgehammer *) |
|
23449 | 800 |
apply (case_tac "0 <= f x - k x") |
45575 | 801 |
apply simp |
23449 | 802 |
apply (subst abs_of_nonneg) |
803 |
apply (drule_tac x = x in spec) back |
|
45705 | 804 |
apply (metis diff_less_0_iff_less linorder_not_le not_leE xt1(12) xt1(6)) |
45575 | 805 |
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
|
806 |
apply (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute) |
23449 | 807 |
done |
808 |
||
45705 | 809 |
lemma bigo_lesso4: |
810 |
"f <o g =o O(k\<Colon>'a=>'b\<Colon>{linordered_field,number_ring}) \<Longrightarrow> |
|
811 |
g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)" |
|
812 |
apply (unfold lesso_def) |
|
813 |
apply (drule set_plus_imp_minus) |
|
814 |
apply (drule bigo_abs5) back |
|
815 |
apply (simp add: fun_diff_def) |
|
816 |
apply (drule bigo_useful_add, assumption) |
|
817 |
apply (erule bigo_lesseq2) back |
|
818 |
apply (rule allI) |
|
819 |
by (auto simp add: func_plus fun_diff_def algebra_simps |
|
23449 | 820 |
split: split_max abs_split) |
821 |
||
45705 | 822 |
lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x <= g x + C * abs (h x)" |
823 |
apply (simp only: lesso_def bigo_alt_def) |
|
824 |
apply clarsimp |
|
825 |
by (metis abs_if abs_mult add_commute diff_le_eq less_not_permute) |
|
23449 | 826 |
|
827 |
end |