src/HOL/Metis_Examples/Big_O.thy
author huffman
Tue Nov 15 12:49:05 2011 +0100 (2011-11-15)
changeset 45504 cad35ed6effa
parent 45270 d5b5c9259afd
child 45532 74b17a0881b3
permissions -rw-r--r--
Metis_Examples/Big_O.thy: add number_ring class constraints, adapt proofs to use cancellation simprocs
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(*  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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  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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declare [[metis_new_skolemizer]]
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subsection {* Definitions *}
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definition bigo :: "('a => 'b::{linordered_idom,number_ring}) => ('a => 'b) set"    ("(1O'(_'))") where
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  "O(f::('a => 'b)) ==   {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_pos_const" ]]
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lemma bigo_pos_const: "(EX (c::'a::linordered_idom).
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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  apply (metis abs_ge_zero abs_of_nonneg Orderings.xt1(6) abs_mult)
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  done
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(*** Now various verions with an increasing shrink factor ***)
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sledgehammer_params [isar_proof, isar_shrink_factor = 1]
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lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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proof -
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  fix c :: 'a and x :: 'b
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  assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
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  have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> \<bar>x\<^isub>1\<bar>" by (metis abs_ge_zero)
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  have F2: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
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  have F3: "\<forall>x\<^isub>1 x\<^isub>3. x\<^isub>3 \<le> \<bar>h x\<^isub>1\<bar> \<longrightarrow> x\<^isub>3 \<le> c * \<bar>f x\<^isub>1\<bar>" by (metis A1 order_trans)
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  have F4: "\<forall>x\<^isub>2 x\<^isub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^isub>3\<bar> * \<bar>x\<^isub>2\<bar> = \<bar>x\<^isub>3 * x\<^isub>2\<bar>"
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    by (metis abs_mult)
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  have F5: "\<forall>x\<^isub>3 x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> x\<^isub>1 \<longrightarrow> \<bar>x\<^isub>3 * x\<^isub>1\<bar> = \<bar>x\<^isub>3\<bar> * x\<^isub>1"
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    by (metis abs_mult_pos)
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  hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = \<bar>1\<bar> * x\<^isub>1" by (metis F2)
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  hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F2 abs_one)
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  hence "\<forall>x\<^isub>3. 0 \<le> \<bar>h x\<^isub>3\<bar> \<longrightarrow> \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>" by (metis F3)
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  hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>" by (metis F1)
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  hence "\<forall>x\<^isub>3. (0\<Colon>'a) \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^isub>3\<bar>" by (metis F5)
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  hence "\<forall>x\<^isub>3. (0\<Colon>'a) \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c * f x\<^isub>3\<bar>" by (metis F4)
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  hence "\<forall>x\<^isub>3. c * \<bar>f x\<^isub>3\<bar> = \<bar>c * f x\<^isub>3\<bar>" by (metis F1)
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  hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1)
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  thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F4)
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qed
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sledgehammer_params [isar_proof, isar_shrink_factor = 2]
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lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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proof -
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  fix c :: 'a and x :: 'b
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  assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
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  have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
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  have F2: "\<forall>x\<^isub>2 x\<^isub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^isub>3\<bar> * \<bar>x\<^isub>2\<bar> = \<bar>x\<^isub>3 * x\<^isub>2\<bar>"
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    by (metis abs_mult)
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  have "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F1 abs_mult_pos abs_one)
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  hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>" by (metis A1 abs_ge_zero order_trans)
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  hence "\<forall>x\<^isub>3. 0 \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c * f x\<^isub>3\<bar>" by (metis F2 abs_mult_pos)
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  hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero)
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  thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F2)
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qed
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sledgehammer_params [isar_proof, isar_shrink_factor = 3]
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lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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proof -
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  fix c :: 'a and x :: 'b
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  assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
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  have F1: "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
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  have F2: "\<forall>x\<^isub>3 x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> x\<^isub>1 \<longrightarrow> \<bar>x\<^isub>3 * x\<^isub>1\<bar> = \<bar>x\<^isub>3\<bar> * x\<^isub>1" by (metis abs_mult_pos)
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  hence "\<forall>x\<^isub>1\<ge>0. \<bar>x\<^isub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^isub>1" by (metis F1 abs_one)
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  hence "\<forall>x\<^isub>3. 0 \<le> \<bar>f x\<^isub>3\<bar> \<longrightarrow> c * \<bar>f x\<^isub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^isub>3\<bar>" by (metis F2 A1 abs_ge_zero order_trans)
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  thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis A1 abs_mult abs_ge_zero)
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qed
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sledgehammer_params [isar_proof, isar_shrink_factor = 4]
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lemma (*bigo_pos_const:*) "(EX (c::'a::linordered_idom).
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    ALL x. (abs (h x)) <= (c * (abs (f x))))
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      = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
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  apply auto
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  apply (case_tac "c = 0", simp)
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  apply (rule_tac x = "1" in exI, simp)
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  apply (rule_tac x = "abs c" in exI, auto)
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proof -
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  fix c :: 'a and x :: 'b
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  assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
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  have "\<forall>x\<^isub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^isub>1 = x\<^isub>1" by (metis mult_1)
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  hence "\<forall>x\<^isub>3. \<bar>c * \<bar>f x\<^isub>3\<bar>\<bar> = c * \<bar>f x\<^isub>3\<bar>"
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    by (metis A1 abs_ge_zero order_trans abs_mult_pos abs_one)
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  hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero abs_mult_pos abs_mult)
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  thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis abs_mult)
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qed
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sledgehammer_params [isar_proof, isar_shrink_factor = 1]
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lemma bigo_alt_def: "O(f) =
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    {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
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by (auto simp add: bigo_def bigo_pos_const)
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_elt_subset" ]]
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lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
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  apply (auto simp add: bigo_alt_def)
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  apply (rule_tac x = "ca * c" in exI)
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  apply (rule conjI)
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  apply (rule mult_pos_pos)
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  apply (assumption)+
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(*sledgehammer*)
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  apply (rule allI)
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  apply (drule_tac x = "xa" in spec)+
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  apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))")
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  apply (erule order_trans)
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  apply (simp add: mult_ac)
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  apply (rule mult_left_mono, assumption)
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  apply (rule order_less_imp_le, assumption)
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done
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_refl" ]]
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lemma bigo_refl [intro]: "f : O(f)"
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apply (auto simp add: bigo_def)
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by (metis mult_1 order_refl)
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_zero" ]]
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lemma bigo_zero: "0 : O(g)"
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apply (auto simp add: bigo_def func_zero)
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by (metis mult_zero_left order_refl)
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lemma bigo_zero2: "O(%x.0) = {%x.0}"
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  by (auto simp add: bigo_def)
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lemma bigo_plus_self_subset [intro]:
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  "O(f) \<oplus> O(f) <= O(f)"
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  apply (auto simp add: bigo_alt_def set_plus_def)
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  apply (rule_tac x = "c + ca" in exI)
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  apply auto
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  apply (simp add: ring_distribs func_plus)
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  apply (blast intro:order_trans abs_triangle_ineq add_mono elim:)
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done
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lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
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  apply (rule equalityI)
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  apply (rule bigo_plus_self_subset)
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  apply (rule set_zero_plus2)
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  apply (rule bigo_zero)
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done
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lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) \<oplus> O(g)"
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  apply (rule subsetI)
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  apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
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  apply (subst bigo_pos_const [symmetric])+
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  apply (rule_tac x =
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    "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
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  apply (rule conjI)
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  apply (rule_tac x = "c + c" in exI)
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  apply (clarsimp)
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  apply (auto)
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  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
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  apply (erule_tac x = xa in allE)
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  apply (erule order_trans)
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  apply (simp)
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  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
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  apply (erule order_trans)
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  apply (simp add: ring_distribs)
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  apply (rule mult_left_mono)
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  apply (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 = "%n. if (abs (f n)) <  abs (g n) then x n else 0"
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     in exI)
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  apply (rule conjI)
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  apply (rule_tac x = "c + c" in exI)
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  apply auto
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  apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
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  apply (erule_tac x = xa in allE)
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  apply (erule order_trans)
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  apply (simp)
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  apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
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  apply (erule order_trans)
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  apply (simp add: ring_distribs)
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  apply (rule mult_left_mono)
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  apply (rule abs_triangle_ineq)
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  apply (simp add: order_less_le)
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apply (metis abs_not_less_zero double_less_0_iff less_not_permute linorder_not_less mult_less_0_iff)
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done
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lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A \<oplus> B <= O(f)"
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  apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> O(f)")
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  apply (erule order_trans)
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  apply simp
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  apply (auto del: subsetI simp del: bigo_plus_idemp)
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done
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declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq" ]]
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lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==>
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  O(f + g) = O(f) \<oplus> O(g)"
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  apply (rule equalityI)
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  apply (rule bigo_plus_subset)
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  apply (simp add: bigo_alt_def set_plus_def func_plus)
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  apply clarify
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(*sledgehammer*)
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  apply (rule_tac x = "max c ca" in exI)
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  apply (rule conjI)
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   apply (metis Orderings.less_max_iff_disj)
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  apply clarify
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  apply (drule_tac x = "xa" in spec)+
paulson@23449
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  apply (subgoal_tac "0 <= f xa + g xa")
nipkow@23477
   242
  apply (simp add: ring_distribs)
paulson@23449
   243
  apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
blanchet@43197
   244
  apply (subgoal_tac "abs(a xa) + abs(b xa) <=
paulson@23449
   245
      max c ca * f xa + max c ca * g xa")
paulson@23449
   246
  apply (blast intro: order_trans)
paulson@23449
   247
  defer 1
paulson@23449
   248
  apply (rule abs_triangle_ineq)
paulson@25087
   249
  apply (metis add_nonneg_nonneg)
paulson@23449
   250
  apply (rule add_mono)
blanchet@39259
   251
using [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq_simpler" ]]
blanchet@39259
   252
  apply (metis le_maxI2 linorder_linear min_max.sup_absorb1 mult_right_mono xt1(6))
blanchet@39259
   253
  apply (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans)
paulson@23449
   254
done
paulson@23449
   255
blanchet@38991
   256
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt" ]]
blanchet@43197
   257
lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==>
blanchet@43197
   258
    f : O(g)"
paulson@23449
   259
  apply (auto simp add: bigo_def)
blanchet@36561
   260
(* Version 1: one-line proof *)
haftmann@35050
   261
  apply (metis abs_le_D1 linorder_class.not_less  order_less_le  Orderings.xt1(12)  abs_mult)
paulson@23449
   262
  done
paulson@23449
   263
blanchet@43197
   264
lemma (*bigo_bounded_alt:*) "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==>
blanchet@36561
   265
    f : O(g)"
blanchet@36561
   266
apply (auto simp add: bigo_def)
blanchet@36561
   267
(* Version 2: structured proof *)
blanchet@36561
   268
proof -
blanchet@36561
   269
  assume "\<forall>x. f x \<le> c * g x"
blanchet@36561
   270
  thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans)
paulson@23449
   271
qed
paulson@23449
   272
blanchet@36561
   273
text{*So here is the easier (and more natural) problem using transitivity*}
blanchet@38991
   274
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]]
blanchet@43197
   275
lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)"
blanchet@36561
   276
apply (auto simp add: bigo_def)
blanchet@36561
   277
(* Version 1: one-line proof *)
blanchet@36561
   278
by (metis abs_ge_self abs_mult order_trans)
paulson@23449
   279
paulson@23449
   280
text{*So here is the easier (and more natural) problem using transitivity*}
blanchet@38991
   281
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]]
blanchet@43197
   282
lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)"
paulson@23449
   283
  apply (auto simp add: bigo_def)
blanchet@36561
   284
(* Version 2: structured proof *)
blanchet@36561
   285
proof -
blanchet@36561
   286
  assume "\<forall>x. f x \<le> c * g x"
blanchet@36561
   287
  thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans)
paulson@23449
   288
qed
paulson@23449
   289
blanchet@43197
   290
lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==>
blanchet@43197
   291
    f : O(g)"
paulson@23449
   292
  apply (erule bigo_bounded_alt [of f 1 g])
paulson@23449
   293
  apply simp
paulson@23449
   294
done
paulson@23449
   295
blanchet@38991
   296
declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded2" ]]
paulson@23449
   297
lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
paulson@23449
   298
    f : lb +o O(g)"
blanchet@36561
   299
apply (rule set_minus_imp_plus)
blanchet@36561
   300
apply (rule bigo_bounded)
blanchet@36561
   301
 apply (auto simp add: diff_minus fun_Compl_def func_plus)
blanchet@36561
   302
 prefer 2
blanchet@36561
   303
 apply (drule_tac x = x in spec)+
hoelzl@36844
   304
 apply (metis add_right_mono add_commute diff_add_cancel diff_minus_eq_add le_less order_trans)
blanchet@36561
   305
proof -
blanchet@36561
   306
  fix x :: 'a
blanchet@36561
   307
  assume "\<forall>x. lb x \<le> f x"
blanchet@36561
   308
  thus "(0\<Colon>'b) \<le> f x + - lb x" by (metis not_leE diff_minus less_iff_diff_less_0 less_le_not_le)
paulson@23449
   309
qed
paulson@23449
   310
blanchet@38991
   311
declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs" ]]
blanchet@43197
   312
lemma bigo_abs: "(%x. abs(f x)) =o O(f)"
blanchet@36561
   313
apply (unfold bigo_def)
blanchet@36561
   314
apply auto
hoelzl@36844
   315
by (metis mult_1 order_refl)
paulson@23449
   316
blanchet@38991
   317
declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs2" ]]
paulson@23449
   318
lemma bigo_abs2: "f =o O(%x. abs(f x))"
blanchet@36561
   319
apply (unfold bigo_def)
blanchet@36561
   320
apply auto
hoelzl@36844
   321
by (metis mult_1 order_refl)
blanchet@43197
   322
paulson@23449
   323
lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
blanchet@36561
   324
proof -
blanchet@36561
   325
  have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset)
blanchet@36561
   326
  have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs)
blanchet@36561
   327
  have "\<forall>u. u \<in> O(\<lambda>R. \<bar>u R\<bar>)" by (metis bigo_abs2)
blanchet@36561
   328
  thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" using F1 F2 by auto
blanchet@43197
   329
qed
paulson@23449
   330
blanchet@43197
   331
lemma bigo_abs4: "f =o g +o O(h) ==>
paulson@23449
   332
    (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
paulson@23449
   333
  apply (drule set_plus_imp_minus)
paulson@23449
   334
  apply (rule set_minus_imp_plus)
berghofe@26814
   335
  apply (subst fun_diff_def)
paulson@23449
   336
proof -
paulson@23449
   337
  assume a: "f - g : O(h)"
paulson@23449
   338
  have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
paulson@23449
   339
    by (rule bigo_abs2)
paulson@23449
   340
  also have "... <= O(%x. abs (f x - g x))"
paulson@23449
   341
    apply (rule bigo_elt_subset)
paulson@23449
   342
    apply (rule bigo_bounded)
paulson@23449
   343
    apply force
paulson@23449
   344
    apply (rule allI)
paulson@23449
   345
    apply (rule abs_triangle_ineq3)
paulson@23449
   346
    done
paulson@23449
   347
  also have "... <= O(f - g)"
paulson@23449
   348
    apply (rule bigo_elt_subset)
berghofe@26814
   349
    apply (subst fun_diff_def)
paulson@23449
   350
    apply (rule bigo_abs)
paulson@23449
   351
    done
paulson@23449
   352
  also have "... <= O(h)"
wenzelm@23464
   353
    using a by (rule bigo_elt_subset)
paulson@23449
   354
  finally show "(%x. abs (f x) - abs (g x)) : O(h)".
paulson@23449
   355
qed
paulson@23449
   356
blanchet@43197
   357
lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)"
paulson@23449
   358
by (unfold bigo_def, auto)
paulson@23449
   359
berghofe@26814
   360
lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) \<oplus> O(h)"
paulson@23449
   361
proof -
paulson@23449
   362
  assume "f : g +o O(h)"
berghofe@26814
   363
  also have "... <= O(g) \<oplus> O(h)"
paulson@23449
   364
    by (auto del: subsetI)
berghofe@26814
   365
  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
paulson@23449
   366
    apply (subst bigo_abs3 [symmetric])+
paulson@23449
   367
    apply (rule refl)
paulson@23449
   368
    done
paulson@23449
   369
  also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
paulson@23449
   370
    by (rule bigo_plus_eq [symmetric], auto)
paulson@23449
   371
  finally have "f : ...".
paulson@23449
   372
  then have "O(f) <= ..."
paulson@23449
   373
    by (elim bigo_elt_subset)
berghofe@26814
   374
  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
paulson@23449
   375
    by (rule bigo_plus_eq, auto)
paulson@23449
   376
  finally show ?thesis
paulson@23449
   377
    by (simp add: bigo_abs3 [symmetric])
paulson@23449
   378
qed
paulson@23449
   379
blanchet@38991
   380
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult" ]]
berghofe@26814
   381
lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)"
paulson@23449
   382
  apply (rule subsetI)
paulson@23449
   383
  apply (subst bigo_def)
paulson@23449
   384
  apply (auto simp del: abs_mult mult_ac
berghofe@26814
   385
              simp add: bigo_alt_def set_times_def func_times)
paulson@41865
   386
(*sledgehammer*)
paulson@23449
   387
  apply (rule_tac x = "c * ca" in exI)
paulson@23449
   388
  apply(rule allI)
paulson@23449
   389
  apply(erule_tac x = x in allE)+
blanchet@43197
   390
  apply(subgoal_tac "c * ca * abs(f x * g x) =
paulson@23449
   391
      (c * abs(f x)) * (ca * abs(g x))")
blanchet@38991
   392
using [[ sledgehammer_problem_prefix = "BigO__bigo_mult_simpler" ]]
blanchet@43197
   393
prefer 2
haftmann@26041
   394
apply (metis mult_assoc mult_left_commute
haftmann@35050
   395
  abs_of_pos mult_left_commute
haftmann@35050
   396
  abs_mult mult_pos_pos)
blanchet@43197
   397
  apply (erule ssubst)
paulson@23449
   398
  apply (subst abs_mult)
blanchet@36561
   399
(* not quite as hard as BigO__bigo_mult_simpler_1 (a hard problem!) since
blanchet@36561
   400
   abs_mult has just been done *)
blanchet@36561
   401
by (metis abs_ge_zero mult_mono')
paulson@23449
   402
blanchet@38991
   403
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult2" ]]
paulson@23449
   404
lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
paulson@23449
   405
  apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
paulson@41865
   406
(*sledgehammer*)
paulson@23449
   407
  apply (rule_tac x = c in exI)
paulson@23449
   408
  apply clarify
paulson@23449
   409
  apply (drule_tac x = x in spec)
blanchet@38991
   410
using [[ sledgehammer_problem_prefix = "BigO__bigo_mult2_simpler" ]]
paulson@41865
   411
(*sledgehammer [no luck]*)
paulson@23449
   412
  apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
paulson@23449
   413
  apply (simp add: mult_ac)
paulson@23449
   414
  apply (rule mult_left_mono, assumption)
paulson@23449
   415
  apply (rule abs_ge_zero)
paulson@23449
   416
done
paulson@23449
   417
blanchet@38991
   418
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult3" ]]
paulson@23449
   419
lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
blanchet@36561
   420
by (metis bigo_mult set_rev_mp set_times_intro)
paulson@23449
   421
blanchet@38991
   422
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult4" ]]
paulson@23449
   423
lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
paulson@23449
   424
by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib)
paulson@23449
   425
paulson@23449
   426
paulson@23449
   427
lemma bigo_mult5: "ALL x. f x ~= 0 ==>
huffman@45504
   428
    O(f * g) <= (f::'a => ('b::{linordered_field,number_ring})) *o O(g)"
paulson@23449
   429
proof -
wenzelm@41541
   430
  assume a: "ALL x. f x ~= 0"
paulson@23449
   431
  show "O(f * g) <= f *o O(g)"
paulson@23449
   432
  proof
paulson@23449
   433
    fix h
wenzelm@41541
   434
    assume h: "h : O(f * g)"
paulson@23449
   435
    then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
paulson@23449
   436
      by auto
paulson@23449
   437
    also have "... <= O((%x. 1 / f x) * (f * g))"
paulson@23449
   438
      by (rule bigo_mult2)
paulson@23449
   439
    also have "(%x. 1 / f x) * (f * g) = g"
blanchet@43197
   440
      apply (simp add: func_times)
paulson@23449
   441
      apply (rule ext)
wenzelm@41541
   442
      apply (simp add: a h nonzero_divide_eq_eq mult_ac)
paulson@23449
   443
      done
paulson@23449
   444
    finally have "(%x. (1::'b) / f x) * h : O(g)".
paulson@23449
   445
    then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
paulson@23449
   446
      by auto
paulson@23449
   447
    also have "f * ((%x. (1::'b) / f x) * h) = h"
blanchet@43197
   448
      apply (simp add: func_times)
paulson@23449
   449
      apply (rule ext)
wenzelm@41541
   450
      apply (simp add: a h nonzero_divide_eq_eq mult_ac)
paulson@23449
   451
      done
paulson@23449
   452
    finally show "h : f *o O(g)".
paulson@23449
   453
  qed
paulson@23449
   454
qed
paulson@23449
   455
blanchet@38991
   456
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult6" ]]
paulson@23449
   457
lemma bigo_mult6: "ALL x. f x ~= 0 ==>
huffman@45504
   458
    O(f * g) = (f::'a => ('b::{linordered_field,number_ring})) *o O(g)"
paulson@23449
   459
by (metis bigo_mult2 bigo_mult5 order_antisym)
paulson@23449
   460
paulson@23449
   461
(*proof requires relaxing relevance: 2007-01-25*)
blanchet@38991
   462
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult7" ]]
paulson@23449
   463
  declare bigo_mult6 [simp]
paulson@23449
   464
lemma bigo_mult7: "ALL x. f x ~= 0 ==>
huffman@45504
   465
    O(f * g) <= O(f::'a => ('b::{linordered_field,number_ring})) \<otimes> O(g)"
paulson@23449
   466
(*sledgehammer*)
paulson@23449
   467
  apply (subst bigo_mult6)
paulson@23449
   468
  apply assumption
blanchet@43197
   469
  apply (rule set_times_mono3)
paulson@23449
   470
  apply (rule bigo_refl)
paulson@23449
   471
done
paulson@23449
   472
  declare bigo_mult6 [simp del]
paulson@23449
   473
blanchet@38991
   474
declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult8" ]]
paulson@23449
   475
  declare bigo_mult7[intro!]
paulson@23449
   476
lemma bigo_mult8: "ALL x. f x ~= 0 ==>
huffman@45504
   477
    O(f * g) = O(f::'a => ('b::{linordered_field,number_ring})) \<otimes> O(g)"
paulson@23449
   478
by (metis bigo_mult bigo_mult7 order_antisym_conv)
paulson@23449
   479
paulson@23449
   480
lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
berghofe@26814
   481
  by (auto simp add: bigo_def fun_Compl_def)
paulson@23449
   482
paulson@23449
   483
lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
paulson@23449
   484
  apply (rule set_minus_imp_plus)
paulson@23449
   485
  apply (drule set_plus_imp_minus)
paulson@23449
   486
  apply (drule bigo_minus)
paulson@23449
   487
  apply (simp add: diff_minus)
paulson@23449
   488
done
paulson@23449
   489
paulson@23449
   490
lemma bigo_minus3: "O(-f) = O(f)"
berghofe@26814
   491
  by (auto simp add: bigo_def fun_Compl_def abs_minus_cancel)
paulson@23449
   492
paulson@23449
   493
lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
paulson@23449
   494
proof -
paulson@23449
   495
  assume a: "f : O(g)"
paulson@23449
   496
  show "f +o O(g) <= O(g)"
paulson@23449
   497
  proof -
paulson@23449
   498
    have "f : O(f)" by auto
berghofe@26814
   499
    then have "f +o O(g) <= O(f) \<oplus> O(g)"
paulson@23449
   500
      by (auto del: subsetI)
berghofe@26814
   501
    also have "... <= O(g) \<oplus> O(g)"
paulson@23449
   502
    proof -
paulson@23449
   503
      from a have "O(f) <= O(g)" by (auto del: subsetI)
paulson@23449
   504
      thus ?thesis by (auto del: subsetI)
paulson@23449
   505
    qed
paulson@23449
   506
    also have "... <= O(g)" by (simp add: bigo_plus_idemp)
paulson@23449
   507
    finally show ?thesis .
paulson@23449
   508
  qed
paulson@23449
   509
qed
paulson@23449
   510
paulson@23449
   511
lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
paulson@23449
   512
proof -
paulson@23449
   513
  assume a: "f : O(g)"
paulson@23449
   514
  show "O(g) <= f +o O(g)"
paulson@23449
   515
  proof -
paulson@23449
   516
    from a have "-f : O(g)" by auto
paulson@23449
   517
    then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
paulson@23449
   518
    then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
paulson@23449
   519
    also have "f +o (-f +o O(g)) = O(g)"
paulson@23449
   520
      by (simp add: set_plus_rearranges)
paulson@23449
   521
    finally show ?thesis .
paulson@23449
   522
  qed
paulson@23449
   523
qed
paulson@23449
   524
blanchet@38991
   525
declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_absorb" ]]
paulson@23449
   526
lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
paulson@41865
   527
by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff)
paulson@23449
   528
paulson@23449
   529
lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
paulson@23449
   530
  apply (subgoal_tac "f +o A <= f +o O(g)")
paulson@23449
   531
  apply force+
paulson@23449
   532
done
paulson@23449
   533
paulson@23449
   534
lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
paulson@23449
   535
  apply (subst set_minus_plus [symmetric])
paulson@23449
   536
  apply (subgoal_tac "g - f = - (f - g)")
paulson@23449
   537
  apply (erule ssubst)
paulson@23449
   538
  apply (rule bigo_minus)
paulson@23449
   539
  apply (subst set_minus_plus)
paulson@23449
   540
  apply assumption
paulson@23449
   541
  apply  (simp add: diff_minus add_ac)
paulson@23449
   542
done
paulson@23449
   543
paulson@23449
   544
lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
paulson@23449
   545
  apply (rule iffI)
paulson@23449
   546
  apply (erule bigo_add_commute_imp)+
paulson@23449
   547
done
paulson@23449
   548
paulson@23449
   549
lemma bigo_const1: "(%x. c) : O(%x. 1)"
paulson@23449
   550
by (auto simp add: bigo_def mult_ac)
paulson@23449
   551
blanchet@38991
   552
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const2" ]]
paulson@23449
   553
lemma (*bigo_const2 [intro]:*) "O(%x. c) <= O(%x. 1)"
paulson@41865
   554
by (metis bigo_const1 bigo_elt_subset)
paulson@23449
   555
huffman@45504
   556
lemma bigo_const2 [intro]: "O(%x. c::'b::{linordered_idom,number_ring}) <= O(%x. 1)"
blanchet@36561
   557
(* "thus" had to be replaced by "show" with an explicit reference to "F1" *)
blanchet@36561
   558
proof -
blanchet@36561
   559
  have F1: "\<forall>u. (\<lambda>Q. u) \<in> O(\<lambda>Q. 1)" by (metis bigo_const1)
blanchet@36561
   560
  show "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)" by (metis F1 bigo_elt_subset)
paulson@23449
   561
qed
paulson@23449
   562
blanchet@38991
   563
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const3" ]]
huffman@45504
   564
lemma bigo_const3: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> (%x. 1) : O(%x. c)"
paulson@23449
   565
apply (simp add: bigo_def)
blanchet@36561
   566
by (metis abs_eq_0 left_inverse order_refl)
paulson@23449
   567
huffman@45504
   568
lemma bigo_const4: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> O(%x. 1) <= O(%x. c)"
paulson@23449
   569
by (rule bigo_elt_subset, rule bigo_const3, assumption)
paulson@23449
   570
huffman@45504
   571
lemma bigo_const [simp]: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
paulson@23449
   572
    O(%x. c) = O(%x. 1)"
paulson@23449
   573
by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
paulson@23449
   574
blanchet@38991
   575
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult1" ]]
paulson@23449
   576
lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
paulson@24937
   577
  apply (simp add: bigo_def abs_mult)
blanchet@36561
   578
by (metis le_less)
paulson@23449
   579
paulson@23449
   580
lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
paulson@23449
   581
by (rule bigo_elt_subset, rule bigo_const_mult1)
paulson@23449
   582
blanchet@38991
   583
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult3" ]]
huffman@45504
   584
lemma bigo_const_mult3: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> f : O(%x. c * f x)"
paulson@23449
   585
  apply (simp add: bigo_def)
blanchet@36561
   586
(*sledgehammer [no luck]*)
paulson@23449
   587
  apply (rule_tac x = "abs(inverse c)" in exI)
paulson@23449
   588
  apply (simp only: abs_mult [symmetric] mult_assoc [symmetric])
blanchet@43197
   589
apply (subst left_inverse)
paulson@41865
   590
apply (auto )
paulson@23449
   591
done
paulson@23449
   592
huffman@45504
   593
lemma bigo_const_mult4: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
paulson@23449
   594
    O(f) <= O(%x. c * f x)"
paulson@23449
   595
by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
paulson@23449
   596
huffman@45504
   597
lemma bigo_const_mult [simp]: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
paulson@23449
   598
    O(%x. c * f x) = O(f)"
paulson@23449
   599
by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
paulson@23449
   600
blanchet@38991
   601
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult5" ]]
huffman@45504
   602
lemma bigo_const_mult5 [simp]: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
paulson@23449
   603
    (%x. c) *o O(f) = O(f)"
paulson@23449
   604
  apply (auto del: subsetI)
paulson@23449
   605
  apply (rule order_trans)
paulson@23449
   606
  apply (rule bigo_mult2)
paulson@23449
   607
  apply (simp add: func_times)
paulson@23449
   608
  apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times)
paulson@23449
   609
  apply (rule_tac x = "%y. inverse c * x y" in exI)
blanchet@43197
   610
  apply (rename_tac g d)
paulson@24942
   611
  apply safe
blanchet@43197
   612
  apply (rule_tac [2] ext)
blanchet@43197
   613
   prefer 2
haftmann@26041
   614
   apply simp
paulson@24942
   615
  apply (simp add: mult_assoc [symmetric] abs_mult)
blanchet@39259
   616
  (* couldn't get this proof without the step above *)
blanchet@39259
   617
proof -
blanchet@39259
   618
  fix g :: "'b \<Rightarrow> 'a" and d :: 'a
blanchet@39259
   619
  assume A1: "c \<noteq> (0\<Colon>'a)"
blanchet@39259
   620
  assume A2: "\<forall>x\<Colon>'b. \<bar>g x\<bar> \<le> d * \<bar>f x\<bar>"
blanchet@39259
   621
  have F1: "inverse \<bar>c\<bar> = \<bar>inverse c\<bar>" using A1 by (metis nonzero_abs_inverse)
blanchet@39259
   622
  have F2: "(0\<Colon>'a) < \<bar>c\<bar>" using A1 by (metis zero_less_abs_iff)
blanchet@39259
   623
  have "(0\<Colon>'a) < \<bar>c\<bar> \<longrightarrow> (0\<Colon>'a) < \<bar>inverse c\<bar>" using F1 by (metis positive_imp_inverse_positive)
blanchet@39259
   624
  hence "(0\<Colon>'a) < \<bar>inverse c\<bar>" using F2 by metis
blanchet@39259
   625
  hence F3: "(0\<Colon>'a) \<le> \<bar>inverse c\<bar>" by (metis order_le_less)
blanchet@39259
   626
  have "\<exists>(u\<Colon>'a) SKF\<^isub>7\<Colon>'a \<Rightarrow> 'b. \<bar>g (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar>"
blanchet@39259
   627
    using A2 by metis
blanchet@39259
   628
  hence F4: "\<exists>(u\<Colon>'a) SKF\<^isub>7\<Colon>'a \<Rightarrow> 'b. \<bar>g (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^isub>7 (\<bar>inverse c\<bar> * u))\<bar> \<and> (0\<Colon>'a) \<le> \<bar>inverse c\<bar>"
blanchet@39259
   629
    using F3 by metis
blanchet@39259
   630
  hence "\<exists>(v\<Colon>'a) (u\<Colon>'a) SKF\<^isub>7\<Colon>'a \<Rightarrow> 'b. \<bar>inverse c\<bar> * \<bar>g (SKF\<^isub>7 (u * v))\<bar> \<le> u * (v * \<bar>f (SKF\<^isub>7 (u * v))\<bar>)"
blanchet@39259
   631
    by (metis comm_mult_left_mono)
blanchet@39259
   632
  thus "\<exists>ca\<Colon>'a. \<forall>x\<Colon>'b. \<bar>inverse c\<bar> * \<bar>g x\<bar> \<le> ca * \<bar>f x\<bar>"
blanchet@39259
   633
    using A2 F4 by (metis ab_semigroup_mult_class.mult_ac(1) comm_mult_left_mono)
blanchet@39259
   634
qed
paulson@23449
   635
paulson@23449
   636
blanchet@38991
   637
declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult6" ]]
paulson@23449
   638
lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
paulson@23449
   639
  apply (auto intro!: subsetI
paulson@23449
   640
    simp add: bigo_def elt_set_times_def func_times
paulson@23449
   641
    simp del: abs_mult mult_ac)
paulson@41865
   642
(*sledgehammer*)
paulson@23449
   643
  apply (rule_tac x = "ca * (abs c)" in exI)
paulson@23449
   644
  apply (rule allI)
paulson@23449
   645
  apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
paulson@23449
   646
  apply (erule ssubst)
paulson@23449
   647
  apply (subst abs_mult)
paulson@23449
   648
  apply (rule mult_left_mono)
paulson@23449
   649
  apply (erule spec)
paulson@23449
   650
  apply simp
paulson@23449
   651
  apply(simp add: mult_ac)
paulson@23449
   652
done
paulson@23449
   653
paulson@23449
   654
lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
paulson@23449
   655
proof -
paulson@23449
   656
  assume "f =o O(g)"
paulson@23449
   657
  then have "(%x. c) * f =o (%x. c) *o O(g)"
paulson@23449
   658
    by auto
paulson@23449
   659
  also have "(%x. c) * f = (%x. c * f x)"
paulson@23449
   660
    by (simp add: func_times)
paulson@23449
   661
  also have "(%x. c) *o O(g) <= O(g)"
paulson@23449
   662
    by (auto del: subsetI)
paulson@23449
   663
  finally show ?thesis .
paulson@23449
   664
qed
paulson@23449
   665
paulson@23449
   666
lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
paulson@23449
   667
by (unfold bigo_def, auto)
paulson@23449
   668
blanchet@43197
   669
lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o
paulson@23449
   670
    O(%x. h(k x))"
berghofe@26814
   671
  apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def
paulson@23449
   672
      func_plus)
paulson@23449
   673
  apply (erule bigo_compose1)
paulson@23449
   674
done
paulson@23449
   675
paulson@23449
   676
subsection {* Setsum *}
paulson@23449
   677
blanchet@43197
   678
lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==>
paulson@23449
   679
    EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
blanchet@43197
   680
      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
paulson@23449
   681
  apply (auto simp add: bigo_def)
paulson@23449
   682
  apply (rule_tac x = "abs c" in exI)
paulson@23449
   683
  apply (subst abs_of_nonneg) back back
paulson@23449
   684
  apply (rule setsum_nonneg)
paulson@23449
   685
  apply force
paulson@23449
   686
  apply (subst setsum_right_distrib)
paulson@23449
   687
  apply (rule allI)
paulson@23449
   688
  apply (rule order_trans)
paulson@23449
   689
  apply (rule setsum_abs)
paulson@23449
   690
  apply (rule setsum_mono)
blanchet@43197
   691
apply (blast intro: order_trans mult_right_mono abs_ge_self)
paulson@23449
   692
done
paulson@23449
   693
blanchet@38991
   694
declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum1" ]]
blanchet@43197
   695
lemma bigo_setsum1: "ALL x y. 0 <= h x y ==>
paulson@23449
   696
    EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
paulson@23449
   697
      (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
paulson@23449
   698
  apply (rule bigo_setsum_main)
paulson@41865
   699
(*sledgehammer*)
paulson@23449
   700
  apply force
paulson@23449
   701
  apply clarsimp
paulson@23449
   702
  apply (rule_tac x = c in exI)
paulson@23449
   703
  apply force
paulson@23449
   704
done
paulson@23449
   705
blanchet@43197
   706
lemma bigo_setsum2: "ALL y. 0 <= h y ==>
paulson@23449
   707
    EX c. ALL y. abs(f y) <= c * (h y) ==>
paulson@23449
   708
      (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
blanchet@43197
   709
by (rule bigo_setsum1, auto)
paulson@23449
   710
blanchet@38991
   711
declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum3" ]]
paulson@23449
   712
lemma bigo_setsum3: "f =o O(h) ==>
paulson@23449
   713
    (%x. SUM y : A x. (l x y) * f(k x y)) =o
paulson@23449
   714
      O(%x. SUM y : A x. abs(l x y * h(k x y)))"
paulson@23449
   715
  apply (rule bigo_setsum1)
paulson@23449
   716
  apply (rule allI)+
paulson@23449
   717
  apply (rule abs_ge_zero)
paulson@23449
   718
  apply (unfold bigo_def)
paulson@41865
   719
  apply (auto simp add: abs_mult)
paulson@41865
   720
(*sledgehammer*)
paulson@23449
   721
  apply (rule_tac x = c in exI)
paulson@23449
   722
  apply (rule allI)+
paulson@23449
   723
  apply (subst mult_left_commute)
paulson@23449
   724
  apply (rule mult_left_mono)
paulson@23449
   725
  apply (erule spec)
paulson@23449
   726
  apply (rule abs_ge_zero)
paulson@23449
   727
done
paulson@23449
   728
paulson@23449
   729
lemma bigo_setsum4: "f =o g +o O(h) ==>
paulson@23449
   730
    (%x. SUM y : A x. l x y * f(k x y)) =o
paulson@23449
   731
      (%x. SUM y : A x. l x y * g(k x y)) +o
paulson@23449
   732
        O(%x. SUM y : A x. abs(l x y * h(k x y)))"
paulson@23449
   733
  apply (rule set_minus_imp_plus)
berghofe@26814
   734
  apply (subst fun_diff_def)
paulson@23449
   735
  apply (subst setsum_subtractf [symmetric])
paulson@23449
   736
  apply (subst right_diff_distrib [symmetric])
paulson@23449
   737
  apply (rule bigo_setsum3)
berghofe@26814
   738
  apply (subst fun_diff_def [symmetric])
paulson@23449
   739
  apply (erule set_plus_imp_minus)
paulson@23449
   740
done
paulson@23449
   741
blanchet@38991
   742
declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum5" ]]
blanchet@43197
   743
lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==>
paulson@23449
   744
    ALL x. 0 <= h x ==>
paulson@23449
   745
      (%x. SUM y : A x. (l x y) * f(k x y)) =o
blanchet@43197
   746
        O(%x. SUM y : A x. (l x y) * h(k x y))"
blanchet@43197
   747
  apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) =
paulson@23449
   748
      (%x. SUM y : A x. abs((l x y) * h(k x y)))")
paulson@23449
   749
  apply (erule ssubst)
paulson@23449
   750
  apply (erule bigo_setsum3)
paulson@23449
   751
  apply (rule ext)
paulson@23449
   752
  apply (rule setsum_cong2)
blanchet@43197
   753
  apply (thin_tac "f \<in> O(h)")
paulson@24942
   754
apply (metis abs_of_nonneg zero_le_mult_iff)
paulson@23449
   755
done
paulson@23449
   756
paulson@23449
   757
lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
paulson@23449
   758
    ALL x. 0 <= h x ==>
paulson@23449
   759
      (%x. SUM y : A x. (l x y) * f(k x y)) =o
paulson@23449
   760
        (%x. SUM y : A x. (l x y) * g(k x y)) +o
blanchet@43197
   761
          O(%x. SUM y : A x. (l x y) * h(k x y))"
paulson@23449
   762
  apply (rule set_minus_imp_plus)
berghofe@26814
   763
  apply (subst fun_diff_def)
paulson@23449
   764
  apply (subst setsum_subtractf [symmetric])
paulson@23449
   765
  apply (subst right_diff_distrib [symmetric])
paulson@23449
   766
  apply (rule bigo_setsum5)
berghofe@26814
   767
  apply (subst fun_diff_def [symmetric])
paulson@23449
   768
  apply (drule set_plus_imp_minus)
paulson@23449
   769
  apply auto
paulson@23449
   770
done
paulson@23449
   771
paulson@23449
   772
subsection {* Misc useful stuff *}
paulson@23449
   773
paulson@23449
   774
lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
berghofe@26814
   775
  A \<oplus> B <= O(f)"
paulson@23449
   776
  apply (subst bigo_plus_idemp [symmetric])
paulson@23449
   777
  apply (rule set_plus_mono2)
paulson@23449
   778
  apply assumption+
paulson@23449
   779
done
paulson@23449
   780
paulson@23449
   781
lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
paulson@23449
   782
  apply (subst bigo_plus_idemp [symmetric])
paulson@23449
   783
  apply (rule set_plus_intro)
paulson@23449
   784
  apply assumption+
paulson@23449
   785
done
blanchet@43197
   786
huffman@45504
   787
lemma bigo_useful_const_mult: "(c::'a::{linordered_field,number_ring}) ~= 0 ==>
paulson@23449
   788
    (%x. c) * f =o O(h) ==> f =o O(h)"
paulson@23449
   789
  apply (rule subsetD)
paulson@23449
   790
  apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
paulson@23449
   791
  apply assumption
paulson@23449
   792
  apply (rule bigo_const_mult6)
paulson@23449
   793
  apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
paulson@23449
   794
  apply (erule ssubst)
paulson@23449
   795
  apply (erule set_times_intro2)
blanchet@43197
   796
  apply (simp add: func_times)
paulson@23449
   797
done
paulson@23449
   798
blanchet@38991
   799
declare [[ sledgehammer_problem_prefix = "BigO__bigo_fix" ]]
paulson@23449
   800
lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
paulson@23449
   801
    f =o O(h)"
paulson@23449
   802
  apply (simp add: bigo_alt_def)
paulson@41865
   803
(*sledgehammer*)
paulson@23449
   804
  apply clarify
paulson@23449
   805
  apply (rule_tac x = c in exI)
paulson@23449
   806
  apply safe
paulson@23449
   807
  apply (case_tac "x = 0")
blanchet@43197
   808
apply (metis abs_ge_zero  abs_zero  order_less_le  split_mult_pos_le)
paulson@23449
   809
  apply (subgoal_tac "x = Suc (x - 1)")
paulson@23816
   810
  apply metis
paulson@23449
   811
  apply simp
paulson@23449
   812
  done
paulson@23449
   813
paulson@23449
   814
blanchet@43197
   815
lemma bigo_fix2:
blanchet@43197
   816
    "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==>
paulson@23449
   817
       f 0 = g 0 ==> f =o g +o O(h)"
paulson@23449
   818
  apply (rule set_minus_imp_plus)
paulson@23449
   819
  apply (rule bigo_fix)
berghofe@26814
   820
  apply (subst fun_diff_def)
berghofe@26814
   821
  apply (subst fun_diff_def [symmetric])
paulson@23449
   822
  apply (rule set_plus_imp_minus)
paulson@23449
   823
  apply simp
berghofe@26814
   824
  apply (simp add: fun_diff_def)
paulson@23449
   825
done
paulson@23449
   826
paulson@23449
   827
subsection {* Less than or equal to *}
paulson@23449
   828
haftmann@35416
   829
definition lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where
paulson@23449
   830
  "f <o g == (%x. max (f x - g x) 0)"
paulson@23449
   831
paulson@23449
   832
lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
paulson@23449
   833
    g =o O(h)"
paulson@23449
   834
  apply (unfold bigo_def)
paulson@23449
   835
  apply clarsimp
blanchet@43197
   836
apply (blast intro: order_trans)
paulson@23449
   837
done
paulson@23449
   838
paulson@23449
   839
lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
paulson@23449
   840
      g =o O(h)"
paulson@23449
   841
  apply (erule bigo_lesseq1)
blanchet@43197
   842
apply (blast intro: abs_ge_self order_trans)
paulson@23449
   843
done
paulson@23449
   844
paulson@23449
   845
lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
paulson@23449
   846
      g =o O(h)"
paulson@23449
   847
  apply (erule bigo_lesseq2)
paulson@23449
   848
  apply (rule allI)
paulson@23449
   849
  apply (subst abs_of_nonneg)
paulson@23449
   850
  apply (erule spec)+
paulson@23449
   851
done
paulson@23449
   852
paulson@23449
   853
lemma bigo_lesseq4: "f =o O(h) ==>
paulson@23449
   854
    ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
paulson@23449
   855
      g =o O(h)"
paulson@23449
   856
  apply (erule bigo_lesseq1)
paulson@23449
   857
  apply (rule allI)
paulson@23449
   858
  apply (subst abs_of_nonneg)
paulson@23449
   859
  apply (erule spec)+
paulson@23449
   860
done
paulson@23449
   861
blanchet@38991
   862
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso1" ]]
paulson@23449
   863
lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
blanchet@36561
   864
apply (unfold lesso_def)
blanchet@36561
   865
apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
blanchet@36561
   866
proof -
blanchet@36561
   867
  assume "(\<lambda>x. max (f x - g x) 0) = 0"
blanchet@36561
   868
  thus "(\<lambda>x. max (f x - g x) 0) \<in> O(h)" by (metis bigo_zero)
blanchet@36561
   869
next
blanchet@36561
   870
  show "\<forall>x\<Colon>'a. f x \<le> g x \<Longrightarrow> (\<lambda>x\<Colon>'a. max (f x - g x) (0\<Colon>'b)) = (0\<Colon>'a \<Rightarrow> 'b)"
paulson@23449
   871
  apply (unfold func_zero)
paulson@23449
   872
  apply (rule ext)
blanchet@36561
   873
  by (simp split: split_max)
blanchet@36561
   874
qed
paulson@23449
   875
blanchet@38991
   876
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso2" ]]
paulson@23449
   877
lemma bigo_lesso2: "f =o g +o O(h) ==>
paulson@23449
   878
    ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
paulson@23449
   879
      k <o g =o O(h)"
paulson@23449
   880
  apply (unfold lesso_def)
paulson@23449
   881
  apply (rule bigo_lesseq4)
paulson@23449
   882
  apply (erule set_plus_imp_minus)
paulson@23449
   883
  apply (rule allI)
paulson@23449
   884
  apply (rule le_maxI2)
paulson@23449
   885
  apply (rule allI)
berghofe@26814
   886
  apply (subst fun_diff_def)
paulson@23449
   887
apply (erule thin_rl)
paulson@41865
   888
(*sledgehammer*)
paulson@23449
   889
  apply (case_tac "0 <= k x - g x")
blanchet@36561
   890
(* apply (metis abs_le_iff add_le_imp_le_right diff_minus le_less
blanchet@36561
   891
                le_max_iff_disj min_max.le_supE min_max.sup_absorb2
blanchet@36561
   892
                min_max.sup_commute) *)
blanchet@37320
   893
proof -
blanchet@37320
   894
  fix x :: 'a
blanchet@37320
   895
  assume "\<forall>x\<Colon>'a. k x \<le> f x"
blanchet@37320
   896
  hence F1: "\<forall>x\<^isub>1\<Colon>'a. max (k x\<^isub>1) (f x\<^isub>1) = f x\<^isub>1" by (metis min_max.sup_absorb2)
blanchet@37320
   897
  assume "(0\<Colon>'b) \<le> k x - g x"
blanchet@37320
   898
  hence F2: "max (0\<Colon>'b) (k x - g x) = k x - g x" by (metis min_max.sup_absorb2)
blanchet@37320
   899
  have F3: "\<forall>x\<^isub>1\<Colon>'b. x\<^isub>1 \<le> \<bar>x\<^isub>1\<bar>" by (metis abs_le_iff le_less)
blanchet@37320
   900
  have "\<forall>(x\<^isub>2\<Colon>'b) x\<^isub>1\<Colon>'b. max x\<^isub>1 x\<^isub>2 \<le> x\<^isub>2 \<or> max x\<^isub>1 x\<^isub>2 \<le> x\<^isub>1" by (metis le_less le_max_iff_disj)
blanchet@37320
   901
  hence "\<forall>(x\<^isub>3\<Colon>'b) (x\<^isub>2\<Colon>'b) x\<^isub>1\<Colon>'b. x\<^isub>1 - x\<^isub>2 \<le> x\<^isub>3 - x\<^isub>2 \<or> x\<^isub>3 \<le> x\<^isub>1" by (metis add_le_imp_le_right diff_minus min_max.le_supE)
blanchet@37320
   902
  hence "k x - g x \<le> f x - g x" by (metis F1 le_less min_max.sup_absorb2 min_max.sup_commute)
blanchet@37320
   903
  hence "k x - g x \<le> \<bar>f x - g x\<bar>" by (metis F3 le_max_iff_disj min_max.sup_absorb2)
blanchet@37320
   904
  thus "max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>" by (metis F2 min_max.sup_commute)
blanchet@36561
   905
next
blanchet@36561
   906
  show "\<And>x\<Colon>'a.
blanchet@36561
   907
       \<lbrakk>\<forall>x\<Colon>'a. (0\<Colon>'b) \<le> k x; \<forall>x\<Colon>'a. k x \<le> f x; \<not> (0\<Colon>'b) \<le> k x - g x\<rbrakk>
blanchet@36561
   908
       \<Longrightarrow> max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>"
blanchet@36561
   909
    by (metis abs_ge_zero le_cases min_max.sup_absorb2)
paulson@24545
   910
qed
paulson@23449
   911
blanchet@38991
   912
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3" ]]
paulson@23449
   913
lemma bigo_lesso3: "f =o g +o O(h) ==>
paulson@23449
   914
    ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
paulson@23449
   915
      f <o k =o O(h)"
paulson@23449
   916
  apply (unfold lesso_def)
paulson@23449
   917
  apply (rule bigo_lesseq4)
paulson@23449
   918
  apply (erule set_plus_imp_minus)
paulson@23449
   919
  apply (rule allI)
paulson@23449
   920
  apply (rule le_maxI2)
paulson@23449
   921
  apply (rule allI)
berghofe@26814
   922
  apply (subst fun_diff_def)
blanchet@43197
   923
apply (erule thin_rl)
paulson@41865
   924
(*sledgehammer*)
paulson@23449
   925
  apply (case_tac "0 <= f x - k x")
nipkow@29667
   926
  apply (simp)
paulson@23449
   927
  apply (subst abs_of_nonneg)
paulson@23449
   928
  apply (drule_tac x = x in spec) back
blanchet@38991
   929
using [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3_simpler" ]]
paulson@24545
   930
apply (metis diff_less_0_iff_less linorder_not_le not_leE uminus_add_conv_diff xt1(12) xt1(6))
paulson@24545
   931
apply (metis add_minus_cancel diff_le_eq le_diff_eq uminus_add_conv_diff)
haftmann@29511
   932
apply (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute)
paulson@23449
   933
done
paulson@23449
   934
huffman@45504
   935
lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::{linordered_field,number_ring}) ==>
paulson@23449
   936
    g =o h +o O(k) ==> f <o h =o O(k)"
paulson@23449
   937
  apply (unfold lesso_def)
paulson@23449
   938
  apply (drule set_plus_imp_minus)
paulson@23449
   939
  apply (drule bigo_abs5) back
berghofe@26814
   940
  apply (simp add: fun_diff_def)
paulson@23449
   941
  apply (drule bigo_useful_add)
paulson@23449
   942
  apply assumption
paulson@23449
   943
  apply (erule bigo_lesseq2) back
paulson@23449
   944
  apply (rule allI)
nipkow@29667
   945
  apply (auto simp add: func_plus fun_diff_def algebra_simps
paulson@23449
   946
    split: split_max abs_split)
paulson@23449
   947
done
paulson@23449
   948
blanchet@38991
   949
declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso5" ]]
paulson@23449
   950
lemma bigo_lesso5: "f <o g =o O(h) ==>
paulson@23449
   951
    EX C. ALL x. f x <= g x + C * abs(h x)"
paulson@23449
   952
  apply (simp only: lesso_def bigo_alt_def)
paulson@23449
   953
  apply clarsimp
blanchet@43197
   954
  apply (metis abs_if abs_mult add_commute diff_le_eq less_not_permute)
paulson@23449
   955
done
paulson@23449
   956
paulson@23449
   957
end