session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
(* Title: HOL/Metis_Examples/Big_O.thy
Author: Lawrence C. Paulson, Cambridge University Computer Laboratory
Author: Jasmin Blanchette, TU Muenchen
Metis example featuring the Big O notation.
*)
section \<open>Metis Example Featuring the Big O Notation\<close>
theory Big_O
imports
"HOL-Decision_Procs.Dense_Linear_Order"
"HOL-Library.Function_Algebras"
"HOL-Library.Set_Algebras"
begin
subsection \<open>Definitions\<close>
definition bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set" ("(1O'(_'))") where
"O(f::('a => 'b)) == {h. \<exists>c. \<forall>x. \<bar>h x\<bar> <= c * \<bar>f x\<bar>}"
lemma bigo_pos_const:
"(\<exists>c::'a::linordered_idom.
\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
by (metis (no_types) abs_ge_zero
algebra_simps mult.comm_neutral
mult_nonpos_nonneg not_le_imp_less order_trans zero_less_one)
(*** Now various verions with an increasing shrink factor ***)
sledgehammer_params [isar_proofs, compress = 1]
lemma
"(\<exists>c::'a::linordered_idom.
\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
apply auto
apply (case_tac "c = 0", simp)
apply (rule_tac x = "1" in exI, simp)
apply (rule_tac x = "\<bar>c\<bar>" in exI, auto)
proof -
fix c :: 'a and x :: 'b
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
have F1: "\<forall>x\<^sub>1::'a::linordered_idom. 0 \<le> \<bar>x\<^sub>1\<bar>" by (metis abs_ge_zero)
have F2: "\<forall>x\<^sub>1::'a::linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1)
have F3: "\<forall>x\<^sub>1 x\<^sub>3. x\<^sub>3 \<le> \<bar>h x\<^sub>1\<bar> \<longrightarrow> x\<^sub>3 \<le> c * \<bar>f x\<^sub>1\<bar>" by (metis A1 order_trans)
have F4: "\<forall>x\<^sub>2 x\<^sub>3::'a::linordered_idom. \<bar>x\<^sub>3\<bar> * \<bar>x\<^sub>2\<bar> = \<bar>x\<^sub>3 * x\<^sub>2\<bar>"
by (metis abs_mult)
have F5: "\<forall>x\<^sub>3 x\<^sub>1::'a::linordered_idom. 0 \<le> x\<^sub>1 \<longrightarrow> \<bar>x\<^sub>3 * x\<^sub>1\<bar> = \<bar>x\<^sub>3\<bar> * x\<^sub>1"
by (metis abs_mult_pos)
hence "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1::'a::linordered_idom\<bar> = \<bar>1\<bar> * x\<^sub>1" by (metis F2)
hence "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1::'a::linordered_idom\<bar> = x\<^sub>1" by (metis F2 abs_one)
hence "\<forall>x\<^sub>3. 0 \<le> \<bar>h x\<^sub>3\<bar> \<longrightarrow> \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" by (metis F3)
hence "\<forall>x\<^sub>3. \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" by (metis F1)
hence "\<forall>x\<^sub>3. (0::'a) \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^sub>3\<bar>" by (metis F5)
hence "\<forall>x\<^sub>3. (0::'a) \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c * f x\<^sub>3\<bar>" by (metis F4)
hence "\<forall>x\<^sub>3. c * \<bar>f x\<^sub>3\<bar> = \<bar>c * f x\<^sub>3\<bar>" by (metis F1)
hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1)
thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F4)
qed
sledgehammer_params [isar_proofs, compress = 2]
lemma
"(\<exists>c::'a::linordered_idom.
\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
apply auto
apply (case_tac "c = 0", simp)
apply (rule_tac x = "1" in exI, simp)
apply (rule_tac x = "\<bar>c\<bar>" in exI, auto)
proof -
fix c :: 'a and x :: 'b
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
have F1: "\<forall>x\<^sub>1::'a::linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1)
have F2: "\<forall>x\<^sub>2 x\<^sub>3::'a::linordered_idom. \<bar>x\<^sub>3\<bar> * \<bar>x\<^sub>2\<bar> = \<bar>x\<^sub>3 * x\<^sub>2\<bar>"
by (metis abs_mult)
have "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1::'a::linordered_idom\<bar> = x\<^sub>1" by (metis F1 abs_mult_pos abs_one)
hence "\<forall>x\<^sub>3. \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" by (metis A1 abs_ge_zero order_trans)
hence "\<forall>x\<^sub>3. 0 \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c * f x\<^sub>3\<bar>" by (metis F2 abs_mult_pos)
hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero)
thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F2)
qed
sledgehammer_params [isar_proofs, compress = 3]
lemma
"(\<exists>c::'a::linordered_idom.
\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
apply auto
apply (case_tac "c = 0", simp)
apply (rule_tac x = "1" in exI, simp)
apply (rule_tac x = "\<bar>c\<bar>" in exI, auto)
proof -
fix c :: 'a and x :: 'b
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
have F1: "\<forall>x\<^sub>1::'a::linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1)
have F2: "\<forall>x\<^sub>3 x\<^sub>1::'a::linordered_idom. 0 \<le> x\<^sub>1 \<longrightarrow> \<bar>x\<^sub>3 * x\<^sub>1\<bar> = \<bar>x\<^sub>3\<bar> * x\<^sub>1" by (metis abs_mult_pos)
hence "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1::'a::linordered_idom\<bar> = x\<^sub>1" by (metis F1 abs_one)
hence "\<forall>x\<^sub>3. 0 \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^sub>3\<bar>" by (metis F2 A1 abs_ge_zero order_trans)
thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis A1 abs_ge_zero)
qed
sledgehammer_params [isar_proofs, compress = 4]
lemma
"(\<exists>c::'a::linordered_idom.
\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)
\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
apply auto
apply (case_tac "c = 0", simp)
apply (rule_tac x = "1" in exI, simp)
apply (rule_tac x = "\<bar>c\<bar>" in exI, auto)
proof -
fix c :: 'a and x :: 'b
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
have "\<forall>x\<^sub>1::'a::linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1)
hence "\<forall>x\<^sub>3. \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>"
by (metis A1 abs_ge_zero order_trans abs_mult_pos abs_one)
hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero abs_mult_pos abs_mult)
thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis abs_mult)
qed
sledgehammer_params [isar_proofs, compress = 1]
lemma bigo_alt_def: "O(f) = {h. \<exists>c. (0 < c \<and> (\<forall>x. \<bar>h x\<bar> <= c * \<bar>f x\<bar>))}"
by (auto simp add: bigo_def bigo_pos_const)
lemma bigo_elt_subset [intro]: "f : O(g) \<Longrightarrow> O(f) \<le> O(g)"
apply (auto simp add: bigo_alt_def)
apply (rule_tac x = "ca * c" in exI)
apply (metis algebra_simps mult_le_cancel_left_pos order_trans mult_pos_pos)
done
lemma bigo_refl [intro]: "f : O(f)"
unfolding bigo_def mem_Collect_eq
by (metis mult_1 order_refl)
lemma bigo_zero: "0 : O(g)"
apply (auto simp add: bigo_def func_zero)
by (metis mult_zero_left order_refl)
lemma bigo_zero2: "O(\<lambda>x. 0) = {\<lambda>x. 0}"
by (auto simp add: bigo_def)
lemma bigo_plus_self_subset [intro]:
"O(f) + O(f) <= O(f)"
apply (auto simp add: bigo_alt_def set_plus_def)
apply (rule_tac x = "c + ca" in exI)
apply auto
apply (simp add: ring_distribs func_plus)
by (metis order_trans abs_triangle_ineq add_mono)
lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
by (metis bigo_plus_self_subset bigo_zero set_eq_subset set_zero_plus2)
lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
apply (rule subsetI)
apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
apply (subst bigo_pos_const [symmetric])+
apply (rule_tac x = "\<lambda>n. if \<bar>g n\<bar> <= \<bar>f n\<bar> then x n else 0" in exI)
apply (rule conjI)
apply (rule_tac x = "c + c" in exI)
apply clarsimp
apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> <= (c + c) * \<bar>f xa\<bar>")
apply (metis mult_2 order_trans)
apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> <= c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
apply (erule order_trans)
apply (simp add: ring_distribs)
apply (rule mult_left_mono)
apply (simp add: abs_triangle_ineq)
apply (simp add: order_less_le)
apply (rule_tac x = "\<lambda>n. if \<bar>f n\<bar> < \<bar>g n\<bar> then x n else 0" in exI)
apply (rule conjI)
apply (rule_tac x = "c + c" in exI)
apply auto
apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> <= (c + c) * \<bar>g xa\<bar>")
apply (metis order_trans mult_2)
apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> <= c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
apply (erule order_trans)
apply (simp add: ring_distribs)
by (metis abs_triangle_ineq mult_le_cancel_left_pos)
lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A + B <= O(f)"
by (metis bigo_plus_idemp set_plus_mono2)
lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
apply (rule equalityI)
apply (rule bigo_plus_subset)
apply (simp add: bigo_alt_def set_plus_def func_plus)
apply clarify
(* sledgehammer *)
apply (rule_tac x = "max c ca" in exI)
apply (rule conjI)
apply (metis less_max_iff_disj)
apply clarify
apply (drule_tac x = "xa" in spec)+
apply (subgoal_tac "0 <= f xa + g xa")
apply (simp add: ring_distribs)
apply (subgoal_tac "\<bar>a xa + b xa\<bar> <= \<bar>a xa\<bar> + \<bar>b xa\<bar>")
apply (subgoal_tac "\<bar>a xa\<bar> + \<bar>b xa\<bar> <= max c ca * f xa + max c ca * g xa")
apply (metis order_trans)
defer 1
apply (metis abs_triangle_ineq)
apply (metis add_nonneg_nonneg)
apply (rule add_mono)
apply (metis max.cobounded2 linorder_linear max.absorb1 mult_right_mono xt1(6))
by (metis max.cobounded2 linorder_not_le mult_le_cancel_right order_trans)
lemma bigo_bounded_alt: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)"
apply (auto simp add: bigo_def)
(* Version 1: one-line proof *)
by (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult)
lemma "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)"
apply (auto simp add: bigo_def)
(* Version 2: structured proof *)
proof -
assume "\<forall>x. f x \<le> c * g x"
thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans)
qed
lemma bigo_bounded: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= g x \<Longrightarrow> f : O(g)"
apply (erule bigo_bounded_alt [of f 1 g])
by (metis mult_1)
lemma bigo_bounded2: "\<forall>x. lb x <= f x \<Longrightarrow> \<forall>x. f x <= lb x + g x \<Longrightarrow> f : lb +o O(g)"
apply (rule set_minus_imp_plus)
apply (rule bigo_bounded)
apply (metis add_le_cancel_left diff_add_cancel diff_self minus_apply
algebra_simps)
by (metis add_le_cancel_left diff_add_cancel func_plus le_fun_def
algebra_simps)
lemma bigo_abs: "(\<lambda>x. \<bar>f x\<bar>) =o O(f)"
apply (unfold bigo_def)
apply auto
by (metis mult_1 order_refl)
lemma bigo_abs2: "f =o O(\<lambda>x. \<bar>f x\<bar>)"
apply (unfold bigo_def)
apply auto
by (metis mult_1 order_refl)
lemma bigo_abs3: "O(f) = O(\<lambda>x. \<bar>f x\<bar>)"
proof -
have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset)
have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs)
have "\<forall>u. u \<in> O(\<lambda>R. \<bar>u R\<bar>)" by (metis bigo_abs2)
thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" using F1 F2 by auto
qed
lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o (\<lambda>x. \<bar>g x\<bar>) +o O(h)"
apply (drule set_plus_imp_minus)
apply (rule set_minus_imp_plus)
apply (subst fun_diff_def)
proof -
assume a: "f - g : O(h)"
have "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) =o O(\<lambda>x. \<bar>\<bar>f x\<bar> - \<bar>g x\<bar>\<bar>)"
by (rule bigo_abs2)
also have "... <= O(\<lambda>x. \<bar>f x - g x\<bar>)"
apply (rule bigo_elt_subset)
apply (rule bigo_bounded)
apply (metis abs_ge_zero)
by (metis abs_triangle_ineq3)
also have "... <= O(f - g)"
apply (rule bigo_elt_subset)
apply (subst fun_diff_def)
apply (rule bigo_abs)
done
also have "... <= O(h)"
using a by (rule bigo_elt_subset)
finally show "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) : O(h)" .
qed
lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o O(g)"
by (unfold bigo_def, auto)
lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) + O(h)"
proof -
assume "f : g +o O(h)"
also have "... <= O(g) + O(h)"
by (auto del: subsetI)
also have "... = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
by (metis bigo_abs3)
also have "... = O((\<lambda>x. \<bar>g x\<bar>) + (\<lambda>x. \<bar>h x\<bar>))"
by (rule bigo_plus_eq [symmetric], auto)
finally have "f : ...".
then have "O(f) <= ..."
by (elim bigo_elt_subset)
also have "... = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
by (rule bigo_plus_eq, auto)
finally show ?thesis
by (simp add: bigo_abs3 [symmetric])
qed
lemma bigo_mult [intro]: "O(f) * O(g) <= O(f * g)"
apply (rule subsetI)
apply (subst bigo_def)
apply (auto simp del: abs_mult ac_simps
simp add: bigo_alt_def set_times_def func_times)
(* sledgehammer *)
apply (rule_tac x = "c * ca" in exI)
apply (rule allI)
apply (erule_tac x = x in allE)+
apply (subgoal_tac "c * ca * \<bar>f x * g x\<bar> = (c * \<bar>f x\<bar>) * (ca * \<bar>g x\<bar>)")
apply (metis (no_types) abs_ge_zero abs_mult mult_mono')
by (metis mult.assoc mult.left_commute abs_of_pos mult.left_commute abs_mult)
lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
by (metis bigo_mult bigo_refl set_times_mono3 subset_trans)
lemma bigo_mult3: "f : O(h) \<Longrightarrow> g : O(j) \<Longrightarrow> f * g : O(h * j)"
by (metis bigo_mult set_rev_mp set_times_intro)
lemma bigo_mult4 [intro]:"f : k +o O(h) \<Longrightarrow> g * f : (g * k) +o O(g * h)"
by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib)
lemma bigo_mult5: "\<forall>x. f x ~= 0 \<Longrightarrow>
O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
proof -
assume a: "\<forall>x. f x ~= 0"
show "O(f * g) <= f *o O(g)"
proof
fix h
assume h: "h : O(f * g)"
then have "(\<lambda>x. 1 / (f x)) * h : (\<lambda>x. 1 / f x) *o O(f * g)"
by auto
also have "... <= O((\<lambda>x. 1 / f x) * (f * g))"
by (rule bigo_mult2)
also have "(\<lambda>x. 1 / f x) * (f * g) = g"
by (simp add: fun_eq_iff a)
finally have "(\<lambda>x. (1::'b) / f x) * h : O(g)".
then have "f * ((\<lambda>x. (1::'b) / f x) * h) : f *o O(g)"
by auto
also have "f * ((\<lambda>x. (1::'b) / f x) * h) = h"
by (simp add: func_times fun_eq_iff a)
finally show "h : f *o O(g)".
qed
qed
lemma bigo_mult6:
"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = (f::'a \<Rightarrow> ('b::linordered_field)) *o O(g)"
by (metis bigo_mult2 bigo_mult5 order_antisym)
(*proof requires relaxing relevance: 2007-01-25*)
declare bigo_mult6 [simp]
lemma bigo_mult7:
"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<le> O(f::'a \<Rightarrow> ('b::linordered_field)) * O(g)"
by (metis bigo_refl bigo_mult6 set_times_mono3)
declare bigo_mult6 [simp del]
declare bigo_mult7 [intro!]
lemma bigo_mult8:
"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f::'a \<Rightarrow> ('b::linordered_field)) * O(g)"
by (metis bigo_mult bigo_mult7 order_antisym_conv)
lemma bigo_minus [intro]: "f : O(g) \<Longrightarrow> - f : O(g)"
by (auto simp add: bigo_def fun_Compl_def)
lemma bigo_minus2: "f : g +o O(h) \<Longrightarrow> -f : -g +o O(h)"
by (metis (no_types, lifting) bigo_minus diff_minus_eq_add minus_add_distrib
minus_minus set_minus_imp_plus set_plus_imp_minus)
lemma bigo_minus3: "O(-f) = O(f)"
by (metis bigo_elt_subset bigo_minus bigo_refl equalityI minus_minus)
lemma bigo_plus_absorb_lemma1: "f : O(g) \<Longrightarrow> f +o O(g) \<le> O(g)"
by (metis bigo_plus_idemp set_plus_mono3)
lemma bigo_plus_absorb_lemma2: "f : O(g) \<Longrightarrow> O(g) \<le> f +o O(g)"
by (metis (no_types) bigo_minus bigo_plus_absorb_lemma1 right_minus
set_plus_mono set_plus_rearrange2 set_zero_plus subsetD subset_refl
subset_trans)
lemma bigo_plus_absorb [simp]: "f : O(g) \<Longrightarrow> f +o O(g) = O(g)"
by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff)
lemma bigo_plus_absorb2 [intro]: "f : O(g) \<Longrightarrow> A <= O(g) \<Longrightarrow> f +o A \<le> O(g)"
by (metis bigo_plus_absorb set_plus_mono)
lemma bigo_add_commute_imp: "f : g +o O(h) \<Longrightarrow> g : f +o O(h)"
by (metis bigo_minus minus_diff_eq set_plus_imp_minus set_minus_plus)
lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
by (metis bigo_add_commute_imp)
lemma bigo_const1: "(\<lambda>x. c) : O(\<lambda>x. 1)"
by (auto simp add: bigo_def ac_simps)
lemma bigo_const2 [intro]: "O(\<lambda>x. c) \<le> O(\<lambda>x. 1)"
by (metis bigo_const1 bigo_elt_subset)
lemma bigo_const3: "(c::'a::linordered_field) ~= 0 \<Longrightarrow> (\<lambda>x. 1) : O(\<lambda>x. c)"
apply (simp add: bigo_def)
by (metis abs_eq_0 left_inverse order_refl)
lemma bigo_const4: "(c::'a::linordered_field) ~= 0 \<Longrightarrow> O(\<lambda>x. 1) <= O(\<lambda>x. c)"
by (metis bigo_elt_subset bigo_const3)
lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 \<Longrightarrow>
O(\<lambda>x. c) = O(\<lambda>x. 1)"
by (metis bigo_const2 bigo_const4 equalityI)
lemma bigo_const_mult1: "(\<lambda>x. c * f x) : O(f)"
apply (simp add: bigo_def abs_mult)
by (metis le_less)
lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<le> O(f)"
by (rule bigo_elt_subset, rule bigo_const_mult1)
lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 \<Longrightarrow> f : O(\<lambda>x. c * f x)"
apply (simp add: bigo_def)
by (metis (no_types) abs_mult mult.assoc mult_1 order_refl left_inverse)
lemma bigo_const_mult4:
"(c::'a::linordered_field) \<noteq> 0 \<Longrightarrow> O(f) \<le> O(\<lambda>x. c * f x)"
by (metis bigo_elt_subset bigo_const_mult3)
lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 \<Longrightarrow>
O(\<lambda>x. c * f x) = O(f)"
by (metis equalityI bigo_const_mult2 bigo_const_mult4)
lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 \<Longrightarrow>
(\<lambda>x. c) *o O(f) = O(f)"
apply (auto del: subsetI)
apply (rule order_trans)
apply (rule bigo_mult2)
apply (simp add: func_times)
apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times)
apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
apply (rename_tac g d)
apply safe
apply (rule_tac [2] ext)
prefer 2
apply simp
apply (simp add: mult.assoc [symmetric] abs_mult)
(* couldn't get this proof without the step above *)
proof -
fix g :: "'b \<Rightarrow> 'a" and d :: 'a
assume A1: "c \<noteq> (0::'a)"
assume A2: "\<forall>x::'b. \<bar>g x\<bar> \<le> d * \<bar>f x\<bar>"
have F1: "inverse \<bar>c\<bar> = \<bar>inverse c\<bar>" using A1 by (metis nonzero_abs_inverse)
have F2: "(0::'a) < \<bar>c\<bar>" using A1 by (metis zero_less_abs_iff)
have "(0::'a) < \<bar>c\<bar> \<longrightarrow> (0::'a) < \<bar>inverse c\<bar>" using F1 by (metis positive_imp_inverse_positive)
hence "(0::'a) < \<bar>inverse c\<bar>" using F2 by metis
hence F3: "(0::'a) \<le> \<bar>inverse c\<bar>" by (metis order_le_less)
have "\<exists>(u::'a) SKF\<^sub>7::'a \<Rightarrow> 'b. \<bar>g (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar>"
using A2 by metis
hence F4: "\<exists>(u::'a) SKF\<^sub>7::'a \<Rightarrow> 'b. \<bar>g (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<and> (0::'a) \<le> \<bar>inverse c\<bar>"
using F3 by metis
hence "\<exists>(v::'a) (u::'a) SKF\<^sub>7::'a \<Rightarrow> 'b. \<bar>inverse c\<bar> * \<bar>g (SKF\<^sub>7 (u * v))\<bar> \<le> u * (v * \<bar>f (SKF\<^sub>7 (u * v))\<bar>)"
by (metis mult_left_mono)
then show "\<exists>ca::'a. \<forall>x::'b. inverse \<bar>c\<bar> * \<bar>g x\<bar> \<le> ca * \<bar>f x\<bar>"
using A2 F4 by (metis F1 \<open>0 < \<bar>inverse c\<bar>\<close> linordered_field_class.sign_simps(23) mult_le_cancel_left_pos)
qed
lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) <= O(f)"
apply (auto intro!: subsetI
simp add: bigo_def elt_set_times_def func_times
simp del: abs_mult ac_simps)
(* sledgehammer *)
apply (rule_tac x = "ca * \<bar>c\<bar>" in exI)
apply (rule allI)
apply (subgoal_tac "ca * \<bar>c\<bar> * \<bar>f x\<bar> = \<bar>c\<bar> * (ca * \<bar>f x\<bar>)")
apply (erule ssubst)
apply (subst abs_mult)
apply (rule mult_left_mono)
apply (erule spec)
apply simp
apply (simp add: ac_simps)
done
lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
by (metis bigo_const_mult1 bigo_elt_subset order_less_le psubsetD)
lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f(k x)) =o O(\<lambda>x. g(k x))"
by (unfold bigo_def, auto)
lemma bigo_compose2:
"f =o g +o O(h) \<Longrightarrow> (\<lambda>x. f(k x)) =o (\<lambda>x. g(k x)) +o O(\<lambda>x. h(k x))"
apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus)
apply (drule bigo_compose1 [of "f - g" h k])
apply (simp add: fun_diff_def)
done
subsection \<open>Sum\<close>
lemma bigo_sum_main: "\<forall>x. \<forall>y \<in> A x. 0 <= h x y \<Longrightarrow>
\<exists>c. \<forall>x. \<forall>y \<in> A x. \<bar>f x y\<bar> <= c * (h x y) \<Longrightarrow>
(\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
apply (auto simp add: bigo_def)
apply (rule_tac x = "\<bar>c\<bar>" in exI)
apply (subst abs_of_nonneg) back back
apply (rule sum_nonneg)
apply force
apply (subst sum_distrib_left)
apply (rule allI)
apply (rule order_trans)
apply (rule sum_abs)
apply (rule sum_mono)
by (metis abs_ge_self abs_mult_pos order_trans)
lemma bigo_sum1: "\<forall>x y. 0 <= h x y \<Longrightarrow>
\<exists>c. \<forall>x y. \<bar>f x y\<bar> <= c * (h x y) \<Longrightarrow>
(\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
by (metis (no_types) bigo_sum_main)
lemma bigo_sum2: "\<forall>y. 0 <= h y \<Longrightarrow>
\<exists>c. \<forall>y. \<bar>f y\<bar> <= c * (h y) \<Longrightarrow>
(\<lambda>x. \<Sum>y \<in> A x. f y) =o O(\<lambda>x. \<Sum>y \<in> A x. h y)"
apply (rule bigo_sum1)
by metis+
lemma bigo_sum3: "f =o O(h) \<Longrightarrow>
(\<lambda>x. \<Sum>y \<in> A x. (l x y) * f(k x y)) =o
O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h(k x y)\<bar>)"
apply (rule bigo_sum1)
apply (rule allI)+
apply (rule abs_ge_zero)
apply (unfold bigo_def)
apply (auto simp add: abs_mult)
by (metis abs_ge_zero mult.left_commute mult_left_mono)
lemma bigo_sum4: "f =o g +o O(h) \<Longrightarrow>
(\<lambda>x. \<Sum>y \<in> A x. l x y * f(k x y)) =o
(\<lambda>x. \<Sum>y \<in> A x. l x y * g(k x y)) +o
O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h(k x y)\<bar>)"
apply (rule set_minus_imp_plus)
apply (subst fun_diff_def)
apply (subst sum_subtractf [symmetric])
apply (subst right_diff_distrib [symmetric])
apply (rule bigo_sum3)
by (metis (lifting, no_types) fun_diff_def set_plus_imp_minus ext)
lemma bigo_sum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
\<forall>x. 0 <= h x \<Longrightarrow>
(\<lambda>x. \<Sum>y \<in> A x. (l x y) * f(k x y)) =o
O(\<lambda>x. \<Sum>y \<in> A x. (l x y) * h(k x y))"
apply (subgoal_tac "(\<lambda>x. \<Sum>y \<in> A x. (l x y) * h(k x y)) =
(\<lambda>x. \<Sum>y \<in> A x. \<bar>(l x y) * h(k x y)\<bar>)")
apply (erule ssubst)
apply (erule bigo_sum3)
apply (rule ext)
apply (rule sum.cong)
apply (rule refl)
by (metis abs_of_nonneg zero_le_mult_iff)
lemma bigo_sum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
\<forall>x. 0 <= h x \<Longrightarrow>
(\<lambda>x. \<Sum>y \<in> A x. (l x y) * f(k x y)) =o
(\<lambda>x. \<Sum>y \<in> A x. (l x y) * g(k x y)) +o
O(\<lambda>x. \<Sum>y \<in> A x. (l x y) * h(k x y))"
apply (rule set_minus_imp_plus)
apply (subst fun_diff_def)
apply (subst sum_subtractf [symmetric])
apply (subst right_diff_distrib [symmetric])
apply (rule bigo_sum5)
apply (subst fun_diff_def [symmetric])
apply (drule set_plus_imp_minus)
apply auto
done
subsection \<open>Misc useful stuff\<close>
lemma bigo_useful_intro: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow>
A + B <= O(f)"
apply (subst bigo_plus_idemp [symmetric])
apply (rule set_plus_mono2)
apply assumption+
done
lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
apply (subst bigo_plus_idemp [symmetric])
apply (rule set_plus_intro)
apply assumption+
done
lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 \<Longrightarrow>
(\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
apply (rule subsetD)
apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) <= O(h)")
apply assumption
apply (rule bigo_const_mult6)
apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
apply (erule ssubst)
apply (erule set_times_intro2)
apply (simp add: func_times)
done
lemma bigo_fix: "(\<lambda>x. f ((x::nat) + 1)) =o O(\<lambda>x. h(x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow>
f =o O(h)"
apply (simp add: bigo_alt_def)
by (metis abs_ge_zero abs_mult abs_of_pos abs_zero not0_implies_Suc)
lemma bigo_fix2:
"(\<lambda>x. f ((x::nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
apply (rule set_minus_imp_plus)
apply (rule bigo_fix)
apply (subst fun_diff_def)
apply (subst fun_diff_def [symmetric])
apply (rule set_plus_imp_minus)
apply simp
apply (simp add: fun_diff_def)
done
subsection \<open>Less than or equal to\<close>
definition lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where
"f <o g == (\<lambda>x. max (f x - g x) 0)"
lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> <= \<bar>f x\<bar> \<Longrightarrow>
g =o O(h)"
apply (unfold bigo_def)
apply clarsimp
apply (blast intro: order_trans)
done
lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> <= f x \<Longrightarrow>
g =o O(h)"
apply (erule bigo_lesseq1)
apply (blast intro: abs_ge_self order_trans)
done
lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= f x \<Longrightarrow>
g =o O(h)"
apply (erule bigo_lesseq2)
apply (rule allI)
apply (subst abs_of_nonneg)
apply (erule spec)+
done
lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
\<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= \<bar>f x\<bar> \<Longrightarrow>
g =o O(h)"
apply (erule bigo_lesseq1)
apply (rule allI)
apply (subst abs_of_nonneg)
apply (erule spec)+
done
lemma bigo_lesso1: "\<forall>x. f x <= g x \<Longrightarrow> f <o g =o O(h)"
apply (unfold lesso_def)
apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
apply (metis bigo_zero)
by (metis (lifting, no_types) func_zero le_fun_def le_iff_diff_le_0
max.absorb2 order_eq_iff)
lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow>
\<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. k x <= f x \<Longrightarrow>
k <o g =o O(h)"
apply (unfold lesso_def)
apply (rule bigo_lesseq4)
apply (erule set_plus_imp_minus)
apply (rule allI)
apply (rule max.cobounded2)
apply (rule allI)
apply (subst fun_diff_def)
apply (erule thin_rl)
(* sledgehammer *)
apply (case_tac "0 <= k x - g x")
apply (metis (lifting) abs_le_D1 linorder_linear min_diff_distrib_left
min.absorb1 min.absorb2 max.absorb1)
by (metis abs_ge_zero le_cases max.absorb2)
lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow>
\<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. g x <= k x \<Longrightarrow>
f <o k =o O(h)"
apply (unfold lesso_def)
apply (rule bigo_lesseq4)
apply (erule set_plus_imp_minus)
apply (rule allI)
apply (rule max.cobounded2)
apply (rule allI)
apply (subst fun_diff_def)
apply (erule thin_rl)
(* sledgehammer *)
apply (case_tac "0 <= f x - k x")
apply simp
apply (subst abs_of_nonneg)
apply (drule_tac x = x in spec) back
apply (metis diff_less_0_iff_less linorder_not_le not_le_imp_less xt1(12) xt1(6))
apply (metis add_minus_cancel diff_le_eq le_diff_eq uminus_add_conv_diff)
by (metis abs_ge_zero linorder_linear max.absorb1 max.commute)
lemma bigo_lesso4:
"f <o g =o O(k::'a=>'b::{linordered_field}) \<Longrightarrow>
g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
apply (unfold lesso_def)
apply (drule set_plus_imp_minus)
apply (drule bigo_abs5) back
apply (simp add: fun_diff_def)
apply (drule bigo_useful_add, assumption)
apply (erule bigo_lesseq2) back
apply (rule allI)
by (auto simp add: func_plus fun_diff_def algebra_simps
split: split_max abs_split)
lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x <= g x + C * \<bar>h x\<bar>"
apply (simp only: lesso_def bigo_alt_def)
apply clarsimp
by (metis add.commute diff_le_eq)
end