--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/BigO.thy Mon Jul 25 18:54:49 2005 +0200
@@ -0,0 +1,916 @@
+(* Title: BigO.thy
+ Authors: Jeremy Avigad and Kevin Donnelly
+*)
+
+header {* Big O notation *}
+
+theory BigO
+imports SetsAndFunctions
+begin
+
+text {*
+This library is designed to support asymptotic ``big O'' calculations,
+i.e.~reasoning with expressions of the form $f = O(g)$ and $f = g + O(h)$.
+An earlier version of this library is described in detail in
+\begin{quote}
+Avigad, Jeremy, and Kevin Donnelly, \emph{Formalizing O notation in
+Isabelle/HOL}, in David Basin and Micha\"el Rusiowitch, editors,
+\emph{Automated Reasoning: second international conference, IJCAR 2004},
+Springer, 357--371, 2004.
+\end{quote}
+The main changes in this version are as follows:
+\begin{itemize}
+\item We have eliminated the $O$ operator on sets. (Most uses of this seem
+ to be inessential.)
+\item We no longer use $+$ as output syntax for $+o$.
+\item Lemmas involving ``sumr-pos'' have been replaced by more
+ general lemmas involving ``setsum''.
+\item The library has been expanded, with e.g.~support for expressions of
+ the form $f < g + O(h)$.
+\end{itemize}
+Note that two lemmas at the end of this file are commented out, as they
+require the HOL-Complex library.
+
+Note also since the Big O library includes rules that demonstrate set
+inclusion, to use the automated reasoners effectively with the library one
+should redeclare the theorem ``subsetI'' as an intro rule, rather than as
+an intro! rule, for example, using ``declare subsetI [del, intro]''.
+*}
+
+subsection {* Definitions *}
+
+constdefs
+
+ bigo :: "('a => 'b::ordered_idom) => ('a => 'b) set" ("(1O'(_'))")
+ "O(f::('a => 'b)) ==
+ {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
+
+lemma bigo_pos_const: "(EX (c::'a::ordered_idom).
+ ALL x. (abs (h x)) <= (c * (abs (f x))))
+ = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
+ apply auto
+ apply (case_tac "c = 0")
+ apply simp
+ apply (rule_tac x = "1" in exI)
+ apply simp
+ apply (rule_tac x = "abs c" in exI)
+ apply auto
+ apply (subgoal_tac "c * abs(f x) <= abs c * abs (f x)")
+ apply (erule_tac x = x in allE)
+ apply force
+ apply (rule mult_right_mono)
+ apply (rule abs_ge_self)
+ apply (rule abs_ge_zero)
+done
+
+lemma bigo_alt_def: "O(f) =
+ {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
+by (auto simp add: bigo_def bigo_pos_const)
+
+lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
+ apply (auto simp add: bigo_alt_def)
+ apply (rule_tac x = "ca * c" in exI)
+ apply (rule conjI)
+ apply (rule mult_pos_pos)
+ apply (assumption)+
+ apply (rule allI)
+ apply (drule_tac x = "xa" in spec)+
+ apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))")
+ apply (erule order_trans)
+ apply (simp add: mult_ac)
+ apply (rule mult_left_mono, assumption)
+ apply (rule order_less_imp_le, assumption)
+done
+
+lemma bigo_refl [intro]: "f : O(f)"
+ apply(auto simp add: bigo_def)
+ apply(rule_tac x = 1 in exI)
+ apply simp
+done
+
+lemma bigo_zero: "0 : O(g)"
+ apply (auto simp add: bigo_def func_zero)
+ apply (rule_tac x = 0 in exI)
+ apply auto
+done
+
+lemma bigo_zero2: "O(%x.0) = {%x.0}"
+ apply (auto simp add: bigo_def)
+ apply (rule ext)
+ apply auto
+done
+
+lemma bigo_plus_self_subset [intro]:
+ "O(f) + O(f) <= O(f)"
+ apply (auto simp add: bigo_alt_def set_plus)
+ apply (rule_tac x = "c + ca" in exI)
+ apply auto
+ apply (simp add: ring_distrib func_plus)
+ apply (rule order_trans)
+ apply (rule abs_triangle_ineq)
+ apply (rule add_mono)
+ apply force
+ apply force
+done
+
+lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
+ apply (rule equalityI)
+ apply (rule bigo_plus_self_subset)
+ apply (rule set_zero_plus2)
+ apply (rule bigo_zero)
+done
+
+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)
+ apply (subst bigo_pos_const [symmetric])+
+ apply (rule_tac x =
+ "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
+ apply (rule conjI)
+ apply (rule_tac x = "c + c" in exI)
+ apply (clarsimp)
+ apply (auto)
+ apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
+ apply (erule_tac x = xa in allE)
+ apply (erule order_trans)
+ apply (simp)
+ apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
+ apply (erule order_trans)
+ apply (simp add: ring_distrib)
+ apply (rule mult_left_mono)
+ apply assumption
+ apply (simp add: order_less_le)
+ apply (rule mult_left_mono)
+ apply (simp add: abs_triangle_ineq)
+ apply (simp add: order_less_le)
+ apply (rule mult_nonneg_nonneg)
+ apply (rule add_nonneg_nonneg)
+ apply auto
+ apply (rule_tac x = "%n. if (abs (f n)) < abs (g n) then x n else 0"
+ in exI)
+ apply (rule conjI)
+ apply (rule_tac x = "c + c" in exI)
+ apply auto
+ apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
+ apply (erule_tac x = xa in allE)
+ apply (erule order_trans)
+ apply (simp)
+ apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
+ apply (erule order_trans)
+ apply (simp add: ring_distrib)
+ apply (rule mult_left_mono)
+ apply (simp add: order_less_le)
+ apply (simp add: order_less_le)
+ apply (rule mult_left_mono)
+ apply (rule abs_triangle_ineq)
+ apply (simp add: order_less_le)
+ apply (rule mult_nonneg_nonneg)
+ apply (rule add_nonneg_nonneg)
+ apply (erule order_less_imp_le)+
+ apply simp
+ apply (rule ext)
+ apply (auto simp add: if_splits linorder_not_le)
+done
+
+lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A + B <= O(f)"
+ apply (subgoal_tac "A + B <= O(f) + O(f)")
+ apply (erule order_trans)
+ apply simp
+ apply (auto del: subsetI simp del: bigo_plus_idemp)
+done
+
+lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==>
+ O(f + g) = O(f) + O(g)"
+ apply (rule equalityI)
+ apply (rule bigo_plus_subset)
+ apply (simp add: bigo_alt_def set_plus func_plus)
+ apply clarify
+ apply (rule_tac x = "max c ca" in exI)
+ apply (rule conjI)
+ apply (subgoal_tac "c <= max c ca")
+ apply (erule order_less_le_trans)
+ apply assumption
+ apply (rule le_maxI1)
+ apply clarify
+ apply (drule_tac x = "xa" in spec)+
+ apply (subgoal_tac "0 <= f xa + g xa")
+ apply (simp add: ring_distrib)
+ apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
+ apply (subgoal_tac "abs(a xa) + abs(b xa) <=
+ max c ca * f xa + max c ca * g xa")
+ apply (force)
+ apply (rule add_mono)
+ apply (subgoal_tac "c * f xa <= max c ca * f xa")
+ apply (force)
+ apply (rule mult_right_mono)
+ apply (rule le_maxI1)
+ apply assumption
+ apply (subgoal_tac "ca * g xa <= max c ca * g xa")
+ apply (force)
+ apply (rule mult_right_mono)
+ apply (rule le_maxI2)
+ apply assumption
+ apply (rule abs_triangle_ineq)
+ apply (rule add_nonneg_nonneg)
+ apply assumption+
+done
+
+lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==>
+ f : O(g)"
+ apply (auto simp add: bigo_def)
+ apply (rule_tac x = "abs c" in exI)
+ apply auto
+ apply (drule_tac x = x in spec)+
+ apply (simp add: abs_mult [symmetric])
+done
+
+lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==>
+ f : O(g)"
+ apply (erule bigo_bounded_alt [of f 1 g])
+ apply simp
+done
+
+lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
+ f : lb +o O(g)"
+ apply (rule set_minus_imp_plus)
+ apply (rule bigo_bounded)
+ apply (auto simp add: diff_minus func_minus func_plus)
+ apply (drule_tac x = x in spec)+
+ apply force
+ apply (drule_tac x = x in spec)+
+ apply force
+done
+
+lemma bigo_abs: "(%x. abs(f x)) =o O(f)"
+ apply (unfold bigo_def)
+ apply auto
+ apply (rule_tac x = 1 in exI)
+ apply auto
+done
+
+lemma bigo_abs2: "f =o O(%x. abs(f x))"
+ apply (unfold bigo_def)
+ apply auto
+ apply (rule_tac x = 1 in exI)
+ apply auto
+done
+
+lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
+ apply (rule equalityI)
+ apply (rule bigo_elt_subset)
+ apply (rule bigo_abs2)
+ apply (rule bigo_elt_subset)
+ apply (rule bigo_abs)
+done
+
+lemma bigo_abs4: "f =o g +o O(h) ==>
+ (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
+ apply (drule set_plus_imp_minus)
+ apply (rule set_minus_imp_plus)
+ apply (subst func_diff)
+proof -
+ assume a: "f - g : O(h)"
+ have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
+ by (rule bigo_abs2)
+ also have "... <= O(%x. abs (f x - g x))"
+ apply (rule bigo_elt_subset)
+ apply (rule bigo_bounded)
+ apply force
+ apply (rule allI)
+ apply (rule abs_triangle_ineq3)
+ done
+ also have "... <= O(f - g)"
+ apply (rule bigo_elt_subset)
+ apply (subst func_diff)
+ apply (rule bigo_abs)
+ done
+ also have "... <= O(h)"
+ by (rule bigo_elt_subset)
+ finally show "(%x. abs (f x) - abs (g x)) : O(h)".
+qed
+
+lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)"
+by (unfold bigo_def, auto)
+
+lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> 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(%x. abs(g x)) + O(%x. abs(h x))"
+ apply (subst bigo_abs3 [symmetric])+
+ apply (rule refl)
+ done
+ also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
+ by (rule bigo_plus_eq [symmetric], auto)
+ finally have "f : ...".
+ then have "O(f) <= ..."
+ by (elim bigo_elt_subset)
+ also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
+ 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 add: bigo_alt_def set_times func_times)
+ apply (rule_tac x = "c * ca" in exI)
+ apply(rule allI)
+ apply(erule_tac x = x in allE)+
+ apply(subgoal_tac "c * ca * abs(f x * g x) =
+ (c * abs(f x)) * (ca * abs(g x))")
+ apply(erule ssubst)
+ apply (subst abs_mult)
+ apply (rule mult_mono)
+ apply assumption+
+ apply (rule mult_nonneg_nonneg)
+ apply auto
+ apply (simp add: mult_ac abs_mult)
+done
+
+lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
+ apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
+ apply (rule_tac x = c in exI)
+ apply auto
+ apply (drule_tac x = x in spec)
+ apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
+ apply (force simp add: mult_ac)
+ apply (rule mult_left_mono, assumption)
+ apply (rule abs_ge_zero)
+done
+
+lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
+ apply (rule subsetD)
+ apply (rule bigo_mult)
+ apply (erule set_times_intro, assumption)
+done
+
+lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
+ apply (drule set_plus_imp_minus)
+ apply (rule set_minus_imp_plus)
+ apply (drule bigo_mult3 [where g = g and j = g])
+ apply (auto simp add: ring_eq_simps mult_ac)
+done
+
+lemma bigo_mult5: "ALL x. f x ~= 0 ==>
+ O(f * g) <= (f::'a => ('b::ordered_field)) *o O(g)"
+proof -
+ assume "ALL x. f x ~= 0"
+ show "O(f * g) <= f *o O(g)"
+ proof
+ fix h
+ assume "h : O(f * g)"
+ then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
+ by auto
+ also have "... <= O((%x. 1 / f x) * (f * g))"
+ by (rule bigo_mult2)
+ also have "(%x. 1 / f x) * (f * g) = g"
+ apply (simp add: func_times)
+ apply (rule ext)
+ apply (simp add: prems nonzero_divide_eq_eq mult_ac)
+ done
+ finally have "(%x. (1::'b) / f x) * h : O(g)".
+ then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
+ by auto
+ also have "f * ((%x. (1::'b) / f x) * h) = h"
+ apply (simp add: func_times)
+ apply (rule ext)
+ apply (simp add: prems nonzero_divide_eq_eq mult_ac)
+ done
+ finally show "h : f *o O(g)".
+ qed
+qed
+
+lemma bigo_mult6: "ALL x. f x ~= 0 ==>
+ O(f * g) = (f::'a => ('b::ordered_field)) *o O(g)"
+ apply (rule equalityI)
+ apply (erule bigo_mult5)
+ apply (rule bigo_mult2)
+done
+
+lemma bigo_mult7: "ALL x. f x ~= 0 ==>
+ O(f * g) <= O(f::'a => ('b::ordered_field)) * O(g)"
+ apply (subst bigo_mult6)
+ apply assumption
+ apply (rule set_times_mono3)
+ apply (rule bigo_refl)
+done
+
+lemma bigo_mult8: "ALL x. f x ~= 0 ==>
+ O(f * g) = O(f::'a => ('b::ordered_field)) * O(g)"
+ apply (rule equalityI)
+ apply (erule bigo_mult7)
+ apply (rule bigo_mult)
+done
+
+lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
+ by (auto simp add: bigo_def func_minus)
+
+lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
+ apply (rule set_minus_imp_plus)
+ apply (drule set_plus_imp_minus)
+ apply (drule bigo_minus)
+ apply (simp add: diff_minus)
+done
+
+lemma bigo_minus3: "O(-f) = O(f)"
+ by (auto simp add: bigo_def func_minus abs_minus_cancel)
+
+lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
+proof -
+ assume a: "f : O(g)"
+ show "f +o O(g) <= O(g)"
+ proof -
+ have "f : O(f)" by auto
+ then have "f +o O(g) <= O(f) + O(g)"
+ by (auto del: subsetI)
+ also have "... <= O(g) + O(g)"
+ proof -
+ from a have "O(f) <= O(g)" by (auto del: subsetI)
+ thus ?thesis by (auto del: subsetI)
+ qed
+ also have "... <= O(g)" by (simp add: bigo_plus_idemp)
+ finally show ?thesis .
+ qed
+qed
+
+lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
+proof -
+ assume a: "f : O(g)"
+ show "O(g) <= f +o O(g)"
+ proof -
+ from a have "-f : O(g)" by auto
+ then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
+ then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
+ also have "f +o (-f +o O(g)) = O(g)"
+ by (simp add: set_plus_rearranges)
+ finally show ?thesis .
+ qed
+qed
+
+lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
+ apply (rule equalityI)
+ apply (erule bigo_plus_absorb_lemma1)
+ apply (erule bigo_plus_absorb_lemma2)
+done
+
+lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
+ apply (subgoal_tac "f +o A <= f +o O(g)")
+ apply force+
+done
+
+lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
+ apply (subst set_minus_plus [symmetric])
+ apply (subgoal_tac "g - f = - (f - g)")
+ apply (erule ssubst)
+ apply (rule bigo_minus)
+ apply (subst set_minus_plus)
+ apply assumption
+ apply (simp add: diff_minus add_ac)
+done
+
+lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
+ apply (rule iffI)
+ apply (erule bigo_add_commute_imp)+
+done
+
+lemma bigo_const1: "(%x. c) : O(%x. 1)"
+by (auto simp add: bigo_def mult_ac)
+
+lemma bigo_const2 [intro]: "O(%x. c) <= O(%x. 1)"
+ apply (rule bigo_elt_subset)
+ apply (rule bigo_const1)
+done
+
+lemma bigo_const3: "(c::'a::ordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
+ apply (simp add: bigo_def)
+ apply (rule_tac x = "abs(inverse c)" in exI)
+ apply (simp add: abs_mult [symmetric])
+done
+
+lemma bigo_const4: "(c::'a::ordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
+by (rule bigo_elt_subset, rule bigo_const3, assumption)
+
+lemma bigo_const [simp]: "(c::'a::ordered_field) ~= 0 ==>
+ O(%x. c) = O(%x. 1)"
+by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
+
+lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
+ apply (simp add: bigo_def)
+ apply (rule_tac x = "abs(c)" in exI)
+ apply (auto simp add: abs_mult [symmetric])
+done
+
+lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
+by (rule bigo_elt_subset, rule bigo_const_mult1)
+
+lemma bigo_const_mult3: "(c::'a::ordered_field) ~= 0 ==> f : O(%x. c * f x)"
+ apply (simp add: bigo_def)
+ apply (rule_tac x = "abs(inverse c)" in exI)
+ apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
+done
+
+lemma bigo_const_mult4: "(c::'a::ordered_field) ~= 0 ==>
+ O(f) <= O(%x. c * f x)"
+by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
+
+lemma bigo_const_mult [simp]: "(c::'a::ordered_field) ~= 0 ==>
+ O(%x. c * f x) = O(f)"
+by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
+
+lemma bigo_const_mult5 [simp]: "(c::'a::ordered_field) ~= 0 ==>
+ (%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 = "%y. inverse c * x y" in exI)
+ apply (simp add: mult_assoc [symmetric] abs_mult)
+ apply (rule_tac x = "abs (inverse c) * ca" in exI)
+ apply (rule allI)
+ apply (subst mult_assoc)
+ apply (rule mult_left_mono)
+ apply (erule spec)
+ apply force
+done
+
+lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
+ apply (auto intro!: subsetI
+ simp add: bigo_def elt_set_times_def func_times)
+ apply (rule_tac x = "ca * (abs c)" in exI)
+ apply (rule allI)
+ apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
+ apply (erule ssubst)
+ apply (subst abs_mult)
+ apply (rule mult_left_mono)
+ apply (erule spec)
+ apply simp
+ apply(simp add: mult_ac)
+done
+
+lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
+proof -
+ assume "f =o O(g)"
+ then have "(%x. c) * f =o (%x. c) *o O(g)"
+ by auto
+ also have "(%x. c) * f = (%x. c * f x)"
+ by (simp add: func_times)
+ also have "(%x. c) *o O(g) <= O(g)"
+ by (auto del: subsetI)
+ finally show ?thesis .
+qed
+
+lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
+by (unfold bigo_def, auto)
+
+lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o
+ O(%x. h(k x))"
+ apply (simp only: set_minus_plus [symmetric] diff_minus func_minus
+ func_plus)
+ apply (erule bigo_compose1)
+done
+
+subsection {* Setsum *}
+
+lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==>
+ EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
+ (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
+ apply (auto simp add: bigo_def)
+ apply (rule_tac x = "abs c" in exI)
+ apply (subst abs_of_nonneg);back;back
+ apply (rule setsum_nonneg)
+ apply force
+ apply (subst setsum_mult)
+ apply (rule allI)
+ apply (rule order_trans)
+ apply (rule setsum_abs)
+ apply (rule setsum_mono)
+ apply (rule order_trans)
+ apply (drule spec)+
+ apply (drule bspec)+
+ apply assumption+
+ apply (drule bspec)
+ apply assumption+
+ apply (rule mult_right_mono)
+ apply (rule abs_ge_self)
+ apply force
+done
+
+lemma bigo_setsum1: "ALL x y. 0 <= h x y ==>
+ EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
+ (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
+ apply (rule bigo_setsum_main)
+ apply force
+ apply clarsimp
+ apply (rule_tac x = c in exI)
+ apply force
+done
+
+lemma bigo_setsum2: "ALL y. 0 <= h y ==>
+ EX c. ALL y. abs(f y) <= c * (h y) ==>
+ (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
+by (rule bigo_setsum1, auto)
+
+lemma bigo_setsum3: "f =o O(h) ==>
+ (%x. SUM y : A x. (l x y) * f(k x y)) =o
+ O(%x. SUM y : A x. abs(l x y * h(k x y)))"
+ apply (rule bigo_setsum1)
+ apply (rule allI)+
+ apply (rule abs_ge_zero)
+ apply (unfold bigo_def)
+ apply auto
+ apply (rule_tac x = c in exI)
+ apply (rule allI)+
+ apply (subst abs_mult)+
+ apply (subst mult_left_commute)
+ apply (rule mult_left_mono)
+ apply (erule spec)
+ apply (rule abs_ge_zero)
+done
+
+lemma bigo_setsum4: "f =o g +o O(h) ==>
+ (%x. SUM y : A x. l x y * f(k x y)) =o
+ (%x. SUM y : A x. l x y * g(k x y)) +o
+ O(%x. SUM y : A x. abs(l x y * h(k x y)))"
+ apply (rule set_minus_imp_plus)
+ apply (subst func_diff)
+ apply (subst setsum_subtractf [symmetric])
+ apply (subst right_diff_distrib [symmetric])
+ apply (rule bigo_setsum3)
+ apply (subst func_diff [symmetric])
+ apply (erule set_plus_imp_minus)
+done
+
+lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==>
+ ALL x. 0 <= h x ==>
+ (%x. SUM y : A x. (l x y) * f(k x y)) =o
+ O(%x. SUM y : A x. (l x y) * h(k x y))"
+ apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) =
+ (%x. SUM y : A x. abs((l x y) * h(k x y)))")
+ apply (erule ssubst)
+ apply (erule bigo_setsum3)
+ apply (rule ext)
+ apply (rule setsum_cong2)
+ apply (subst abs_of_nonneg)
+ apply (rule mult_nonneg_nonneg)
+ apply auto
+done
+
+lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
+ ALL x. 0 <= h x ==>
+ (%x. SUM y : A x. (l x y) * f(k x y)) =o
+ (%x. SUM y : A x. (l x y) * g(k x y)) +o
+ O(%x. SUM y : A x. (l x y) * h(k x y))"
+ apply (rule set_minus_imp_plus)
+ apply (subst func_diff)
+ apply (subst setsum_subtractf [symmetric])
+ apply (subst right_diff_distrib [symmetric])
+ apply (rule bigo_setsum5)
+ apply (subst func_diff [symmetric])
+ apply (drule set_plus_imp_minus)
+ apply auto
+done
+
+subsection {* Misc useful stuff *}
+
+lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
+ 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) ==> g =o O(h) ==> 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::ordered_field) ~= 0 ==>
+ (%x. c) * f =o O(h) ==> f =o O(h)"
+ apply (rule subsetD)
+ apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
+ apply assumption
+ apply (rule bigo_const_mult6)
+ apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
+ apply (erule ssubst)
+ apply (erule set_times_intro2)
+ apply (simp add: func_times)
+ apply (rule ext)
+ apply (subst times_divide_eq_left [symmetric])
+ apply (subst divide_self)
+ apply (assumption, simp)
+done
+
+lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
+ f =o O(h)"
+ apply (simp add: bigo_alt_def)
+ apply auto
+ apply (rule_tac x = c in exI)
+ apply auto
+ apply (case_tac "x = 0")
+ apply simp
+ apply (rule mult_nonneg_nonneg)
+ apply force
+ apply force
+ apply (subgoal_tac "x = Suc (x - 1)")
+ apply (erule ssubst)back
+ apply (erule spec)
+ apply simp
+done
+
+lemma bigo_fix2:
+ "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==>
+ f 0 = g 0 ==> f =o g +o O(h)"
+ apply (rule set_minus_imp_plus)
+ apply (rule bigo_fix)
+ apply (subst func_diff)
+ apply (subst func_diff [symmetric])
+ apply (rule set_plus_imp_minus)
+ apply simp
+ apply (simp add: func_diff)
+done
+
+subsection {* Less than or equal to *}
+
+constdefs
+ lesso :: "('a => 'b::ordered_idom) => ('a => 'b) => ('a => 'b)"
+ (infixl "<o" 70)
+ "f <o g == (%x. max (f x - g x) 0)"
+
+lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
+ g =o O(h)"
+ apply (unfold bigo_def)
+ apply clarsimp
+ apply (rule_tac x = c in exI)
+ apply (rule allI)
+ apply (rule order_trans)
+ apply (erule spec)+
+done
+
+lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
+ g =o O(h)"
+ apply (erule bigo_lesseq1)
+ apply (rule allI)
+ apply (drule_tac x = x in spec)
+ apply (rule order_trans)
+ apply assumption
+ apply (rule abs_ge_self)
+done
+
+lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
+ 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) ==>
+ ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
+ g =o O(h)"
+ apply (erule bigo_lesseq1)
+ apply (rule allI)
+ apply (subst abs_of_nonneg)
+ apply (erule spec)+
+done
+
+lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
+ apply (unfold lesso_def)
+ apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
+ apply (erule ssubst)
+ apply (rule bigo_zero)
+ apply (unfold func_zero)
+ apply (rule ext)
+ apply (simp split: split_max)
+done
+
+lemma bigo_lesso2: "f =o g +o O(h) ==>
+ ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
+ 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 le_maxI2)
+ apply (rule allI)
+ apply (subst func_diff)
+ apply (case_tac "0 <= k x - g x")
+ apply simp
+ apply (subst abs_of_nonneg)
+ apply (drule_tac x = x in spec)back
+ apply (simp add: compare_rls)
+ apply (subst diff_minus)+
+ apply (rule add_right_mono)
+ apply (erule spec)
+ apply (rule order_trans)
+ prefer 2
+ apply (rule abs_ge_zero)
+ apply (simp add: compare_rls)
+done
+
+lemma bigo_lesso3: "f =o g +o O(h) ==>
+ ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
+ 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 le_maxI2)
+ apply (rule allI)
+ apply (subst func_diff)
+ apply (case_tac "0 <= f x - k x")
+ apply simp
+ apply (subst abs_of_nonneg)
+ apply (drule_tac x = x in spec)back
+ apply (simp add: compare_rls)
+ apply (subst diff_minus)+
+ apply (rule add_left_mono)
+ apply (rule le_imp_neg_le)
+ apply (erule spec)
+ apply (rule order_trans)
+ prefer 2
+ apply (rule abs_ge_zero)
+ apply (simp add: compare_rls)
+done
+
+lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::ordered_field) ==>
+ g =o h +o O(k) ==> f <o h =o O(k)"
+ apply (unfold lesso_def)
+ apply (drule set_plus_imp_minus)
+ apply (drule bigo_abs5)back
+ apply (simp add: func_diff)
+ apply (drule bigo_useful_add)
+ apply assumption
+ apply (erule bigo_lesseq2)back
+ apply (rule allI)
+ apply (auto simp add: func_plus func_diff compare_rls
+ split: split_max abs_split)
+done
+
+lemma bigo_lesso5: "f <o g =o O(h) ==>
+ EX C. ALL x. f x <= g x + C * abs(h x)"
+ apply (simp only: lesso_def bigo_alt_def)
+ apply clarsimp
+ apply (rule_tac x = c in exI)
+ apply (rule allI)
+ apply (drule_tac x = x in spec)
+ apply (subgoal_tac "abs(max (f x - g x) 0) = max (f x - g x) 0")
+ apply (clarsimp simp add: compare_rls add_ac)
+ apply (rule abs_of_nonneg)
+ apply (rule le_maxI2)
+done
+
+lemma lesso_add: "f <o g =o O(h) ==>
+ k <o l =o O(h) ==> (f + k) <o (g + l) =o O(h)"
+ apply (unfold lesso_def)
+ apply (rule bigo_lesseq3)
+ apply (erule bigo_useful_add)
+ apply assumption
+ apply (force split: split_max)
+ apply (auto split: split_max simp add: func_plus)
+done
+
+(*
+These last two lemmas require the HOL-Complex library.
+
+lemma bigo_LIMSEQ1: "f =o O(g) ==> g ----> 0 ==> f ----> 0"
+ apply (simp add: LIMSEQ_def bigo_alt_def)
+ apply clarify
+ apply (drule_tac x = "r / c" in spec)
+ apply (drule mp)
+ apply (erule divide_pos_pos)
+ apply assumption
+ apply clarify
+ apply (rule_tac x = no in exI)
+ apply (rule allI)
+ apply (drule_tac x = n in spec)+
+ apply (rule impI)
+ apply (drule mp)
+ apply assumption
+ apply (rule order_le_less_trans)
+ apply assumption
+ apply (rule order_less_le_trans)
+ apply (subgoal_tac "c * abs(g n) < c * (r / c)")
+ apply assumption
+ apply (erule mult_strict_left_mono)
+ apply assumption
+ apply simp
+done
+
+lemma bigo_LIMSEQ2: "f =o g +o O(h) ==> h ----> 0 ==> f ----> a
+ ==> g ----> a"
+ apply (drule set_plus_imp_minus)
+ apply (drule bigo_LIMSEQ1)
+ apply assumption
+ apply (simp only: func_diff)
+ apply (erule LIMSEQ_diff_approach_zero2)
+ apply assumption
+done
+
+*)
+
+end