Theory BigO (Isabelle2005: October 2005)

# Theory BigO

Up to index of Isabelle/HOL/HOL-Complex/ex

theory BigO
imports SetsAndFunctions
begin

    (*  Title:      HOL/Library/BigO.thy
        ID:         $Id: BigO.thy,v 1.4 2005/08/31 13:46:36 wenzelm Exp $
        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
    \cite{Avigad-Donnelly}.

    The main changes in this version are as follows:
    \begin{itemize}
    \item We have eliminated the @{text O} operator on sets. (Most uses of this seem
      to be inessential.)
    \item We no longer use @{text "+"} as output syntax for @{text "+o"}
    \item Lemmas involving @{text "sumr"} have been replaced by more general lemmas
      involving `@{text "setsum"}.
    \item The library has been expanded, with e.g.~support for expressions of
      the form @{text "f < g + O(h)"}.
    \end{itemize}

    See \verb,Complex/ex/BigO_Complex.thy, for additional lemmas that
    require the \verb,HOL-Complex, logic image.

    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 @{text "subsetI"} as an intro rule,
    rather than as an @{text "intro!"} rule, for example, using
    \isa{\isakeyword{declare}}~@{text "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

    end

### Definitions

lemma bigo_pos_const:

      (∃c. ∀x. ¦h x¦ ≤ c * ¦f x¦) = (∃c>0::'a. ∀x. ¦h x¦ ≤ c * ¦f x¦)

lemma bigo_alt_def:

      O(f) = {h. ∃c>0::'b. ∀x. ¦h x¦ ≤ c * ¦f x¦}

lemma bigo_elt_subset:

      f ∈ O(g) ==> O(f) ⊆ O(g)

lemma bigo_refl:

      f ∈ O(f)

lemma bigo_zero:

      0 ∈ O(g)

lemma bigo_zero2:

      O(%x. 0::'b) = {%x. 0::'b}

lemma bigo_plus_self_subset:

      O(f) + O(f) ⊆ O(f)

lemma bigo_plus_idemp:

      O(f) + O(f) = O(f)

lemma bigo_plus_subset:

      O(f + g) ⊆ O(f) + O(g)

lemma bigo_plus_subset2:

      [| A ⊆ O(f); B ⊆ O(f) |] ==> A + B ⊆ O(f)

lemma bigo_plus_eq:

      [| ∀x. (0::'b) ≤ f x; ∀x. (0::'b) ≤ g x |] ==> O(f + g) = O(f) + O(g)

lemma bigo_bounded_alt:

      [| ∀x. (0::'b) ≤ f x; ∀x. f x ≤ c * g x |] ==> f ∈ O(g)

lemma bigo_bounded:

      [| ∀x. (0::'b) ≤ f x; ∀x. f x ≤ g x |] ==> f ∈ O(g)

lemma bigo_bounded2:

      [| ∀x. lb x ≤ f x; ∀x. f x ≤ lb x + g x |] ==> f ∈ lb +o O(g)

lemma bigo_abs:

      (%x. ¦f x¦) ∈ O(f)

lemma bigo_abs2:

      f ∈ O(%x. ¦f x¦)

lemma bigo_abs3:

      O(f) = O(%x. ¦f x¦)

lemma bigo_abs4:

      f ∈ g +o O(h) ==> (%x. ¦f x¦) ∈ (%x. ¦g x¦) +o O(h)

lemma bigo_abs5:

      f ∈ O(g) ==> (%x. ¦f x¦) ∈ O(g)

lemma bigo_elt_subset2:

      f ∈ g +o O(h) ==> O(f) ⊆ O(g) + O(h)

lemma bigo_mult:

      O(f) * O(g) ⊆ O(f * g)

lemma bigo_mult2:

      f *o O(g) ⊆ O(f * g)

lemma bigo_mult3:

      [| f ∈ O(h); g ∈ O(j) |] ==> f * g ∈ O(h * j)

lemma bigo_mult4:

      f ∈ k +o O(h) ==> g * f ∈ g * k +o O(g * h)

lemma bigo_mult5:

      ∀x. f x ≠ (0::'b) ==> O(f * g) ⊆ f *o O(g)

lemma bigo_mult6:

      ∀x. f x ≠ (0::'b) ==> O(f * g) = f *o O(g)

lemma bigo_mult7:

      ∀x. f x ≠ (0::'b) ==> O(f * g) ⊆ O(f) * O(g)

lemma bigo_mult8:

      ∀x. f x ≠ (0::'b) ==> O(f * g) = O(f) * O(g)

lemma bigo_minus:

      f ∈ O(g) ==> - f ∈ O(g)

lemma bigo_minus2:

      f ∈ g +o O(h) ==> - f ∈ - g +o O(h)

lemma bigo_minus3:

      O(- f) = O(f)

lemma bigo_plus_absorb_lemma1:

      f ∈ O(g) ==> f +o O(g) ⊆ O(g)

lemma bigo_plus_absorb_lemma2:

      f ∈ O(g) ==> O(g) ⊆ f +o O(g)

lemma bigo_plus_absorb:

      f ∈ O(g) ==> f +o O(g) = O(g)

lemma bigo_plus_absorb2:

      [| f ∈ O(g); A ⊆ O(g) |] ==> f +o A ⊆ O(g)

lemma bigo_add_commute_imp:

      f ∈ g +o O(h) ==> g ∈ f +o O(h)

lemma bigo_add_commute:

      (f ∈ g +o O(h)) = (g ∈ f +o O(h))

lemma bigo_const1:

      (%x. c) ∈ O(%x. 1::'b)

lemma bigo_const2:

      O(%x. c) ⊆ O(%x. 1::'b)

lemma bigo_const3:

      c ≠ (0::'a) ==> (%x. 1::'a) ∈ O(%x. c)

lemma bigo_const4:

      c ≠ (0::'a) ==> O(%x. 1::'a) ⊆ O(%x. c)

lemma bigo_const:

      c ≠ (0::'a) ==> O(%x. c) = O(%x. 1::'a)

lemma bigo_const_mult1:

      (%x. c * f x) ∈ O(f)

lemma bigo_const_mult2:

      O(%x. c * f x) ⊆ O(f)

lemma bigo_const_mult3:

      c ≠ (0::'a) ==> f ∈ O(%x. c * f x)

lemma bigo_const_mult4:

      c ≠ (0::'a) ==> O(f) ⊆ O(%x. c * f x)

lemma bigo_const_mult:

      c ≠ (0::'a) ==> O(%x. c * f x) = O(f)

lemma bigo_const_mult5:

      c ≠ (0::'a) ==> (%x. c) *o O(f) = O(f)

lemma bigo_const_mult6:

      (%x. c) *o O(f) ⊆ O(f)

lemma bigo_const_mult7:

      f ∈ O(g) ==> (%x. c * f x) ∈ O(g)

lemma bigo_compose1:

      f ∈ O(g) ==> (%x. f (k x)) ∈ O(%x. g (k x))

lemma bigo_compose2:

      f ∈ g +o O(h) ==> (%x. f (k x)) ∈ (%x. g (k x)) +o O(%x. h (k x))

### Setsum

lemma bigo_setsum_main:

      [| ∀x. ∀y∈A x. (0::'c) ≤ h x y; ∃c. ∀x. ∀y∈A x. ¦f x y¦ ≤ c * h x y |]
      ==> (%x. setsum (f x) (A x)) ∈ O(%x. setsum (h x) (A x))

lemma bigo_setsum1:

      [| ∀x y. (0::'c) ≤ h x y; ∃c. ∀x y. ¦f x y¦ ≤ c * h x y |]
      ==> (%x. setsum (f x) (A x)) ∈ O(%x. setsum (h x) (A x))

lemma bigo_setsum2:

      [| ∀y. (0::'b) ≤ h y; ∃c. ∀y. ¦f y¦ ≤ c * h y |]
      ==> (%x. setsum f (A x)) ∈ O(%x. setsum h (A x))

lemma bigo_setsum3:

      f ∈ O(h)
      ==> (%x. ∑y∈A x. l x y * f (k x y)) ∈ O(%x. ∑y∈A x. ¦l x y * h (k x y)¦)

lemma bigo_setsum4:

      f ∈ g +o O(h)
      ==> (%x. ∑y∈A x. l x y * f (k x y))
          ∈ (%x. ∑y∈A x. l x y * g (k x y)) +o O(%x. ∑y∈A x. ¦l x y * h (k x y)¦)

lemma bigo_setsum5:

      [| f ∈ O(h); ∀x y. (0::'b) ≤ l x y; ∀x. (0::'b) ≤ h x |]
      ==> (%x. ∑y∈A x. l x y * f (k x y)) ∈ O(%x. ∑y∈A x. l x y * h (k x y))

lemma bigo_setsum6:

      [| f ∈ g +o O(h); ∀x y. (0::'b) ≤ l x y; ∀x. (0::'b) ≤ h x |]
      ==> (%x. ∑y∈A x. l x y * f (k x y))
          ∈ (%x. ∑y∈A x. l x y * g (k x y)) +o O(%x. ∑y∈A x. l x y * h (k x y))

### Misc useful stuff

lemma bigo_useful_intro:

      [| A ⊆ O(f); B ⊆ O(f) |] ==> A + B ⊆ O(f)

lemma bigo_useful_add:

      [| f ∈ O(h); g ∈ O(h) |] ==> f + g ∈ O(h)

lemma bigo_useful_const_mult:

      [| c ≠ (0::'a); (%x. c) * f ∈ O(h) |] ==> f ∈ O(h)

lemma bigo_fix:

      [| (%x. f (x + 1)) ∈ O(%x. h (x + 1)); f 0 = (0::'a) |] ==> f ∈ O(h)

lemma bigo_fix2:

      [| (%x. f (x + 1)) ∈ (%x. g (x + 1)) +o O(%x. h (x + 1)); f 0 = g 0 |]
      ==> f ∈ g +o O(h)

### Less than or equal to

lemma bigo_lesseq1:

      [| f ∈ O(h); ∀x. ¦g x¦ ≤ ¦f x¦ |] ==> g ∈ O(h)

lemma bigo_lesseq2:

      [| f ∈ O(h); ∀x. ¦g x¦ ≤ f x |] ==> g ∈ O(h)

lemma bigo_lesseq3:

      [| f ∈ O(h); ∀x. (0::'b) ≤ g x; ∀x. g x ≤ f x |] ==> g ∈ O(h)

lemma bigo_lesseq4:

      [| f ∈ O(h); ∀x. (0::'b) ≤ g x; ∀x. g x ≤ ¦f x¦ |] ==> g ∈ O(h)

lemma bigo_lesso1:

      ∀x. f x ≤ g x ==> f <o g ∈ O(h)

lemma bigo_lesso2:

      [| f ∈ g +o O(h); ∀x. (0::'b) ≤ k x; ∀x. k x ≤ f x |] ==> k <o g ∈ O(h)

lemma bigo_lesso3:

      [| f ∈ g +o O(h); ∀x. (0::'b) ≤ k x; ∀x. g x ≤ k x |] ==> f <o k ∈ O(h)

lemma bigo_lesso4:

      [| f <o g ∈ O(k); g ∈ h +o O(k) |] ==> f <o h ∈ O(k)

lemma bigo_lesso5:

      f <o g ∈ O(h) ==> ∃C. ∀x. f x ≤ g x + C * ¦h x¦

lemma lesso_add:

      [| f <o g ∈ O(h); k <o l ∈ O(h) |] ==> (f + k) <o (g + l) ∈ O(h)