# HG changeset patch
# User avigad
# Date 1122310489 -7200
# Node ID d374530bfaaab5a6f5220543b092614c9f8ce550
# Parent  2187e3f947614071d4211dbadf35bbf5009dd7d6
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy

diff -r 2187e3f94761 -r d374530bfaaa NEWS
--- a/NEWS	Mon Jul 25 15:51:30 2005 +0200
+++ b/NEWS	Mon Jul 25 18:54:49 2005 +0200
@@ -389,6 +389,12 @@
 * Theory RComplete: expanded support for floor and ceiling functions.
 
 
+*** HOL-Library ***
+
+* Theories SetsAndFunctions and BigO support asymptotic "big O" calculations.
+See the notes in BigO.thy.
+
+
 *** HOLCF ***
 
 * HOLCF: discontinued special version of 'constdefs' (which used to
diff -r 2187e3f94761 -r d374530bfaaa src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Mon Jul 25 15:51:30 2005 +0200
+++ b/src/HOL/IsaMakefile	Mon Jul 25 18:54:49 2005 +0200
@@ -176,6 +176,7 @@
 HOL-Library: HOL $(LOG)/HOL-Library.gz
 
 $(LOG)/HOL-Library.gz: $(OUT)/HOL Library/Accessible_Part.thy \
+  Library/SetsAndFunctions.thy Library/BigO.thy \
   Library/EfficientNat.thy Library/FuncSet.thy Library/Library.thy \
   Library/List_Prefix.thy Library/Multiset.thy Library/NatPair.thy \
   Library/Permutation.thy Library/Primes.thy Library/Quotient.thy \
diff -r 2187e3f94761 -r d374530bfaaa src/HOL/Library/BigO.thy
--- /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+\item Lemmas involving ``sumr-pos'' have been replaced by more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
diff -r 2187e3f94761 -r d374530bfaaa src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Mon Jul 25 15:51:30 2005 +0200
+++ b/src/HOL/Library/Library.thy	Mon Jul 25 18:54:49 2005 +0200
@@ -2,6 +2,7 @@
 theory Library
 imports
   Accessible_Part
+  BigO
   Continuity
   EfficientNat
   FuncSet
diff -r 2187e3f94761 -r d374530bfaaa src/HOL/Library/SetsAndFunctions.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/SetsAndFunctions.thy	Mon Jul 25 18:54:49 2005 +0200
@@ -0,0 +1,376 @@
+(*  Title:      SetsAndFunctions.thy
+    Author:     Jeremy Avigad and Kevin Donnelly
+*)
+
+header {* Operations on sets and functions *}
+
+theory SetsAndFunctions
+imports Main
+begin
+
+text {* 
+This library lifts operations like addition and muliplication to sets and
+functions of appropriate types. It was designed to support asymptotic
+calculations. See the comments at the top of BigO.thy
+*}
+
+subsection {* Basic definitions *} 
+
+instance set :: (plus)plus
+by intro_classes
+
+instance fun :: (type,plus)plus
+by intro_classes
+
+defs (overloaded)
+  func_plus: "f + g == (%x. f x + g x)"
+  set_plus: "A + B == {c. EX a:A. EX b:B. c = a + b}"
+
+instance set :: (times)times
+by intro_classes
+
+instance fun :: (type,times)times
+by intro_classes
+
+defs (overloaded)
+  func_times: "f * g == (%x. f x * g x)" 
+  set_times:"A * B == {c. EX a:A. EX b:B. c = a * b}"
+
+instance fun :: (type,minus)minus
+by intro_classes
+
+defs (overloaded)
+  func_minus: "- f == (%x. - f x)"
+  func_diff: "f - g == %x. f x - g x"                 
+
+instance fun :: (type,zero)zero
+by intro_classes
+
+instance set :: (zero)zero
+by(intro_classes)
+
+defs (overloaded)
+  func_zero: "0::(('a::type) => ('b::zero)) == %x. 0"
+  set_zero: "0::('a::zero)set == {0}"
+
+instance fun :: (type,one)one
+by intro_classes
+
+instance set :: (one)one
+by intro_classes
+
+defs (overloaded)
+  func_one: "1::(('a::type) => ('b::one)) == %x. 1"
+  set_one: "1::('a::one)set == {1}"
+
+constdefs 
+  elt_set_plus :: "'a::plus => 'a set => 'a set"    (infixl "+o" 70)
+  "a +o B == {c. EX b:B. c = a + b}"
+
+  elt_set_times :: "'a::times => 'a set => 'a set"  (infixl "*o" 80)
+  "a *o B == {c. EX b:B. c = a * b}"
+
+syntax
+  "elt_set_eq" :: "'a => 'a set => bool"      (infix "=o" 50)
+
+translations
+  "x =o A" => "x : A"
+
+instance fun :: (type,semigroup_add)semigroup_add
+  apply intro_classes
+  apply (auto simp add: func_plus add_assoc)
+done
+
+instance fun :: (type,comm_monoid_add)comm_monoid_add
+  apply intro_classes
+  apply (auto simp add: func_zero func_plus add_ac)
+done
+
+instance fun :: (type,ab_group_add)ab_group_add
+  apply intro_classes
+  apply (simp add: func_minus func_plus func_zero)
+  apply (simp add: func_minus func_plus func_diff diff_minus)
+done
+
+instance fun :: (type,semigroup_mult)semigroup_mult
+  apply intro_classes
+  apply (auto simp add: func_times mult_assoc)
+done
+
+instance fun :: (type,comm_monoid_mult)comm_monoid_mult
+  apply intro_classes
+  apply (auto simp add: func_one func_times mult_ac)
+done
+
+instance fun :: (type,comm_ring_1)comm_ring_1
+  apply intro_classes
+  apply (auto simp add: func_plus func_times func_minus func_diff ext 
+    func_one func_zero ring_eq_simps) 
+  apply (drule fun_cong)
+  apply simp
+done
+
+instance set :: (semigroup_add)semigroup_add
+  apply intro_classes
+  apply (unfold set_plus)
+  apply (force simp add: add_assoc)
+done
+
+instance set :: (semigroup_mult)semigroup_mult
+  apply intro_classes
+  apply (unfold set_times)
+  apply (force simp add: mult_assoc)
+done
+
+instance set :: (comm_monoid_add)comm_monoid_add
+  apply intro_classes
+  apply (unfold set_plus)
+  apply (force simp add: add_ac)
+  apply (unfold set_zero)
+  apply force
+done
+
+instance set :: (comm_monoid_mult)comm_monoid_mult
+  apply intro_classes
+  apply (unfold set_times)
+  apply (force simp add: mult_ac)
+  apply (unfold set_one)
+  apply force
+done
+
+subsection {* Basic properties *}
+
+lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C + D" 
+by (auto simp add: set_plus)
+
+lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C"
+by (auto simp add: elt_set_plus_def)
+
+lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) + 
+  (b +o D) = (a + b) +o (C + D)"
+  apply (auto simp add: elt_set_plus_def set_plus add_ac)
+  apply (rule_tac x = "ba + bb" in exI)
+  apply (auto simp add: add_ac)
+  apply (rule_tac x = "aa + a" in exI)
+  apply (auto simp add: add_ac)
+done
+
+lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = 
+  (a + b) +o C"
+by (auto simp add: elt_set_plus_def add_assoc)
+
+lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C = 
+  a +o (B + C)"
+  apply (auto simp add: elt_set_plus_def set_plus)
+  apply (blast intro: add_ac)
+  apply (rule_tac x = "a + aa" in exI)
+  apply (rule conjI)
+  apply (rule_tac x = "aa" in bexI)
+  apply auto
+  apply (rule_tac x = "ba" in bexI)
+  apply (auto simp add: add_ac)
+done
+
+theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) = 
+    a +o (C + D)" 
+  apply (auto intro!: subsetI simp add: elt_set_plus_def set_plus add_ac)
+  apply (rule_tac x = "aa + ba" in exI)
+  apply (auto simp add: add_ac)
+done
+
+theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
+  set_plus_rearrange3 set_plus_rearrange4
+
+lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D"
+by (auto simp add: elt_set_plus_def)
+
+lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==> 
+    C + E <= D + F"
+by (auto simp add: set_plus)
+
+lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C + D"
+by (auto simp add: elt_set_plus_def set_plus)
+
+lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==> 
+  a +o D <= D + C" 
+by (auto simp add: elt_set_plus_def set_plus add_ac)
+
+lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C + D"
+  apply (subgoal_tac "a +o B <= a +o D")
+  apply (erule order_trans)
+  apply (erule set_plus_mono3)
+  apply (erule set_plus_mono)
+done
+
+lemma set_plus_mono_b: "C <= D ==> x : a +o C 
+    ==> x : a +o D"
+  apply (frule set_plus_mono)
+  apply auto
+done
+
+lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C + E ==> 
+    x : D + F"
+  apply (frule set_plus_mono2)
+  prefer 2
+  apply force
+  apply assumption
+done
+
+lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C + D"
+  apply (frule set_plus_mono3)
+  apply auto
+done
+
+lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==> 
+  x : a +o D ==> x : D + C" 
+  apply (frule set_plus_mono4)
+  apply auto
+done
+
+lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
+by (auto simp add: elt_set_plus_def)
+
+lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B"
+  apply (auto intro!: subsetI simp add: set_plus)
+  apply (rule_tac x = 0 in bexI)
+  apply (rule_tac x = x in bexI)
+  apply (auto simp add: add_ac)
+done
+
+lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"
+by (auto simp add: elt_set_plus_def add_ac diff_minus)
+
+lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
+  apply (auto simp add: elt_set_plus_def add_ac diff_minus)
+  apply (subgoal_tac "a = (a + - b) + b")
+  apply (rule bexI, assumption, assumption)
+  apply (auto simp add: add_ac)
+done
+
+lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
+by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus, 
+    assumption)
+
+lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C * D" 
+by (auto simp add: set_times)
+
+lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
+by (auto simp add: elt_set_times_def)
+
+lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) * 
+  (b *o D) = (a * b) *o (C * D)"
+  apply (auto simp add: elt_set_times_def set_times)
+  apply (rule_tac x = "ba * bb" in exI)
+  apply (auto simp add: mult_ac)
+  apply (rule_tac x = "aa * a" in exI)
+  apply (auto simp add: mult_ac)
+done
+
+lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) = 
+  (a * b) *o C"
+by (auto simp add: elt_set_times_def mult_assoc)
+
+lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) * C = 
+  a *o (B * C)"
+  apply (auto simp add: elt_set_times_def set_times)
+  apply (blast intro: mult_ac)
+  apply (rule_tac x = "a * aa" in exI)
+  apply (rule conjI)
+  apply (rule_tac x = "aa" in bexI)
+  apply auto
+  apply (rule_tac x = "ba" in bexI)
+  apply (auto simp add: mult_ac)
+done
+
+theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) = 
+    a *o (C * D)" 
+  apply (auto intro!: subsetI simp add: elt_set_times_def set_times 
+    mult_ac)
+  apply (rule_tac x = "aa * ba" in exI)
+  apply (auto simp add: mult_ac)
+done
+
+theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
+  set_times_rearrange3 set_times_rearrange4
+
+lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"
+by (auto simp add: elt_set_times_def)
+
+lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==> 
+    C * E <= D * F"
+by (auto simp add: set_times)
+
+lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D"
+by (auto simp add: elt_set_times_def set_times)
+
+lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==> 
+  a *o D <= D * C" 
+by (auto simp add: elt_set_times_def set_times mult_ac)
+
+lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C * D"
+  apply (subgoal_tac "a *o B <= a *o D")
+  apply (erule order_trans)
+  apply (erule set_times_mono3)
+  apply (erule set_times_mono)
+done
+
+lemma set_times_mono_b: "C <= D ==> x : a *o C 
+    ==> x : a *o D"
+  apply (frule set_times_mono)
+  apply auto
+done
+
+lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C * E ==> 
+    x : D * F"
+  apply (frule set_times_mono2)
+  prefer 2
+  apply force
+  apply assumption
+done
+
+lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C * D"
+  apply (frule set_times_mono3)
+  apply auto
+done
+
+lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==> 
+  x : a *o D ==> x : D * C" 
+  apply (frule set_times_mono4)
+  apply auto
+done
+
+lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
+by (auto simp add: elt_set_times_def)
+
+lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)= 
+  (a * b) +o (a *o C)"
+by (auto simp add: elt_set_plus_def elt_set_times_def ring_distrib)
+
+lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) = 
+  (a *o B) + (a *o C)"
+  apply (auto simp add: set_plus elt_set_times_def ring_distrib)
+  apply blast
+  apply (rule_tac x = "b + bb" in exI)
+  apply (auto simp add: ring_distrib)
+done
+
+lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <= 
+    a *o D + C * D"
+  apply (auto intro!: subsetI simp add: 
+    elt_set_plus_def elt_set_times_def set_times 
+    set_plus ring_distrib)
+  apply auto
+done
+
+theorems set_times_plus_distribs = set_times_plus_distrib
+  set_times_plus_distrib2
+
+lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==> 
+    - a : C" 
+by (auto simp add: elt_set_times_def)
+
+lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
+    - a : (- 1) *o C"
+by (auto simp add: elt_set_times_def)
+  
+end