Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
authorkrauss
Thu Apr 12 23:07:01 2012 +0200 (2012-04-12)
changeset 4744569e96e5500df
parent 47444 d21c95af2df7
child 47446 ed0795caec95
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
src/HOL/Hahn_Banach/Hahn_Banach.thy
src/HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy
src/HOL/Hahn_Banach/Subspace.thy
src/HOL/Library/BigO.thy
src/HOL/Library/Set_Algebras.thy
src/HOL/Metis_Examples/Big_O.thy
src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
     1.1 --- a/src/HOL/Hahn_Banach/Hahn_Banach.thy	Thu Apr 12 22:55:11 2012 +0200
     1.2 +++ b/src/HOL/Hahn_Banach/Hahn_Banach.thy	Thu Apr 12 23:07:01 2012 +0200
     1.3 @@ -151,12 +151,12 @@
     1.4          qed
     1.5        qed
     1.6  
     1.7 -      def H' \<equiv> "H \<oplus> lin x'"
     1.8 +      def H' \<equiv> "H + lin x'"
     1.9          -- {* Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \skp *}
    1.10        have HH': "H \<unlhd> H'"
    1.11        proof (unfold H'_def)
    1.12          from x'E have "vectorspace (lin x')" ..
    1.13 -        with H show "H \<unlhd> H \<oplus> lin x'" ..
    1.14 +        with H show "H \<unlhd> H + lin x'" ..
    1.15        qed
    1.16  
    1.17        obtain xi where
     2.1 --- a/src/HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy	Thu Apr 12 22:55:11 2012 +0200
     2.2 +++ b/src/HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy	Thu Apr 12 23:07:01 2012 +0200
     2.3 @@ -90,7 +90,7 @@
     2.4  lemma h'_lf:
     2.5    assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =
     2.6        SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"
     2.7 -    and H'_def: "H' \<equiv> H \<oplus> lin x0"
     2.8 +    and H'_def: "H' \<equiv> H + lin x0"
     2.9      and HE: "H \<unlhd> E"
    2.10    assumes "linearform H h"
    2.11    assumes x0: "x0 \<notin> H"  "x0 \<in> E"  "x0 \<noteq> 0"
    2.12 @@ -106,7 +106,7 @@
    2.13      proof (unfold H'_def)
    2.14        from `x0 \<in> E`
    2.15        have "lin x0 \<unlhd> E" ..
    2.16 -      with HE show "vectorspace (H \<oplus> lin x0)" using E ..
    2.17 +      with HE show "vectorspace (H + lin x0)" using E ..
    2.18      qed
    2.19      {
    2.20        fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
    2.21 @@ -194,7 +194,7 @@
    2.22  lemma h'_norm_pres:
    2.23    assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =
    2.24        SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"
    2.25 -    and H'_def: "H' \<equiv> H \<oplus> lin x0"
    2.26 +    and H'_def: "H' \<equiv> H + lin x0"
    2.27      and x0: "x0 \<notin> H"  "x0 \<in> E"  "x0 \<noteq> 0"
    2.28    assumes E: "vectorspace E" and HE: "subspace H E"
    2.29      and "seminorm E p" and "linearform H h"
     3.1 --- a/src/HOL/Hahn_Banach/Subspace.thy	Thu Apr 12 22:55:11 2012 +0200
     3.2 +++ b/src/HOL/Hahn_Banach/Subspace.thy	Thu Apr 12 23:07:01 2012 +0200
     3.3 @@ -228,38 +228,38 @@
     3.4    set of all sums of elements from @{text U} and @{text V}.
     3.5  *}
     3.6  
     3.7 -lemma sum_def: "U \<oplus> V = {u + v | u v. u \<in> U \<and> v \<in> V}"
     3.8 +lemma sum_def: "U + V = {u + v | u v. u \<in> U \<and> v \<in> V}"
     3.9    unfolding set_plus_def by auto
    3.10  
    3.11  lemma sumE [elim]:
    3.12 -    "x \<in> U \<oplus> V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"
    3.13 +    "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"
    3.14    unfolding sum_def by blast
    3.15  
    3.16  lemma sumI [intro]:
    3.17 -    "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U \<oplus> V"
    3.18 +    "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V"
    3.19    unfolding sum_def by blast
    3.20  
    3.21  lemma sumI' [intro]:
    3.22 -    "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U \<oplus> V"
    3.23 +    "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"
    3.24    unfolding sum_def by blast
    3.25  
    3.26 -text {* @{text U} is a subspace of @{text "U \<oplus> V"}. *}
    3.27 +text {* @{text U} is a subspace of @{text "U + V"}. *}
    3.28  
    3.29  lemma subspace_sum1 [iff]:
    3.30    assumes "vectorspace U" "vectorspace V"
    3.31 -  shows "U \<unlhd> U \<oplus> V"
    3.32 +  shows "U \<unlhd> U + V"
    3.33  proof -
    3.34    interpret vectorspace U by fact
    3.35    interpret vectorspace V by fact
    3.36    show ?thesis
    3.37    proof
    3.38      show "U \<noteq> {}" ..
    3.39 -    show "U \<subseteq> U \<oplus> V"
    3.40 +    show "U \<subseteq> U + V"
    3.41      proof
    3.42        fix x assume x: "x \<in> U"
    3.43        moreover have "0 \<in> V" ..
    3.44 -      ultimately have "x + 0 \<in> U \<oplus> V" ..
    3.45 -      with x show "x \<in> U \<oplus> V" by simp
    3.46 +      ultimately have "x + 0 \<in> U + V" ..
    3.47 +      with x show "x \<in> U + V" by simp
    3.48      qed
    3.49      fix x y assume x: "x \<in> U" and "y \<in> U"
    3.50      then show "x + y \<in> U" by simp
    3.51 @@ -271,30 +271,30 @@
    3.52  
    3.53  lemma sum_subspace [intro?]:
    3.54    assumes "subspace U E" "vectorspace E" "subspace V E"
    3.55 -  shows "U \<oplus> V \<unlhd> E"
    3.56 +  shows "U + V \<unlhd> E"
    3.57  proof -
    3.58    interpret subspace U E by fact
    3.59    interpret vectorspace E by fact
    3.60    interpret subspace V E by fact
    3.61    show ?thesis
    3.62    proof
    3.63 -    have "0 \<in> U \<oplus> V"
    3.64 +    have "0 \<in> U + V"
    3.65      proof
    3.66        show "0 \<in> U" using `vectorspace E` ..
    3.67        show "0 \<in> V" using `vectorspace E` ..
    3.68        show "(0::'a) = 0 + 0" by simp
    3.69      qed
    3.70 -    then show "U \<oplus> V \<noteq> {}" by blast
    3.71 -    show "U \<oplus> V \<subseteq> E"
    3.72 +    then show "U + V \<noteq> {}" by blast
    3.73 +    show "U + V \<subseteq> E"
    3.74      proof
    3.75 -      fix x assume "x \<in> U \<oplus> V"
    3.76 +      fix x assume "x \<in> U + V"
    3.77        then obtain u v where "x = u + v" and
    3.78          "u \<in> U" and "v \<in> V" ..
    3.79        then show "x \<in> E" by simp
    3.80      qed
    3.81    next
    3.82 -    fix x y assume x: "x \<in> U \<oplus> V" and y: "y \<in> U \<oplus> V"
    3.83 -    show "x + y \<in> U \<oplus> V"
    3.84 +    fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V"
    3.85 +    show "x + y \<in> U + V"
    3.86      proof -
    3.87        from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" ..
    3.88        moreover
    3.89 @@ -306,7 +306,7 @@
    3.90          using x y by (simp_all add: add_ac)
    3.91        then show ?thesis ..
    3.92      qed
    3.93 -    fix a show "a \<cdot> x \<in> U \<oplus> V"
    3.94 +    fix a show "a \<cdot> x \<in> U + V"
    3.95      proof -
    3.96        from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" ..
    3.97        then have "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V"
    3.98 @@ -319,7 +319,7 @@
    3.99  text{* The sum of two subspaces is a vectorspace. *}
   3.100  
   3.101  lemma sum_vs [intro?]:
   3.102 -    "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U \<oplus> V)"
   3.103 +    "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)"
   3.104    by (rule subspace.vectorspace) (rule sum_subspace)
   3.105  
   3.106  
   3.107 @@ -481,7 +481,7 @@
   3.108  proof -
   3.109    interpret vectorspace E by fact
   3.110    interpret subspace H E by fact
   3.111 -  from x y x' have "x \<in> H \<oplus> lin x'" by auto
   3.112 +  from x y x' have "x \<in> H + lin x'" by auto
   3.113    have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p")
   3.114    proof (rule ex_ex1I)
   3.115      from x y show "\<exists>p. ?P p" by blast
     4.1 --- a/src/HOL/Library/BigO.thy	Thu Apr 12 22:55:11 2012 +0200
     4.2 +++ b/src/HOL/Library/BigO.thy	Thu Apr 12 23:07:01 2012 +0200
     4.3 @@ -92,7 +92,7 @@
     4.4    by (auto simp add: bigo_def) 
     4.5  
     4.6  lemma bigo_plus_self_subset [intro]: 
     4.7 -  "O(f) \<oplus> O(f) <= O(f)"
     4.8 +  "O(f) + O(f) <= O(f)"
     4.9    apply (auto simp add: bigo_alt_def set_plus_def)
    4.10    apply (rule_tac x = "c + ca" in exI)
    4.11    apply auto
    4.12 @@ -104,14 +104,14 @@
    4.13    apply force
    4.14  done
    4.15  
    4.16 -lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
    4.17 +lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
    4.18    apply (rule equalityI)
    4.19    apply (rule bigo_plus_self_subset)
    4.20    apply (rule set_zero_plus2) 
    4.21    apply (rule bigo_zero)
    4.22    done
    4.23  
    4.24 -lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) \<oplus> O(g)"
    4.25 +lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
    4.26    apply (rule subsetI)
    4.27    apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
    4.28    apply (subst bigo_pos_const [symmetric])+
    4.29 @@ -153,15 +153,15 @@
    4.30    apply simp
    4.31    done
    4.32  
    4.33 -lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A \<oplus> B <= O(f)"
    4.34 -  apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> O(f)")
    4.35 +lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A + B <= O(f)"
    4.36 +  apply (subgoal_tac "A + B <= O(f) + O(f)")
    4.37    apply (erule order_trans)
    4.38    apply simp
    4.39    apply (auto del: subsetI simp del: bigo_plus_idemp)
    4.40    done
    4.41  
    4.42  lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> 
    4.43 -    O(f + g) = O(f) \<oplus> O(g)"
    4.44 +    O(f + g) = O(f) + O(g)"
    4.45    apply (rule equalityI)
    4.46    apply (rule bigo_plus_subset)
    4.47    apply (simp add: bigo_alt_def set_plus_def func_plus)
    4.48 @@ -273,12 +273,12 @@
    4.49  lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" 
    4.50    by (unfold bigo_def, auto)
    4.51  
    4.52 -lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) \<oplus> O(h)"
    4.53 +lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) + O(h)"
    4.54  proof -
    4.55    assume "f : g +o O(h)"
    4.56 -  also have "... <= O(g) \<oplus> O(h)"
    4.57 +  also have "... <= O(g) + O(h)"
    4.58      by (auto del: subsetI)
    4.59 -  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
    4.60 +  also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
    4.61      apply (subst bigo_abs3 [symmetric])+
    4.62      apply (rule refl)
    4.63      done
    4.64 @@ -287,13 +287,13 @@
    4.65    finally have "f : ...".
    4.66    then have "O(f) <= ..."
    4.67      by (elim bigo_elt_subset)
    4.68 -  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
    4.69 +  also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
    4.70      by (rule bigo_plus_eq, auto)
    4.71    finally show ?thesis
    4.72      by (simp add: bigo_abs3 [symmetric])
    4.73  qed
    4.74  
    4.75 -lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)"
    4.76 +lemma bigo_mult [intro]: "O(f)*O(g) <= O(f * g)"
    4.77    apply (rule subsetI)
    4.78    apply (subst bigo_def)
    4.79    apply (auto simp add: bigo_alt_def set_times_def func_times)
    4.80 @@ -369,7 +369,7 @@
    4.81    done
    4.82  
    4.83  lemma bigo_mult7: "ALL x. f x ~= 0 ==>
    4.84 -    O(f * g) <= O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
    4.85 +    O(f * g) <= O(f::'a => ('b::linordered_field)) * O(g)"
    4.86    apply (subst bigo_mult6)
    4.87    apply assumption
    4.88    apply (rule set_times_mono3)
    4.89 @@ -377,7 +377,7 @@
    4.90    done
    4.91  
    4.92  lemma bigo_mult8: "ALL x. f x ~= 0 ==>
    4.93 -    O(f * g) = O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
    4.94 +    O(f * g) = O(f::'a => ('b::linordered_field)) * O(g)"
    4.95    apply (rule equalityI)
    4.96    apply (erule bigo_mult7)
    4.97    apply (rule bigo_mult)
    4.98 @@ -402,9 +402,9 @@
    4.99    show "f +o O(g) <= O(g)"
   4.100    proof -
   4.101      have "f : O(f)" by auto
   4.102 -    then have "f +o O(g) <= O(f) \<oplus> O(g)"
   4.103 +    then have "f +o O(g) <= O(f) + O(g)"
   4.104        by (auto del: subsetI)
   4.105 -    also have "... <= O(g) \<oplus> O(g)"
   4.106 +    also have "... <= O(g) + O(g)"
   4.107      proof -
   4.108        from a have "O(f) <= O(g)" by (auto del: subsetI)
   4.109        thus ?thesis by (auto del: subsetI)
   4.110 @@ -656,7 +656,7 @@
   4.111  subsection {* Misc useful stuff *}
   4.112  
   4.113  lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
   4.114 -  A \<oplus> B <= O(f)"
   4.115 +  A + B <= O(f)"
   4.116    apply (subst bigo_plus_idemp [symmetric])
   4.117    apply (rule set_plus_mono2)
   4.118    apply assumption+
     5.1 --- a/src/HOL/Library/Set_Algebras.thy	Thu Apr 12 22:55:11 2012 +0200
     5.2 +++ b/src/HOL/Library/Set_Algebras.thy	Thu Apr 12 23:07:01 2012 +0200
     5.3 @@ -34,14 +34,6 @@
     5.4  
     5.5  end
     5.6  
     5.7 -
     5.8 -text {* Legacy syntax: *}
     5.9 -
    5.10 -abbreviation (input) set_plus :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<oplus>" 65) where
    5.11 -  "A \<oplus> B \<equiv> A + B"
    5.12 -abbreviation (input) set_times :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<otimes>" 70) where
    5.13 -  "A \<otimes> B \<equiv> A * B"
    5.14 -
    5.15  instantiation set :: (zero) zero
    5.16  begin
    5.17  
    5.18 @@ -95,14 +87,14 @@
    5.19  instance set :: (comm_monoid_mult) comm_monoid_mult
    5.20  by default (simp_all add: set_times_def)
    5.21  
    5.22 -lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C \<oplus> D"
    5.23 +lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C + D"
    5.24    by (auto simp add: set_plus_def)
    5.25  
    5.26  lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C"
    5.27    by (auto simp add: elt_set_plus_def)
    5.28  
    5.29 -lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) \<oplus>
    5.30 -    (b +o D) = (a + b) +o (C \<oplus> D)"
    5.31 +lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) +
    5.32 +    (b +o D) = (a + b) +o (C + D)"
    5.33    apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
    5.34     apply (rule_tac x = "ba + bb" in exI)
    5.35    apply (auto simp add: add_ac)
    5.36 @@ -114,8 +106,8 @@
    5.37      (a + b) +o C"
    5.38    by (auto simp add: elt_set_plus_def add_assoc)
    5.39  
    5.40 -lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) \<oplus> C =
    5.41 -    a +o (B \<oplus> C)"
    5.42 +lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C =
    5.43 +    a +o (B + C)"
    5.44    apply (auto simp add: elt_set_plus_def set_plus_def)
    5.45     apply (blast intro: add_ac)
    5.46    apply (rule_tac x = "a + aa" in exI)
    5.47 @@ -126,8 +118,8 @@
    5.48     apply (auto simp add: add_ac)
    5.49    done
    5.50  
    5.51 -theorem set_plus_rearrange4: "C \<oplus> ((a::'a::comm_monoid_add) +o D) =
    5.52 -    a +o (C \<oplus> D)"
    5.53 +theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) =
    5.54 +    a +o (C + D)"
    5.55    apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
    5.56     apply (rule_tac x = "aa + ba" in exI)
    5.57     apply (auto simp add: add_ac)
    5.58 @@ -140,17 +132,17 @@
    5.59    by (auto simp add: elt_set_plus_def)
    5.60  
    5.61  lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>
    5.62 -    C \<oplus> E <= D \<oplus> F"
    5.63 +    C + E <= D + F"
    5.64    by (auto simp add: set_plus_def)
    5.65  
    5.66 -lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C \<oplus> D"
    5.67 +lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C + D"
    5.68    by (auto simp add: elt_set_plus_def set_plus_def)
    5.69  
    5.70  lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>
    5.71 -    a +o D <= D \<oplus> C"
    5.72 +    a +o D <= D + C"
    5.73    by (auto simp add: elt_set_plus_def set_plus_def add_ac)
    5.74  
    5.75 -lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C \<oplus> D"
    5.76 +lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C + D"
    5.77    apply (subgoal_tac "a +o B <= a +o D")
    5.78     apply (erule order_trans)
    5.79     apply (erule set_plus_mono3)
    5.80 @@ -163,21 +155,21 @@
    5.81    apply auto
    5.82    done
    5.83  
    5.84 -lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C \<oplus> E ==>
    5.85 -    x : D \<oplus> F"
    5.86 +lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C + E ==>
    5.87 +    x : D + F"
    5.88    apply (frule set_plus_mono2)
    5.89     prefer 2
    5.90     apply force
    5.91    apply assumption
    5.92    done
    5.93  
    5.94 -lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C \<oplus> D"
    5.95 +lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C + D"
    5.96    apply (frule set_plus_mono3)
    5.97    apply auto
    5.98    done
    5.99  
   5.100  lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==>
   5.101 -    x : a +o D ==> x : D \<oplus> C"
   5.102 +    x : a +o D ==> x : D + C"
   5.103    apply (frule set_plus_mono4)
   5.104    apply auto
   5.105    done
   5.106 @@ -185,7 +177,7 @@
   5.107  lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
   5.108    by (auto simp add: elt_set_plus_def)
   5.109  
   5.110 -lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A \<oplus> B"
   5.111 +lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B"
   5.112    apply (auto simp add: set_plus_def)
   5.113    apply (rule_tac x = 0 in bexI)
   5.114     apply (rule_tac x = x in bexI)
   5.115 @@ -206,14 +198,14 @@
   5.116    by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
   5.117      assumption)
   5.118  
   5.119 -lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C \<otimes> D"
   5.120 +lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C * D"
   5.121    by (auto simp add: set_times_def)
   5.122  
   5.123  lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
   5.124    by (auto simp add: elt_set_times_def)
   5.125  
   5.126 -lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) \<otimes>
   5.127 -    (b *o D) = (a * b) *o (C \<otimes> D)"
   5.128 +lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) *
   5.129 +    (b *o D) = (a * b) *o (C * D)"
   5.130    apply (auto simp add: elt_set_times_def set_times_def)
   5.131     apply (rule_tac x = "ba * bb" in exI)
   5.132     apply (auto simp add: mult_ac)
   5.133 @@ -225,8 +217,8 @@
   5.134      (a * b) *o C"
   5.135    by (auto simp add: elt_set_times_def mult_assoc)
   5.136  
   5.137 -lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) \<otimes> C =
   5.138 -    a *o (B \<otimes> C)"
   5.139 +lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) * C =
   5.140 +    a *o (B * C)"
   5.141    apply (auto simp add: elt_set_times_def set_times_def)
   5.142     apply (blast intro: mult_ac)
   5.143    apply (rule_tac x = "a * aa" in exI)
   5.144 @@ -237,8 +229,8 @@
   5.145     apply (auto simp add: mult_ac)
   5.146    done
   5.147  
   5.148 -theorem set_times_rearrange4: "C \<otimes> ((a::'a::comm_monoid_mult) *o D) =
   5.149 -    a *o (C \<otimes> D)"
   5.150 +theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) =
   5.151 +    a *o (C * D)"
   5.152    apply (auto simp add: elt_set_times_def set_times_def
   5.153      mult_ac)
   5.154     apply (rule_tac x = "aa * ba" in exI)
   5.155 @@ -252,17 +244,17 @@
   5.156    by (auto simp add: elt_set_times_def)
   5.157  
   5.158  lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
   5.159 -    C \<otimes> E <= D \<otimes> F"
   5.160 +    C * E <= D * F"
   5.161    by (auto simp add: set_times_def)
   5.162  
   5.163 -lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C \<otimes> D"
   5.164 +lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D"
   5.165    by (auto simp add: elt_set_times_def set_times_def)
   5.166  
   5.167  lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
   5.168 -    a *o D <= D \<otimes> C"
   5.169 +    a *o D <= D * C"
   5.170    by (auto simp add: elt_set_times_def set_times_def mult_ac)
   5.171  
   5.172 -lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C \<otimes> D"
   5.173 +lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C * D"
   5.174    apply (subgoal_tac "a *o B <= a *o D")
   5.175     apply (erule order_trans)
   5.176     apply (erule set_times_mono3)
   5.177 @@ -275,21 +267,21 @@
   5.178    apply auto
   5.179    done
   5.180  
   5.181 -lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C \<otimes> E ==>
   5.182 -    x : D \<otimes> F"
   5.183 +lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C * E ==>
   5.184 +    x : D * F"
   5.185    apply (frule set_times_mono2)
   5.186     prefer 2
   5.187     apply force
   5.188    apply assumption
   5.189    done
   5.190  
   5.191 -lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C \<otimes> D"
   5.192 +lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C * D"
   5.193    apply (frule set_times_mono3)
   5.194    apply auto
   5.195    done
   5.196  
   5.197  lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>
   5.198 -    x : a *o D ==> x : D \<otimes> C"
   5.199 +    x : a *o D ==> x : D * C"
   5.200    apply (frule set_times_mono4)
   5.201    apply auto
   5.202    done
   5.203 @@ -301,16 +293,16 @@
   5.204      (a * b) +o (a *o C)"
   5.205    by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
   5.206  
   5.207 -lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B \<oplus> C) =
   5.208 -    (a *o B) \<oplus> (a *o C)"
   5.209 +lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) =
   5.210 +    (a *o B) + (a *o C)"
   5.211    apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
   5.212     apply blast
   5.213    apply (rule_tac x = "b + bb" in exI)
   5.214    apply (auto simp add: ring_distribs)
   5.215    done
   5.216  
   5.217 -lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) \<otimes> D <=
   5.218 -    a *o D \<oplus> C \<otimes> D"
   5.219 +lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <=
   5.220 +    a *o D + C * D"
   5.221    apply (auto simp add:
   5.222      elt_set_plus_def elt_set_times_def set_times_def
   5.223      set_plus_def ring_distribs)
   5.224 @@ -330,7 +322,7 @@
   5.225    by (auto simp add: elt_set_times_def)
   5.226  
   5.227  lemma set_plus_image:
   5.228 -  fixes S T :: "'n::semigroup_add set" shows "S \<oplus> T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
   5.229 +  fixes S T :: "'n::semigroup_add set" shows "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
   5.230    unfolding set_plus_def by (fastforce simp: image_iff)
   5.231  
   5.232  lemma set_setsum_alt:
   5.233 @@ -339,7 +331,7 @@
   5.234      (is "_ = ?setsum I")
   5.235  using fin proof induct
   5.236    case (insert x F)
   5.237 -  have "setsum S (insert x F) = S x \<oplus> ?setsum F"
   5.238 +  have "setsum S (insert x F) = S x + ?setsum F"
   5.239      using insert.hyps by auto
   5.240    also have "...= {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
   5.241      unfolding set_plus_def
   5.242 @@ -355,8 +347,8 @@
   5.243  
   5.244  lemma setsum_set_cond_linear:
   5.245    fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"
   5.246 -  assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A \<oplus> B)" "P {0}"
   5.247 -    and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A \<oplus> B) = f A \<oplus> f B" "f {0} = {0}"
   5.248 +  assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A + B)" "P {0}"
   5.249 +    and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}"
   5.250    assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
   5.251    shows "f (setsum S I) = setsum (f \<circ> S) I"
   5.252  proof cases
   5.253 @@ -372,7 +364,7 @@
   5.254  
   5.255  lemma setsum_set_linear:
   5.256    fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"
   5.257 -  assumes "\<And>A B. f(A) \<oplus> f(B) = f(A \<oplus> B)" "f {0} = {0}"
   5.258 +  assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"
   5.259    shows "f (setsum S I) = setsum (f \<circ> S) I"
   5.260    using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
   5.261  
     6.1 --- a/src/HOL/Metis_Examples/Big_O.thy	Thu Apr 12 22:55:11 2012 +0200
     6.2 +++ b/src/HOL/Metis_Examples/Big_O.thy	Thu Apr 12 23:07:01 2012 +0200
     6.3 @@ -146,17 +146,17 @@
     6.4  by (auto simp add: bigo_def)
     6.5  
     6.6  lemma bigo_plus_self_subset [intro]:
     6.7 -  "O(f) \<oplus> O(f) <= O(f)"
     6.8 +  "O(f) + O(f) <= O(f)"
     6.9  apply (auto simp add: bigo_alt_def set_plus_def)
    6.10  apply (rule_tac x = "c + ca" in exI)
    6.11  apply auto
    6.12  apply (simp add: ring_distribs func_plus)
    6.13  by (metis order_trans abs_triangle_ineq add_mono)
    6.14  
    6.15 -lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
    6.16 +lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
    6.17  by (metis bigo_plus_self_subset bigo_zero set_eq_subset set_zero_plus2)
    6.18  
    6.19 -lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) \<oplus> O(g)"
    6.20 +lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
    6.21  apply (rule subsetI)
    6.22  apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
    6.23  apply (subst bigo_pos_const [symmetric])+
    6.24 @@ -187,10 +187,10 @@
    6.25   apply (metis abs_triangle_ineq mult_le_cancel_left_pos)
    6.26  by (metis abs_ge_zero abs_of_pos zero_le_mult_iff)
    6.27  
    6.28 -lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A \<oplus> B <= O(f)"
    6.29 +lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A + B <= O(f)"
    6.30  by (metis bigo_plus_idemp set_plus_mono2)
    6.31  
    6.32 -lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) \<oplus> O(g)"
    6.33 +lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
    6.34  apply (rule equalityI)
    6.35  apply (rule bigo_plus_subset)
    6.36  apply (simp add: bigo_alt_def set_plus_def func_plus)
    6.37 @@ -284,25 +284,25 @@
    6.38  lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. abs(f x)) =o O(g)"
    6.39  by (unfold bigo_def, auto)
    6.40  
    6.41 -lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) \<oplus> O(h)"
    6.42 +lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) + O(h)"
    6.43  proof -
    6.44    assume "f : g +o O(h)"
    6.45 -  also have "... <= O(g) \<oplus> O(h)"
    6.46 +  also have "... <= O(g) + O(h)"
    6.47      by (auto del: subsetI)
    6.48 -  also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))"
    6.49 +  also have "... = O(\<lambda>x. abs(g x)) + O(\<lambda>x. abs(h x))"
    6.50      by (metis bigo_abs3)
    6.51    also have "... = O((\<lambda>x. abs(g x)) + (\<lambda>x. abs(h x)))"
    6.52      by (rule bigo_plus_eq [symmetric], auto)
    6.53    finally have "f : ...".
    6.54    then have "O(f) <= ..."
    6.55      by (elim bigo_elt_subset)
    6.56 -  also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))"
    6.57 +  also have "... = O(\<lambda>x. abs(g x)) + O(\<lambda>x. abs(h x))"
    6.58      by (rule bigo_plus_eq, auto)
    6.59    finally show ?thesis
    6.60      by (simp add: bigo_abs3 [symmetric])
    6.61  qed
    6.62  
    6.63 -lemma bigo_mult [intro]: "O(f) \<otimes> O(g) <= O(f * g)"
    6.64 +lemma bigo_mult [intro]: "O(f) * O(g) <= O(f * g)"
    6.65  apply (rule subsetI)
    6.66  apply (subst bigo_def)
    6.67  apply (auto simp del: abs_mult mult_ac
    6.68 @@ -358,14 +358,14 @@
    6.69  declare bigo_mult6 [simp]
    6.70  
    6.71  lemma bigo_mult7:
    6.72 -"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<le> O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) \<otimes> O(g)"
    6.73 +"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<le> O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) * O(g)"
    6.74  by (metis bigo_refl bigo_mult6 set_times_mono3)
    6.75  
    6.76  declare bigo_mult6 [simp del]
    6.77  declare bigo_mult7 [intro!]
    6.78  
    6.79  lemma bigo_mult8:
    6.80 -"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) \<otimes> O(g)"
    6.81 +"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) * O(g)"
    6.82  by (metis bigo_mult bigo_mult7 order_antisym_conv)
    6.83  
    6.84  lemma bigo_minus [intro]: "f : O(g) \<Longrightarrow> - f : O(g)"
    6.85 @@ -575,7 +575,7 @@
    6.86  subsection {* Misc useful stuff *}
    6.87  
    6.88  lemma bigo_useful_intro: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow>
    6.89 -  A \<oplus> B <= O(f)"
    6.90 +  A + B <= O(f)"
    6.91    apply (subst bigo_plus_idemp [symmetric])
    6.92    apply (rule set_plus_mono2)
    6.93    apply assumption+
     7.1 --- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Thu Apr 12 22:55:11 2012 +0200
     7.2 +++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Thu Apr 12 23:07:01 2012 +0200
     7.3 @@ -5428,13 +5428,13 @@
     7.4  
     7.5  lemma closure_sum:
     7.6    fixes S T :: "('n::euclidean_space) set"
     7.7 -  shows "closure S \<oplus> closure T \<subseteq> closure (S \<oplus> T)"
     7.8 +  shows "closure S + closure T \<subseteq> closure (S + T)"
     7.9  proof-
    7.10 -  have "(closure S) \<oplus> (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
    7.11 +  have "(closure S) + (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
    7.12      by (simp add: set_plus_image)
    7.13    also have "... = (\<lambda>(x,y). x + y) ` closure (S \<times> T)"
    7.14      using closure_direct_sum by auto
    7.15 -  also have "... \<subseteq> closure (S \<oplus> T)"
    7.16 +  also have "... \<subseteq> closure (S + T)"
    7.17      using fst_snd_linear closure_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
    7.18      by (auto simp: set_plus_image)
    7.19    finally show ?thesis
    7.20 @@ -5444,7 +5444,7 @@
    7.21  lemma convex_oplus:
    7.22  fixes S T :: "('n::euclidean_space) set"
    7.23  assumes "convex S" "convex T"
    7.24 -shows "convex (S \<oplus> T)"
    7.25 +shows "convex (S + T)"
    7.26  proof-
    7.27  have "{x + y |x y. x : S & y : T} = {c. EX a:S. EX b:T. c = a + b}" by auto
    7.28  thus ?thesis unfolding set_plus_def using convex_sums[of S T] assms by auto
    7.29 @@ -5452,13 +5452,13 @@
    7.30  
    7.31  lemma convex_hull_sum:
    7.32  fixes S T :: "('n::euclidean_space) set"
    7.33 -shows "convex hull (S \<oplus> T) = (convex hull S) \<oplus> (convex hull T)"
    7.34 +shows "convex hull (S + T) = (convex hull S) + (convex hull T)"
    7.35  proof-
    7.36 -have "(convex hull S) \<oplus> (convex hull T) =
    7.37 +have "(convex hull S) + (convex hull T) =
    7.38        (%(x,y). x + y) ` ((convex hull S) <*> (convex hull T))"
    7.39     by (simp add: set_plus_image)
    7.40  also have "... = (%(x,y). x + y) ` (convex hull (S <*> T))" using convex_hull_direct_sum by auto
    7.41 -also have "...= convex hull (S \<oplus> T)" using fst_snd_linear linear_conv_bounded_linear
    7.42 +also have "...= convex hull (S + T)" using fst_snd_linear linear_conv_bounded_linear
    7.43     convex_hull_linear_image[of "(%(x,y). x + y)" "S <*> T"] by (auto simp add: set_plus_image)
    7.44  finally show ?thesis by auto
    7.45  qed
    7.46 @@ -5466,12 +5466,12 @@
    7.47  lemma rel_interior_sum:
    7.48  fixes S T :: "('n::euclidean_space) set"
    7.49  assumes "convex S" "convex T"
    7.50 -shows "rel_interior (S \<oplus> T) = (rel_interior S) \<oplus> (rel_interior T)"
    7.51 +shows "rel_interior (S + T) = (rel_interior S) + (rel_interior T)"
    7.52  proof-
    7.53 -have "(rel_interior S) \<oplus> (rel_interior T) = (%(x,y). x + y) ` (rel_interior S <*> rel_interior T)"
    7.54 +have "(rel_interior S) + (rel_interior T) = (%(x,y). x + y) ` (rel_interior S <*> rel_interior T)"
    7.55     by (simp add: set_plus_image)
    7.56  also have "... = (%(x,y). x + y) ` rel_interior (S <*> T)" using rel_interior_direct_sum assms by auto
    7.57 -also have "...= rel_interior (S \<oplus> T)" using fst_snd_linear convex_direct_sum assms
    7.58 +also have "...= rel_interior (S + T)" using fst_snd_linear convex_direct_sum assms
    7.59     rel_interior_convex_linear_image[of "(%(x,y). x + y)" "S <*> T"] by (auto simp add: set_plus_image)
    7.60  finally show ?thesis by auto
    7.61  qed
    7.62 @@ -5507,7 +5507,7 @@
    7.63  lemma convex_rel_open_sum:
    7.64  fixes S T :: "('n::euclidean_space) set"
    7.65  assumes "convex S" "rel_open S" "convex T" "rel_open T"
    7.66 -shows "convex (S \<oplus> T) & rel_open (S \<oplus> T)"
    7.67 +shows "convex (S + T) & rel_open (S + T)"
    7.68  by (metis assms convex_oplus rel_interior_sum rel_open_def)
    7.69  
    7.70  lemma convex_hull_finite_union_cones:
    7.71 @@ -5547,7 +5547,7 @@
    7.72  fixes S T :: "('m::euclidean_space) set"
    7.73  assumes "convex S" "cone S" "S ~= {}"
    7.74  assumes "convex T" "cone T" "T ~= {}"
    7.75 -shows "convex hull (S Un T) = S \<oplus> T"
    7.76 +shows "convex hull (S Un T) = S + T"
    7.77  proof-
    7.78  def I == "{(1::nat),2}"
    7.79  def A == "(%i. (if i=(1::nat) then S else T))"
    7.80 @@ -5556,7 +5556,7 @@
    7.81  moreover have "convex hull Union (A ` I) = setsum A I"
    7.82      apply (subst convex_hull_finite_union_cones[of I A]) using assms A_def I_def by auto
    7.83  moreover have
    7.84 -  "setsum A I = S \<oplus> T" using A_def I_def
    7.85 +  "setsum A I = S + T" using A_def I_def
    7.86       unfolding set_plus_def apply auto unfolding set_plus_def by auto
    7.87  ultimately show ?thesis by auto
    7.88  qed