--- a/src/HOL/Hahn_Banach/Hahn_Banach.thy Thu Apr 12 22:55:11 2012 +0200
+++ b/src/HOL/Hahn_Banach/Hahn_Banach.thy Thu Apr 12 23:07:01 2012 +0200
@@ -151,12 +151,12 @@
qed
qed
- def H' \<equiv> "H \<oplus> lin x'"
+ def H' \<equiv> "H + lin x'"
-- {* Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \skp *}
have HH': "H \<unlhd> H'"
proof (unfold H'_def)
from x'E have "vectorspace (lin x')" ..
- with H show "H \<unlhd> H \<oplus> lin x'" ..
+ with H show "H \<unlhd> H + lin x'" ..
qed
obtain xi where
--- a/src/HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy Thu Apr 12 22:55:11 2012 +0200
+++ b/src/HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy Thu Apr 12 23:07:01 2012 +0200
@@ -90,7 +90,7 @@
lemma h'_lf:
assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =
SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"
- and H'_def: "H' \<equiv> H \<oplus> lin x0"
+ and H'_def: "H' \<equiv> H + lin x0"
and HE: "H \<unlhd> E"
assumes "linearform H h"
assumes x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"
@@ -106,7 +106,7 @@
proof (unfold H'_def)
from `x0 \<in> E`
have "lin x0 \<unlhd> E" ..
- with HE show "vectorspace (H \<oplus> lin x0)" using E ..
+ with HE show "vectorspace (H + lin x0)" using E ..
qed
{
fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
@@ -194,7 +194,7 @@
lemma h'_norm_pres:
assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =
SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"
- and H'_def: "H' \<equiv> H \<oplus> lin x0"
+ and H'_def: "H' \<equiv> H + lin x0"
and x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"
assumes E: "vectorspace E" and HE: "subspace H E"
and "seminorm E p" and "linearform H h"
--- a/src/HOL/Hahn_Banach/Subspace.thy Thu Apr 12 22:55:11 2012 +0200
+++ b/src/HOL/Hahn_Banach/Subspace.thy Thu Apr 12 23:07:01 2012 +0200
@@ -228,38 +228,38 @@
set of all sums of elements from @{text U} and @{text V}.
*}
-lemma sum_def: "U \<oplus> V = {u + v | u v. u \<in> U \<and> v \<in> V}"
+lemma sum_def: "U + V = {u + v | u v. u \<in> U \<and> v \<in> V}"
unfolding set_plus_def by auto
lemma sumE [elim]:
- "x \<in> U \<oplus> V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"
+ "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"
unfolding sum_def by blast
lemma sumI [intro]:
- "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U \<oplus> V"
+ "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V"
unfolding sum_def by blast
lemma sumI' [intro]:
- "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U \<oplus> V"
+ "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"
unfolding sum_def by blast
-text {* @{text U} is a subspace of @{text "U \<oplus> V"}. *}
+text {* @{text U} is a subspace of @{text "U + V"}. *}
lemma subspace_sum1 [iff]:
assumes "vectorspace U" "vectorspace V"
- shows "U \<unlhd> U \<oplus> V"
+ shows "U \<unlhd> U + V"
proof -
interpret vectorspace U by fact
interpret vectorspace V by fact
show ?thesis
proof
show "U \<noteq> {}" ..
- show "U \<subseteq> U \<oplus> V"
+ show "U \<subseteq> U + V"
proof
fix x assume x: "x \<in> U"
moreover have "0 \<in> V" ..
- ultimately have "x + 0 \<in> U \<oplus> V" ..
- with x show "x \<in> U \<oplus> V" by simp
+ ultimately have "x + 0 \<in> U + V" ..
+ with x show "x \<in> U + V" by simp
qed
fix x y assume x: "x \<in> U" and "y \<in> U"
then show "x + y \<in> U" by simp
@@ -271,30 +271,30 @@
lemma sum_subspace [intro?]:
assumes "subspace U E" "vectorspace E" "subspace V E"
- shows "U \<oplus> V \<unlhd> E"
+ shows "U + V \<unlhd> E"
proof -
interpret subspace U E by fact
interpret vectorspace E by fact
interpret subspace V E by fact
show ?thesis
proof
- have "0 \<in> U \<oplus> V"
+ have "0 \<in> U + V"
proof
show "0 \<in> U" using `vectorspace E` ..
show "0 \<in> V" using `vectorspace E` ..
show "(0::'a) = 0 + 0" by simp
qed
- then show "U \<oplus> V \<noteq> {}" by blast
- show "U \<oplus> V \<subseteq> E"
+ then show "U + V \<noteq> {}" by blast
+ show "U + V \<subseteq> E"
proof
- fix x assume "x \<in> U \<oplus> V"
+ fix x assume "x \<in> U + V"
then obtain u v where "x = u + v" and
"u \<in> U" and "v \<in> V" ..
then show "x \<in> E" by simp
qed
next
- fix x y assume x: "x \<in> U \<oplus> V" and y: "y \<in> U \<oplus> V"
- show "x + y \<in> U \<oplus> V"
+ fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V"
+ show "x + y \<in> U + V"
proof -
from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" ..
moreover
@@ -306,7 +306,7 @@
using x y by (simp_all add: add_ac)
then show ?thesis ..
qed
- fix a show "a \<cdot> x \<in> U \<oplus> V"
+ fix a show "a \<cdot> x \<in> U + V"
proof -
from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" ..
then have "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V"
@@ -319,7 +319,7 @@
text{* The sum of two subspaces is a vectorspace. *}
lemma sum_vs [intro?]:
- "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U \<oplus> V)"
+ "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)"
by (rule subspace.vectorspace) (rule sum_subspace)
@@ -481,7 +481,7 @@
proof -
interpret vectorspace E by fact
interpret subspace H E by fact
- from x y x' have "x \<in> H \<oplus> lin x'" by auto
+ from x y x' have "x \<in> H + lin x'" by auto
have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p")
proof (rule ex_ex1I)
from x y show "\<exists>p. ?P p" by blast
--- a/src/HOL/Library/BigO.thy Thu Apr 12 22:55:11 2012 +0200
+++ b/src/HOL/Library/BigO.thy Thu Apr 12 23:07:01 2012 +0200
@@ -92,7 +92,7 @@
by (auto simp add: bigo_def)
lemma bigo_plus_self_subset [intro]:
- "O(f) \<oplus> O(f) <= O(f)"
+ "O(f) + O(f) <= O(f)"
apply (auto simp add: bigo_alt_def set_plus_def)
apply (rule_tac x = "c + ca" in exI)
apply auto
@@ -104,14 +104,14 @@
apply force
done
-lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
+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) \<oplus> O(g)"
+lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
apply (rule subsetI)
apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
apply (subst bigo_pos_const [symmetric])+
@@ -153,15 +153,15 @@
apply simp
done
-lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A \<oplus> B <= O(f)"
- apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> O(f)")
+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) \<oplus> O(g)"
+ O(f + g) = O(f) + O(g)"
apply (rule equalityI)
apply (rule bigo_plus_subset)
apply (simp add: bigo_alt_def set_plus_def func_plus)
@@ -273,12 +273,12 @@
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) \<oplus> O(h)"
+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) \<oplus> O(h)"
+ also have "... <= O(g) + O(h)"
by (auto del: subsetI)
- also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
+ also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
apply (subst bigo_abs3 [symmetric])+
apply (rule refl)
done
@@ -287,13 +287,13 @@
finally have "f : ...".
then have "O(f) <= ..."
by (elim bigo_elt_subset)
- also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
+ 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)\<otimes>O(g) <= O(f * g)"
+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_def func_times)
@@ -369,7 +369,7 @@
done
lemma bigo_mult7: "ALL x. f x ~= 0 ==>
- O(f * g) <= O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
+ O(f * g) <= O(f::'a => ('b::linordered_field)) * O(g)"
apply (subst bigo_mult6)
apply assumption
apply (rule set_times_mono3)
@@ -377,7 +377,7 @@
done
lemma bigo_mult8: "ALL x. f x ~= 0 ==>
- O(f * g) = O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
+ O(f * g) = O(f::'a => ('b::linordered_field)) * O(g)"
apply (rule equalityI)
apply (erule bigo_mult7)
apply (rule bigo_mult)
@@ -402,9 +402,9 @@
show "f +o O(g) <= O(g)"
proof -
have "f : O(f)" by auto
- then have "f +o O(g) <= O(f) \<oplus> O(g)"
+ then have "f +o O(g) <= O(f) + O(g)"
by (auto del: subsetI)
- also have "... <= O(g) \<oplus> O(g)"
+ also have "... <= O(g) + O(g)"
proof -
from a have "O(f) <= O(g)" by (auto del: subsetI)
thus ?thesis by (auto del: subsetI)
@@ -656,7 +656,7 @@
subsection {* Misc useful stuff *}
lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
- A \<oplus> B <= O(f)"
+ A + B <= O(f)"
apply (subst bigo_plus_idemp [symmetric])
apply (rule set_plus_mono2)
apply assumption+
--- a/src/HOL/Library/Set_Algebras.thy Thu Apr 12 22:55:11 2012 +0200
+++ b/src/HOL/Library/Set_Algebras.thy Thu Apr 12 23:07:01 2012 +0200
@@ -34,14 +34,6 @@
end
-
-text {* Legacy syntax: *}
-
-abbreviation (input) set_plus :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<oplus>" 65) where
- "A \<oplus> B \<equiv> A + B"
-abbreviation (input) set_times :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<otimes>" 70) where
- "A \<otimes> B \<equiv> A * B"
-
instantiation set :: (zero) zero
begin
@@ -95,14 +87,14 @@
instance set :: (comm_monoid_mult) comm_monoid_mult
by default (simp_all add: set_times_def)
-lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C \<oplus> D"
+lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C + D"
by (auto simp add: set_plus_def)
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) \<oplus>
- (b +o D) = (a + b) +o (C \<oplus> D)"
+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_def add_ac)
apply (rule_tac x = "ba + bb" in exI)
apply (auto simp add: add_ac)
@@ -114,8 +106,8 @@
(a + b) +o C"
by (auto simp add: elt_set_plus_def add_assoc)
-lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) \<oplus> C =
- a +o (B \<oplus> C)"
+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_def)
apply (blast intro: add_ac)
apply (rule_tac x = "a + aa" in exI)
@@ -126,8 +118,8 @@
apply (auto simp add: add_ac)
done
-theorem set_plus_rearrange4: "C \<oplus> ((a::'a::comm_monoid_add) +o D) =
- a +o (C \<oplus> D)"
+theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) =
+ a +o (C + D)"
apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
apply (rule_tac x = "aa + ba" in exI)
apply (auto simp add: add_ac)
@@ -140,17 +132,17 @@
by (auto simp add: elt_set_plus_def)
lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>
- C \<oplus> E <= D \<oplus> F"
+ C + E <= D + F"
by (auto simp add: set_plus_def)
-lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C \<oplus> D"
+lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C + D"
by (auto simp add: elt_set_plus_def set_plus_def)
lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>
- a +o D <= D \<oplus> C"
+ a +o D <= D + C"
by (auto simp add: elt_set_plus_def set_plus_def add_ac)
-lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C \<oplus> D"
+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)
@@ -163,21 +155,21 @@
apply auto
done
-lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C \<oplus> E ==>
- x : D \<oplus> F"
+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 \<oplus> D"
+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 \<oplus> C"
+ x : a +o D ==> x : D + C"
apply (frule set_plus_mono4)
apply auto
done
@@ -185,7 +177,7 @@
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 \<oplus> B"
+lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B"
apply (auto simp add: set_plus_def)
apply (rule_tac x = 0 in bexI)
apply (rule_tac x = x in bexI)
@@ -206,14 +198,14 @@
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 \<otimes> D"
+lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C * D"
by (auto simp add: set_times_def)
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) \<otimes>
- (b *o D) = (a * b) *o (C \<otimes> D)"
+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_def)
apply (rule_tac x = "ba * bb" in exI)
apply (auto simp add: mult_ac)
@@ -225,8 +217,8 @@
(a * b) *o C"
by (auto simp add: elt_set_times_def mult_assoc)
-lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) \<otimes> C =
- a *o (B \<otimes> C)"
+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_def)
apply (blast intro: mult_ac)
apply (rule_tac x = "a * aa" in exI)
@@ -237,8 +229,8 @@
apply (auto simp add: mult_ac)
done
-theorem set_times_rearrange4: "C \<otimes> ((a::'a::comm_monoid_mult) *o D) =
- a *o (C \<otimes> D)"
+theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) =
+ a *o (C * D)"
apply (auto simp add: elt_set_times_def set_times_def
mult_ac)
apply (rule_tac x = "aa * ba" in exI)
@@ -252,17 +244,17 @@
by (auto simp add: elt_set_times_def)
lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
- C \<otimes> E <= D \<otimes> F"
+ C * E <= D * F"
by (auto simp add: set_times_def)
-lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C \<otimes> D"
+lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D"
by (auto simp add: elt_set_times_def set_times_def)
lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
- a *o D <= D \<otimes> C"
+ a *o D <= D * C"
by (auto simp add: elt_set_times_def set_times_def mult_ac)
-lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C \<otimes> D"
+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)
@@ -275,21 +267,21 @@
apply auto
done
-lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C \<otimes> E ==>
- x : D \<otimes> F"
+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 \<otimes> D"
+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 \<otimes> C"
+ x : a *o D ==> x : D * C"
apply (frule set_times_mono4)
apply auto
done
@@ -301,16 +293,16 @@
(a * b) +o (a *o C)"
by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
-lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B \<oplus> C) =
- (a *o B) \<oplus> (a *o C)"
+lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) =
+ (a *o B) + (a *o C)"
apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
apply blast
apply (rule_tac x = "b + bb" in exI)
apply (auto simp add: ring_distribs)
done
-lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) \<otimes> D <=
- a *o D \<oplus> C \<otimes> D"
+lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <=
+ a *o D + C * D"
apply (auto simp add:
elt_set_plus_def elt_set_times_def set_times_def
set_plus_def ring_distribs)
@@ -330,7 +322,7 @@
by (auto simp add: elt_set_times_def)
lemma set_plus_image:
- fixes S T :: "'n::semigroup_add set" shows "S \<oplus> T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
+ fixes S T :: "'n::semigroup_add set" shows "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
unfolding set_plus_def by (fastforce simp: image_iff)
lemma set_setsum_alt:
@@ -339,7 +331,7 @@
(is "_ = ?setsum I")
using fin proof induct
case (insert x F)
- have "setsum S (insert x F) = S x \<oplus> ?setsum F"
+ have "setsum S (insert x F) = S x + ?setsum F"
using insert.hyps by auto
also have "...= {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
unfolding set_plus_def
@@ -355,8 +347,8 @@
lemma setsum_set_cond_linear:
fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"
- assumes [intro!]: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> P (A \<oplus> B)" "P {0}"
- and f: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> f (A \<oplus> B) = f A \<oplus> f B" "f {0} = {0}"
+ assumes [intro!]: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> P (A + B)" "P {0}"
+ and f: "\<And>A B. P A \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}"
assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
shows "f (setsum S I) = setsum (f \<circ> S) I"
proof cases
@@ -372,7 +364,7 @@
lemma setsum_set_linear:
fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"
- assumes "\<And>A B. f(A) \<oplus> f(B) = f(A \<oplus> B)" "f {0} = {0}"
+ assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"
shows "f (setsum S I) = setsum (f \<circ> S) I"
using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
--- a/src/HOL/Metis_Examples/Big_O.thy Thu Apr 12 22:55:11 2012 +0200
+++ b/src/HOL/Metis_Examples/Big_O.thy Thu Apr 12 23:07:01 2012 +0200
@@ -146,17 +146,17 @@
by (auto simp add: bigo_def)
lemma bigo_plus_self_subset [intro]:
- "O(f) \<oplus> O(f) <= O(f)"
+ "O(f) + O(f) <= O(f)"
apply (auto simp add: bigo_alt_def set_plus_def)
apply (rule_tac x = "c + ca" in exI)
apply auto
apply (simp add: ring_distribs func_plus)
by (metis order_trans abs_triangle_ineq add_mono)
-lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
+lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
by (metis bigo_plus_self_subset bigo_zero set_eq_subset set_zero_plus2)
-lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) \<oplus> O(g)"
+lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
apply (rule subsetI)
apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
apply (subst bigo_pos_const [symmetric])+
@@ -187,10 +187,10 @@
apply (metis abs_triangle_ineq mult_le_cancel_left_pos)
by (metis abs_ge_zero abs_of_pos zero_le_mult_iff)
-lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A \<oplus> B <= O(f)"
+lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A + B <= O(f)"
by (metis bigo_plus_idemp set_plus_mono2)
-lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) \<oplus> O(g)"
+lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
apply (rule equalityI)
apply (rule bigo_plus_subset)
apply (simp add: bigo_alt_def set_plus_def func_plus)
@@ -284,25 +284,25 @@
lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. abs(f x)) =o O(g)"
by (unfold bigo_def, auto)
-lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) \<oplus> O(h)"
+lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) + O(h)"
proof -
assume "f : g +o O(h)"
- also have "... <= O(g) \<oplus> O(h)"
+ also have "... <= O(g) + O(h)"
by (auto del: subsetI)
- also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))"
+ also have "... = O(\<lambda>x. abs(g x)) + O(\<lambda>x. abs(h x))"
by (metis bigo_abs3)
also have "... = O((\<lambda>x. abs(g x)) + (\<lambda>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(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))"
+ also have "... = O(\<lambda>x. abs(g x)) + O(\<lambda>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) \<otimes> O(g) <= O(f * g)"
+lemma bigo_mult [intro]: "O(f) * O(g) <= O(f * g)"
apply (rule subsetI)
apply (subst bigo_def)
apply (auto simp del: abs_mult mult_ac
@@ -358,14 +358,14 @@
declare bigo_mult6 [simp]
lemma bigo_mult7:
-"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<le> O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) \<otimes> O(g)"
+"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<le> O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) * O(g)"
by (metis bigo_refl bigo_mult6 set_times_mono3)
declare bigo_mult6 [simp del]
declare bigo_mult7 [intro!]
lemma bigo_mult8:
-"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) \<otimes> O(g)"
+"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) * O(g)"
by (metis bigo_mult bigo_mult7 order_antisym_conv)
lemma bigo_minus [intro]: "f : O(g) \<Longrightarrow> - f : O(g)"
@@ -575,7 +575,7 @@
subsection {* Misc useful stuff *}
lemma bigo_useful_intro: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow>
- A \<oplus> B <= O(f)"
+ A + B <= O(f)"
apply (subst bigo_plus_idemp [symmetric])
apply (rule set_plus_mono2)
apply assumption+
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Thu Apr 12 22:55:11 2012 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Thu Apr 12 23:07:01 2012 +0200
@@ -5428,13 +5428,13 @@
lemma closure_sum:
fixes S T :: "('n::euclidean_space) set"
- shows "closure S \<oplus> closure T \<subseteq> closure (S \<oplus> T)"
+ shows "closure S + closure T \<subseteq> closure (S + T)"
proof-
- have "(closure S) \<oplus> (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
+ have "(closure S) + (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
by (simp add: set_plus_image)
also have "... = (\<lambda>(x,y). x + y) ` closure (S \<times> T)"
using closure_direct_sum by auto
- also have "... \<subseteq> closure (S \<oplus> T)"
+ also have "... \<subseteq> closure (S + T)"
using fst_snd_linear closure_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
by (auto simp: set_plus_image)
finally show ?thesis
@@ -5444,7 +5444,7 @@
lemma convex_oplus:
fixes S T :: "('n::euclidean_space) set"
assumes "convex S" "convex T"
-shows "convex (S \<oplus> T)"
+shows "convex (S + T)"
proof-
have "{x + y |x y. x : S & y : T} = {c. EX a:S. EX b:T. c = a + b}" by auto
thus ?thesis unfolding set_plus_def using convex_sums[of S T] assms by auto
@@ -5452,13 +5452,13 @@
lemma convex_hull_sum:
fixes S T :: "('n::euclidean_space) set"
-shows "convex hull (S \<oplus> T) = (convex hull S) \<oplus> (convex hull T)"
+shows "convex hull (S + T) = (convex hull S) + (convex hull T)"
proof-
-have "(convex hull S) \<oplus> (convex hull T) =
+have "(convex hull S) + (convex hull T) =
(%(x,y). x + y) ` ((convex hull S) <*> (convex hull T))"
by (simp add: set_plus_image)
also have "... = (%(x,y). x + y) ` (convex hull (S <*> T))" using convex_hull_direct_sum by auto
-also have "...= convex hull (S \<oplus> T)" using fst_snd_linear linear_conv_bounded_linear
+also have "...= convex hull (S + T)" using fst_snd_linear linear_conv_bounded_linear
convex_hull_linear_image[of "(%(x,y). x + y)" "S <*> T"] by (auto simp add: set_plus_image)
finally show ?thesis by auto
qed
@@ -5466,12 +5466,12 @@
lemma rel_interior_sum:
fixes S T :: "('n::euclidean_space) set"
assumes "convex S" "convex T"
-shows "rel_interior (S \<oplus> T) = (rel_interior S) \<oplus> (rel_interior T)"
+shows "rel_interior (S + T) = (rel_interior S) + (rel_interior T)"
proof-
-have "(rel_interior S) \<oplus> (rel_interior T) = (%(x,y). x + y) ` (rel_interior S <*> rel_interior T)"
+have "(rel_interior S) + (rel_interior T) = (%(x,y). x + y) ` (rel_interior S <*> rel_interior T)"
by (simp add: set_plus_image)
also have "... = (%(x,y). x + y) ` rel_interior (S <*> T)" using rel_interior_direct_sum assms by auto
-also have "...= rel_interior (S \<oplus> T)" using fst_snd_linear convex_direct_sum assms
+also have "...= rel_interior (S + T)" using fst_snd_linear convex_direct_sum assms
rel_interior_convex_linear_image[of "(%(x,y). x + y)" "S <*> T"] by (auto simp add: set_plus_image)
finally show ?thesis by auto
qed
@@ -5507,7 +5507,7 @@
lemma convex_rel_open_sum:
fixes S T :: "('n::euclidean_space) set"
assumes "convex S" "rel_open S" "convex T" "rel_open T"
-shows "convex (S \<oplus> T) & rel_open (S \<oplus> T)"
+shows "convex (S + T) & rel_open (S + T)"
by (metis assms convex_oplus rel_interior_sum rel_open_def)
lemma convex_hull_finite_union_cones:
@@ -5547,7 +5547,7 @@
fixes S T :: "('m::euclidean_space) set"
assumes "convex S" "cone S" "S ~= {}"
assumes "convex T" "cone T" "T ~= {}"
-shows "convex hull (S Un T) = S \<oplus> T"
+shows "convex hull (S Un T) = S + T"
proof-
def I == "{(1::nat),2}"
def A == "(%i. (if i=(1::nat) then S else T))"
@@ -5556,7 +5556,7 @@
moreover have "convex hull Union (A ` I) = setsum A I"
apply (subst convex_hull_finite_union_cones[of I A]) using assms A_def I_def by auto
moreover have
- "setsum A I = S \<oplus> T" using A_def I_def
+ "setsum A I = S + T" using A_def I_def
unfolding set_plus_def apply auto unfolding set_plus_def by auto
ultimately show ?thesis by auto
qed