--- a/src/HOL/Hahn_Banach/Bounds.thy Sun Sep 11 13:49:42 2011 -0700
+++ b/src/HOL/Hahn_Banach/Bounds.thy Sun Sep 11 22:56:05 2011 +0200
@@ -15,9 +15,8 @@
lemmas [elim?] = lub.least lub.upper
-definition
- the_lub :: "'a::order set \<Rightarrow> 'a" where
- "the_lub A = The (lub A)"
+definition the_lub :: "'a::order set \<Rightarrow> 'a"
+ where "the_lub A = The (lub A)"
notation (xsymbols)
the_lub ("\<Squnion>_" [90] 90)
--- a/src/HOL/Hahn_Banach/Function_Norm.thy Sun Sep 11 13:49:42 2011 -0700
+++ b/src/HOL/Hahn_Banach/Function_Norm.thy Sun Sep 11 22:56:05 2011 +0200
@@ -143,7 +143,7 @@
also
from gt have "\<parallel>x\<parallel> \<noteq> 0" by simp
then have "\<parallel>x\<parallel> * inverse \<parallel>x\<parallel> = 1" by simp
- also have "c * 1 \<le> b" by (simp add: b_def le_maxI1)
+ also have "c * 1 \<le> b" by (simp add: b_def)
finally show "y \<le> b" .
qed
qed
--- a/src/HOL/Hahn_Banach/Function_Order.thy Sun Sep 11 13:49:42 2011 -0700
+++ b/src/HOL/Hahn_Banach/Function_Order.thy Sun Sep 11 22:56:05 2011 +0200
@@ -23,9 +23,8 @@
type_synonym 'a graph = "('a \<times> real) set"
-definition
- graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph" where
- "graph F f = {(x, f x) | x. x \<in> F}"
+definition graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph"
+ where "graph F f = {(x, f x) | x. x \<in> F}"
lemma graphI [intro]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"
unfolding graph_def by blast
@@ -34,8 +33,9 @@
unfolding graph_def by blast
lemma graphE [elim?]:
- "(x, y) \<in> graph F f \<Longrightarrow> (x \<in> F \<Longrightarrow> y = f x \<Longrightarrow> C) \<Longrightarrow> C"
- unfolding graph_def by blast
+ assumes "(x, y) \<in> graph F f"
+ obtains "x \<in> F" and "y = f x"
+ using assms unfolding graph_def by blast
subsection {* Functions ordered by domain extension *}
@@ -50,12 +50,10 @@
\<Longrightarrow> graph H h \<subseteq> graph H' h'"
unfolding graph_def by blast
-lemma graph_extD1 [dest?]:
- "graph H h \<subseteq> graph H' h' \<Longrightarrow> x \<in> H \<Longrightarrow> h x = h' x"
+lemma graph_extD1 [dest?]: "graph H h \<subseteq> graph H' h' \<Longrightarrow> x \<in> H \<Longrightarrow> h x = h' x"
unfolding graph_def by blast
-lemma graph_extD2 [dest?]:
- "graph H h \<subseteq> graph H' h' \<Longrightarrow> H \<subseteq> H'"
+lemma graph_extD2 [dest?]: "graph H h \<subseteq> graph H' h' \<Longrightarrow> H \<subseteq> H'"
unfolding graph_def by blast
@@ -66,13 +64,11 @@
funct}.
*}
-definition
- "domain" :: "'a graph \<Rightarrow> 'a set" where
- "domain g = {x. \<exists>y. (x, y) \<in> g}"
+definition domain :: "'a graph \<Rightarrow> 'a set"
+ where "domain g = {x. \<exists>y. (x, y) \<in> g}"
-definition
- funct :: "'a graph \<Rightarrow> ('a \<Rightarrow> real)" where
- "funct g = (\<lambda>x. (SOME y. (x, y) \<in> g))"
+definition funct :: "'a graph \<Rightarrow> ('a \<Rightarrow> real)"
+ where "funct g = (\<lambda>x. (SOME y. (x, y) \<in> g))"
text {*
The following lemma states that @{text g} is the graph of a function
@@ -107,21 +103,26 @@
definition
norm_pres_extensions ::
"'a::{plus, minus, uminus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real)
- \<Rightarrow> 'a graph set" where
- "norm_pres_extensions E p F f
- = {g. \<exists>H h. g = graph H h
- \<and> linearform H h
- \<and> H \<unlhd> E
- \<and> F \<unlhd> H
- \<and> graph F f \<subseteq> graph H h
- \<and> (\<forall>x \<in> H. h x \<le> p x)}"
+ \<Rightarrow> 'a graph set"
+where
+ "norm_pres_extensions E p F f
+ = {g. \<exists>H h. g = graph H h
+ \<and> linearform H h
+ \<and> H \<unlhd> E
+ \<and> F \<unlhd> H
+ \<and> graph F f \<subseteq> graph H h
+ \<and> (\<forall>x \<in> H. h x \<le> p x)}"
lemma norm_pres_extensionE [elim]:
- "g \<in> norm_pres_extensions E p F f
- \<Longrightarrow> (\<And>H h. g = graph H h \<Longrightarrow> linearform H h
- \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H \<Longrightarrow> graph F f \<subseteq> graph H h
- \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x \<Longrightarrow> C) \<Longrightarrow> C"
- unfolding norm_pres_extensions_def by blast
+ assumes "g \<in> norm_pres_extensions E p F f"
+ obtains H h
+ where "g = graph H h"
+ and "linearform H h"
+ and "H \<unlhd> E"
+ and "F \<unlhd> H"
+ and "graph F f \<subseteq> graph H h"
+ and "\<forall>x \<in> H. h x \<le> p x"
+ using assms unfolding norm_pres_extensions_def by blast
lemma norm_pres_extensionI2 [intro]:
"linearform H h \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H
--- a/src/HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy Sun Sep 11 13:49:42 2011 -0700
+++ b/src/HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy Sun Sep 11 22:56:05 2011 +0200
@@ -334,7 +334,7 @@
proof
fix x assume "x \<in> H"
with M cM graph
- obtain H' h' where x: "x \<in> H'" and H'E: "H' \<unlhd> E"
+ obtain H' where x: "x \<in> H'" and H'E: "H' \<unlhd> E"
by (rule some_H'h' [elim_format]) blast
from H'E have "H' \<subseteq> E" ..
with x show "x \<in> E" ..
--- a/src/HOL/Hahn_Banach/Normed_Space.thy Sun Sep 11 13:49:42 2011 -0700
+++ b/src/HOL/Hahn_Banach/Normed_Space.thy Sun Sep 11 22:56:05 2011 +0200
@@ -50,8 +50,7 @@
interpret vectorspace V by fact
assume x: "x \<in> V"
then have "- x = - 1 \<cdot> x" by (simp only: negate_eq1)
- also from x have "\<parallel>\<dots>\<parallel> = \<bar>- 1\<bar> * \<parallel>x\<parallel>"
- by (rule abs_homogenous)
+ also from x have "\<parallel>\<dots>\<parallel> = \<bar>- 1\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous)
also have "\<dots> = \<parallel>x\<parallel>" by simp
finally show ?thesis .
qed
@@ -80,13 +79,13 @@
declare normed_vectorspace.intro [intro?]
lemma (in normed_vectorspace) gt_zero [intro?]:
- "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> 0 < \<parallel>x\<parallel>"
+ assumes x: "x \<in> V" and neq: "x \<noteq> 0"
+ shows "0 < \<parallel>x\<parallel>"
proof -
- assume x: "x \<in> V" and neq: "x \<noteq> 0"
from x have "0 \<le> \<parallel>x\<parallel>" ..
- also have [symmetric]: "\<dots> \<noteq> 0"
+ also have "0 \<noteq> \<parallel>x\<parallel>"
proof
- assume "\<parallel>x\<parallel> = 0"
+ assume "0 = \<parallel>x\<parallel>"
with x have "x = 0" by simp
with neq show False by contradiction
qed
--- a/src/HOL/Hahn_Banach/Subspace.thy Sun Sep 11 13:49:42 2011 -0700
+++ b/src/HOL/Hahn_Banach/Subspace.thy Sun Sep 11 22:56:05 2011 +0200
@@ -110,6 +110,7 @@
proof
show "V \<noteq> {}" ..
show "V \<subseteq> V" ..
+next
fix x y assume x: "x \<in> V" and y: "y \<in> V"
fix a :: real
from x y show "x + y \<in> V" by simp
@@ -142,9 +143,8 @@
scalar multiples of @{text x}.
*}
-definition
- lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set" where
- "lin x = {a \<cdot> x | a. True}"
+definition lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set"
+ where "lin x = {a \<cdot> x | a. True}"
lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x"
unfolding lin_def by blast
@@ -175,16 +175,18 @@
text {* Any linear closure is a subspace. *}
lemma (in vectorspace) lin_subspace [intro]:
- "x \<in> V \<Longrightarrow> lin x \<unlhd> V"
+ assumes x: "x \<in> V"
+ shows "lin x \<unlhd> V"
proof
- assume x: "x \<in> V"
- then show "lin x \<noteq> {}" by (auto simp add: x_lin_x)
+ from x show "lin x \<noteq> {}" by auto
+next
show "lin x \<subseteq> V"
proof
fix x' assume "x' \<in> lin x"
then obtain a where "x' = a \<cdot> x" ..
with x show "x' \<in> V" by simp
qed
+next
fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x"
show "x' + x'' \<in> lin x"
proof -
@@ -290,6 +292,7 @@
"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"
proof -
@@ -467,8 +470,9 @@
lemma h'_definite:
fixes H
assumes h'_def:
- "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
- in (h y) + a * xi)"
+ "h' \<equiv> \<lambda>x.
+ let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
+ in (h y) + a * xi"
and x: "x = y + a \<cdot> x'"
assumes "vectorspace E" "subspace H E"
assumes y: "y \<in> H"
--- a/src/HOL/Hahn_Banach/Vector_Space.thy Sun Sep 11 13:49:42 2011 -0700
+++ b/src/HOL/Hahn_Banach/Vector_Space.thy Sun Sep 11 22:56:05 2011 +0200
@@ -5,7 +5,7 @@
header {* Vector spaces *}
theory Vector_Space
-imports Real Bounds "~~/src/HOL/Library/Zorn"
+imports Complex_Main Bounds "~~/src/HOL/Library/Zorn"
begin
subsection {* Signature *}
@@ -54,24 +54,24 @@
and mult_1 [simp]: "x \<in> V \<Longrightarrow> 1 \<cdot> x = x"
and negate_eq1: "x \<in> V \<Longrightarrow> - x = (- 1) \<cdot> x"
and diff_eq1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = x + - y"
+begin
-lemma (in vectorspace) negate_eq2: "x \<in> V \<Longrightarrow> (- 1) \<cdot> x = - x"
+lemma negate_eq2: "x \<in> V \<Longrightarrow> (- 1) \<cdot> x = - x"
by (rule negate_eq1 [symmetric])
-lemma (in vectorspace) negate_eq2a: "x \<in> V \<Longrightarrow> -1 \<cdot> x = - x"
+lemma negate_eq2a: "x \<in> V \<Longrightarrow> -1 \<cdot> x = - x"
by (simp add: negate_eq1)
-lemma (in vectorspace) diff_eq2: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + - y = x - y"
+lemma diff_eq2: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + - y = x - y"
by (rule diff_eq1 [symmetric])
-lemma (in vectorspace) diff_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y \<in> V"
+lemma diff_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y \<in> V"
by (simp add: diff_eq1 negate_eq1)
-lemma (in vectorspace) neg_closed [iff]: "x \<in> V \<Longrightarrow> - x \<in> V"
+lemma neg_closed [iff]: "x \<in> V \<Longrightarrow> - x \<in> V"
by (simp add: negate_eq1)
-lemma (in vectorspace) add_left_commute:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> x + (y + z) = y + (x + z)"
+lemma add_left_commute: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> x + (y + z) = y + (x + z)"
proof -
assume xyz: "x \<in> V" "y \<in> V" "z \<in> V"
then have "x + (y + z) = (x + y) + z"
@@ -81,14 +81,13 @@
finally show ?thesis .
qed
-theorems (in vectorspace) add_ac =
- add_assoc add_commute add_left_commute
+theorems add_ac = add_assoc add_commute add_left_commute
text {* The existence of the zero element of a vector space
follows from the non-emptiness of carrier set. *}
-lemma (in vectorspace) zero [iff]: "0 \<in> V"
+lemma zero [iff]: "0 \<in> V"
proof -
from non_empty obtain x where x: "x \<in> V" by blast
then have "0 = x - x" by (rule diff_self [symmetric])
@@ -96,8 +95,7 @@
finally show ?thesis .
qed
-lemma (in vectorspace) add_zero_right [simp]:
- "x \<in> V \<Longrightarrow> x + 0 = x"
+lemma add_zero_right [simp]: "x \<in> V \<Longrightarrow> x + 0 = x"
proof -
assume x: "x \<in> V"
from this and zero have "x + 0 = 0 + x" by (rule add_commute)
@@ -105,16 +103,13 @@
finally show ?thesis .
qed
-lemma (in vectorspace) mult_assoc2:
- "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
+lemma mult_assoc2: "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
by (simp only: mult_assoc)
-lemma (in vectorspace) diff_mult_distrib1:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"
+lemma diff_mult_distrib1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"
by (simp add: diff_eq1 negate_eq1 add_mult_distrib1 mult_assoc2)
-lemma (in vectorspace) diff_mult_distrib2:
- "x \<in> V \<Longrightarrow> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"
+lemma diff_mult_distrib2: "x \<in> V \<Longrightarrow> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"
proof -
assume x: "x \<in> V"
have " (a - b) \<cdot> x = (a + - b) \<cdot> x"
@@ -128,15 +123,14 @@
finally show ?thesis .
qed
-lemmas (in vectorspace) distrib =
+lemmas distrib =
add_mult_distrib1 add_mult_distrib2
diff_mult_distrib1 diff_mult_distrib2
text {* \medskip Further derived laws: *}
-lemma (in vectorspace) mult_zero_left [simp]:
- "x \<in> V \<Longrightarrow> 0 \<cdot> x = 0"
+lemma mult_zero_left [simp]: "x \<in> V \<Longrightarrow> 0 \<cdot> x = 0"
proof -
assume x: "x \<in> V"
have "0 \<cdot> x = (1 - 1) \<cdot> x" by simp
@@ -150,8 +144,7 @@
finally show ?thesis .
qed
-lemma (in vectorspace) mult_zero_right [simp]:
- "a \<cdot> 0 = (0::'a)"
+lemma mult_zero_right [simp]: "a \<cdot> 0 = (0::'a)"
proof -
have "a \<cdot> 0 = a \<cdot> (0 - (0::'a))" by simp
also have "\<dots> = a \<cdot> 0 - a \<cdot> 0"
@@ -160,12 +153,10 @@
finally show ?thesis .
qed
-lemma (in vectorspace) minus_mult_cancel [simp]:
- "x \<in> V \<Longrightarrow> (- a) \<cdot> - x = a \<cdot> x"
+lemma minus_mult_cancel [simp]: "x \<in> V \<Longrightarrow> (- a) \<cdot> - x = a \<cdot> x"
by (simp add: negate_eq1 mult_assoc2)
-lemma (in vectorspace) add_minus_left_eq_diff:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + y = y - x"
+lemma add_minus_left_eq_diff: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + y = y - x"
proof -
assume xy: "x \<in> V" "y \<in> V"
then have "- x + y = y + - x" by (simp add: add_commute)
@@ -173,95 +164,78 @@
finally show ?thesis .
qed
-lemma (in vectorspace) add_minus [simp]:
- "x \<in> V \<Longrightarrow> x + - x = 0"
+lemma add_minus [simp]: "x \<in> V \<Longrightarrow> x + - x = 0"
by (simp add: diff_eq2)
-lemma (in vectorspace) add_minus_left [simp]:
- "x \<in> V \<Longrightarrow> - x + x = 0"
+lemma add_minus_left [simp]: "x \<in> V \<Longrightarrow> - x + x = 0"
by (simp add: diff_eq2 add_commute)
-lemma (in vectorspace) minus_minus [simp]:
- "x \<in> V \<Longrightarrow> - (- x) = x"
+lemma minus_minus [simp]: "x \<in> V \<Longrightarrow> - (- x) = x"
by (simp add: negate_eq1 mult_assoc2)
-lemma (in vectorspace) minus_zero [simp]:
- "- (0::'a) = 0"
+lemma minus_zero [simp]: "- (0::'a) = 0"
by (simp add: negate_eq1)
-lemma (in vectorspace) minus_zero_iff [simp]:
- "x \<in> V \<Longrightarrow> (- x = 0) = (x = 0)"
+lemma minus_zero_iff [simp]:
+ assumes x: "x \<in> V"
+ shows "(- x = 0) = (x = 0)"
proof
- assume x: "x \<in> V"
- {
- from x have "x = - (- x)" by (simp add: minus_minus)
- also assume "- x = 0"
- also have "- \<dots> = 0" by (rule minus_zero)
- finally show "x = 0" .
- next
- assume "x = 0"
- then show "- x = 0" by simp
- }
+ from x have "x = - (- x)" by simp
+ also assume "- x = 0"
+ also have "- \<dots> = 0" by (rule minus_zero)
+ finally show "x = 0" .
+next
+ assume "x = 0"
+ then show "- x = 0" by simp
qed
-lemma (in vectorspace) add_minus_cancel [simp]:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + (- x + y) = y"
- by (simp add: add_assoc [symmetric] del: add_commute)
+lemma add_minus_cancel [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + (- x + y) = y"
+ by (simp add: add_assoc [symmetric])
-lemma (in vectorspace) minus_add_cancel [simp]:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + (x + y) = y"
- by (simp add: add_assoc [symmetric] del: add_commute)
+lemma minus_add_cancel [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + (x + y) = y"
+ by (simp add: add_assoc [symmetric])
-lemma (in vectorspace) minus_add_distrib [simp]:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - (x + y) = - x + - y"
+lemma minus_add_distrib [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - (x + y) = - x + - y"
by (simp add: negate_eq1 add_mult_distrib1)
-lemma (in vectorspace) diff_zero [simp]:
- "x \<in> V \<Longrightarrow> x - 0 = x"
+lemma diff_zero [simp]: "x \<in> V \<Longrightarrow> x - 0 = x"
by (simp add: diff_eq1)
-lemma (in vectorspace) diff_zero_right [simp]:
- "x \<in> V \<Longrightarrow> 0 - x = - x"
+lemma diff_zero_right [simp]: "x \<in> V \<Longrightarrow> 0 - x = - x"
by (simp add: diff_eq1)
-lemma (in vectorspace) add_left_cancel:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + y = x + z) = (y = z)"
+lemma add_left_cancel:
+ assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
+ shows "(x + y = x + z) = (y = z)"
proof
- assume x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
- {
- from y have "y = 0 + y" by simp
- also from x y have "\<dots> = (- x + x) + y" by simp
- also from x y have "\<dots> = - x + (x + y)"
- by (simp add: add_assoc neg_closed)
- also assume "x + y = x + z"
- also from x z have "- x + (x + z) = - x + x + z"
- by (simp add: add_assoc [symmetric] neg_closed)
- also from x z have "\<dots> = z" by simp
- finally show "y = z" .
- next
- assume "y = z"
- then show "x + y = x + z" by (simp only:)
- }
+ from y have "y = 0 + y" by simp
+ also from x y have "\<dots> = (- x + x) + y" by simp
+ also from x y have "\<dots> = - x + (x + y)" by (simp add: add_assoc)
+ also assume "x + y = x + z"
+ also from x z have "- x + (x + z) = - x + x + z" by (simp add: add_assoc)
+ also from x z have "\<dots> = z" by simp
+ finally show "y = z" .
+next
+ assume "y = z"
+ then show "x + y = x + z" by (simp only:)
qed
-lemma (in vectorspace) add_right_cancel:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (y + x = z + x) = (y = z)"
+lemma add_right_cancel: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (y + x = z + x) = (y = z)"
by (simp only: add_commute add_left_cancel)
-lemma (in vectorspace) add_assoc_cong:
+lemma add_assoc_cong:
"x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x' \<in> V \<Longrightarrow> y' \<in> V \<Longrightarrow> z \<in> V
\<Longrightarrow> x + y = x' + y' \<Longrightarrow> x + (y + z) = x' + (y' + z)"
by (simp only: add_assoc [symmetric])
-lemma (in vectorspace) mult_left_commute:
- "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
+lemma mult_left_commute: "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
by (simp add: mult_commute mult_assoc2)
-lemma (in vectorspace) mult_zero_uniq:
- "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> a \<cdot> x = 0 \<Longrightarrow> a = 0"
+lemma mult_zero_uniq:
+ assumes x: "x \<in> V" "x \<noteq> 0" and ax: "a \<cdot> x = 0"
+ shows "a = 0"
proof (rule classical)
assume a: "a \<noteq> 0"
- assume x: "x \<in> V" "x \<noteq> 0" and ax: "a \<cdot> x = 0"
from x a have "x = (inverse a * a) \<cdot> x" by simp
also from `x \<in> V` have "\<dots> = inverse a \<cdot> (a \<cdot> x)" by (rule mult_assoc)
also from ax have "\<dots> = inverse a \<cdot> 0" by simp
@@ -270,10 +244,10 @@
with `x \<noteq> 0` show "a = 0" by contradiction
qed
-lemma (in vectorspace) mult_left_cancel:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (a \<cdot> x = a \<cdot> y) = (x = y)"
+lemma mult_left_cancel:
+ assumes x: "x \<in> V" and y: "y \<in> V" and a: "a \<noteq> 0"
+ shows "(a \<cdot> x = a \<cdot> y) = (x = y)"
proof
- assume x: "x \<in> V" and y: "y \<in> V" and a: "a \<noteq> 0"
from x have "x = 1 \<cdot> x" by simp
also from a have "\<dots> = (inverse a * a) \<cdot> x" by simp
also from x have "\<dots> = inverse a \<cdot> (a \<cdot> x)"
@@ -287,81 +261,75 @@
then show "a \<cdot> x = a \<cdot> y" by (simp only:)
qed
-lemma (in vectorspace) mult_right_cancel:
- "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> (a \<cdot> x = b \<cdot> x) = (a = b)"
+lemma mult_right_cancel:
+ assumes x: "x \<in> V" and neq: "x \<noteq> 0"
+ shows "(a \<cdot> x = b \<cdot> x) = (a = b)"
proof
- assume x: "x \<in> V" and neq: "x \<noteq> 0"
- {
- from x have "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
- by (simp add: diff_mult_distrib2)
- also assume "a \<cdot> x = b \<cdot> x"
- with x have "a \<cdot> x - b \<cdot> x = 0" by simp
- finally have "(a - b) \<cdot> x = 0" .
- with x neq have "a - b = 0" by (rule mult_zero_uniq)
- then show "a = b" by simp
- next
- assume "a = b"
- then show "a \<cdot> x = b \<cdot> x" by (simp only:)
- }
+ from x have "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
+ by (simp add: diff_mult_distrib2)
+ also assume "a \<cdot> x = b \<cdot> x"
+ with x have "a \<cdot> x - b \<cdot> x = 0" by simp
+ finally have "(a - b) \<cdot> x = 0" .
+ with x neq have "a - b = 0" by (rule mult_zero_uniq)
+ then show "a = b" by simp
+next
+ assume "a = b"
+ then show "a \<cdot> x = b \<cdot> x" by (simp only:)
qed
-lemma (in vectorspace) eq_diff_eq:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x = z - y) = (x + y = z)"
+lemma eq_diff_eq:
+ assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
+ shows "(x = z - y) = (x + y = z)"
proof
- assume x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
- {
- assume "x = z - y"
- then have "x + y = z - y + y" by simp
- also from y z have "\<dots> = z + - y + y"
- by (simp add: diff_eq1)
- also have "\<dots> = z + (- y + y)"
- by (rule add_assoc) (simp_all add: y z)
- also from y z have "\<dots> = z + 0"
- by (simp only: add_minus_left)
- also from z have "\<dots> = z"
- by (simp only: add_zero_right)
- finally show "x + y = z" .
- next
- assume "x + y = z"
- then have "z - y = (x + y) - y" by simp
- also from x y have "\<dots> = x + y + - y"
- by (simp add: diff_eq1)
- also have "\<dots> = x + (y + - y)"
- by (rule add_assoc) (simp_all add: x y)
- also from x y have "\<dots> = x" by simp
- finally show "x = z - y" ..
- }
+ assume "x = z - y"
+ then have "x + y = z - y + y" by simp
+ also from y z have "\<dots> = z + - y + y"
+ by (simp add: diff_eq1)
+ also have "\<dots> = z + (- y + y)"
+ by (rule add_assoc) (simp_all add: y z)
+ also from y z have "\<dots> = z + 0"
+ by (simp only: add_minus_left)
+ also from z have "\<dots> = z"
+ by (simp only: add_zero_right)
+ finally show "x + y = z" .
+next
+ assume "x + y = z"
+ then have "z - y = (x + y) - y" by simp
+ also from x y have "\<dots> = x + y + - y"
+ by (simp add: diff_eq1)
+ also have "\<dots> = x + (y + - y)"
+ by (rule add_assoc) (simp_all add: x y)
+ also from x y have "\<dots> = x" by simp
+ finally show "x = z - y" ..
qed
-lemma (in vectorspace) add_minus_eq_minus:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = 0 \<Longrightarrow> x = - y"
+lemma add_minus_eq_minus:
+ assumes x: "x \<in> V" and y: "y \<in> V" and xy: "x + y = 0"
+ shows "x = - y"
proof -
- assume x: "x \<in> V" and y: "y \<in> V"
from x y have "x = (- y + y) + x" by simp
also from x y have "\<dots> = - y + (x + y)" by (simp add: add_ac)
- also assume "x + y = 0"
+ also note xy
also from y have "- y + 0 = - y" by simp
finally show "x = - y" .
qed
-lemma (in vectorspace) add_minus_eq:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = 0 \<Longrightarrow> x = y"
+lemma add_minus_eq:
+ assumes x: "x \<in> V" and y: "y \<in> V" and xy: "x - y = 0"
+ shows "x = y"
proof -
- assume x: "x \<in> V" and y: "y \<in> V"
- assume "x - y = 0"
- with x y have eq: "x + - y = 0" by (simp add: diff_eq1)
+ from x y xy have eq: "x + - y = 0" by (simp add: diff_eq1)
with _ _ have "x = - (- y)"
by (rule add_minus_eq_minus) (simp_all add: x y)
with x y show "x = y" by simp
qed
-lemma (in vectorspace) add_diff_swap:
- "a \<in> V \<Longrightarrow> b \<in> V \<Longrightarrow> c \<in> V \<Longrightarrow> d \<in> V \<Longrightarrow> a + b = c + d
- \<Longrightarrow> a - c = d - b"
+lemma add_diff_swap:
+ assumes vs: "a \<in> V" "b \<in> V" "c \<in> V" "d \<in> V"
+ and eq: "a + b = c + d"
+ shows "a - c = d - b"
proof -
- assume vs: "a \<in> V" "b \<in> V" "c \<in> V" "d \<in> V"
- and eq: "a + b = c + d"
- then have "- c + (a + b) = - c + (c + d)"
+ from assms have "- c + (a + b) = - c + (c + d)"
by (simp add: add_left_cancel)
also have "\<dots> = d" using `c \<in> V` `d \<in> V` by (rule minus_add_cancel)
finally have eq: "- c + (a + b) = d" .
@@ -373,46 +341,41 @@
finally show "a - c = d - b" .
qed
-lemma (in vectorspace) vs_add_cancel_21:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> u \<in> V
- \<Longrightarrow> (x + (y + z) = y + u) = (x + z = u)"
+lemma vs_add_cancel_21:
+ assumes vs: "x \<in> V" "y \<in> V" "z \<in> V" "u \<in> V"
+ shows "(x + (y + z) = y + u) = (x + z = u)"
proof
- assume vs: "x \<in> V" "y \<in> V" "z \<in> V" "u \<in> V"
- {
- from vs have "x + z = - y + y + (x + z)" by simp
- also have "\<dots> = - y + (y + (x + z))"
- by (rule add_assoc) (simp_all add: vs)
- also from vs have "y + (x + z) = x + (y + z)"
- by (simp add: add_ac)
- also assume "x + (y + z) = y + u"
- also from vs have "- y + (y + u) = u" by simp
- finally show "x + z = u" .
- next
- assume "x + z = u"
- with vs show "x + (y + z) = y + u"
- by (simp only: add_left_commute [of x])
- }
+ from vs have "x + z = - y + y + (x + z)" by simp
+ also have "\<dots> = - y + (y + (x + z))"
+ by (rule add_assoc) (simp_all add: vs)
+ also from vs have "y + (x + z) = x + (y + z)"
+ by (simp add: add_ac)
+ also assume "x + (y + z) = y + u"
+ also from vs have "- y + (y + u) = u" by simp
+ finally show "x + z = u" .
+next
+ assume "x + z = u"
+ with vs show "x + (y + z) = y + u"
+ by (simp only: add_left_commute [of x])
qed
-lemma (in vectorspace) add_cancel_end:
- "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + (y + z) = y) = (x = - z)"
+lemma add_cancel_end:
+ assumes vs: "x \<in> V" "y \<in> V" "z \<in> V"
+ shows "(x + (y + z) = y) = (x = - z)"
proof
- assume vs: "x \<in> V" "y \<in> V" "z \<in> V"
- {
- assume "x + (y + z) = y"
- with vs have "(x + z) + y = 0 + y"
- by (simp add: add_ac)
- with vs have "x + z = 0"
- by (simp only: add_right_cancel add_closed zero)
- with vs show "x = - z" by (simp add: add_minus_eq_minus)
- next
- assume eq: "x = - z"
- then have "x + (y + z) = - z + (y + z)" by simp
- also have "\<dots> = y + (- z + z)"
- by (rule add_left_commute) (simp_all add: vs)
- also from vs have "\<dots> = y" by simp
- finally show "x + (y + z) = y" .
- }
+ assume "x + (y + z) = y"
+ with vs have "(x + z) + y = 0 + y" by (simp add: add_ac)
+ with vs have "x + z = 0" by (simp only: add_right_cancel add_closed zero)
+ with vs show "x = - z" by (simp add: add_minus_eq_minus)
+next
+ assume eq: "x = - z"
+ then have "x + (y + z) = - z + (y + z)" by simp
+ also have "\<dots> = y + (- z + z)" by (rule add_left_commute) (simp_all add: vs)
+ also from vs have "\<dots> = y" by simp
+ finally show "x + (y + z) = y" .
qed
end
+
+end
+
--- a/src/HOL/Hahn_Banach/Zorn_Lemma.thy Sun Sep 11 13:49:42 2011 -0700
+++ b/src/HOL/Hahn_Banach/Zorn_Lemma.thy Sun Sep 11 22:56:05 2011 +0200
@@ -5,7 +5,7 @@
header {* Zorn's Lemma *}
theory Zorn_Lemma
-imports Zorn
+imports "~~/src/HOL/Library/Zorn"
begin
text {*