--- a/Admin/isatest/annomaly.ML Sat Jun 13 10:01:01 2009 +0200
+++ b/Admin/isatest/annomaly.ML Sat Jun 13 16:32:38 2009 +0200
@@ -1,5 +1,4 @@
(* Title: Admin/isatest/annomaly.ML
- ID: $Id$
Author: Martin von Gagern <martin@von-gagern.net>
*)
--- a/Admin/isatest/isatest-annomaly Sat Jun 13 10:01:01 2009 +0200
+++ b/Admin/isatest/isatest-annomaly Sat Jun 13 16:32:38 2009 +0200
@@ -1,7 +1,5 @@
#!/usr/bin/env bash
#
-# $Id$
-#
# Create AnnoMaLy documentation for Isabelle
#
# Based on http://martin.von-gagern.net/projects/annomaly/
--- a/Admin/isatest/isatest-check Sat Jun 13 10:01:01 2009 +0200
+++ b/Admin/isatest/isatest-check Sat Jun 13 16:32:38 2009 +0200
@@ -1,6 +1,5 @@
#!/usr/bin/env bash
#
-# $Id$
# Author: Gerwin Klein, TU Muenchen
#
# DESCRIPTION: sends email for failed tests, checks for error.log,
--- a/Admin/isatest/isatest-doc Sat Jun 13 10:01:01 2009 +0200
+++ b/Admin/isatest/isatest-doc Sat Jun 13 16:32:38 2009 +0200
@@ -1,6 +1,5 @@
#!/usr/bin/env bash
#
-# $Id$
# Author: Gerwin Klein, NICTA
#
# Run IsaMakefile for every Doc/ subdirectory.
--- a/Admin/isatest/isatest-lint Sat Jun 13 10:01:01 2009 +0200
+++ b/Admin/isatest/isatest-lint Sat Jun 13 16:32:38 2009 +0200
@@ -1,6 +1,5 @@
#!/usr/bin/env perl
#
-# $Id$
# Author: Florian Haftmann, TUM
#
# Do consistency and quality checks on the isabelle sources
--- a/Admin/isatest/isatest-makeall Sat Jun 13 10:01:01 2009 +0200
+++ b/Admin/isatest/isatest-makeall Sat Jun 13 16:32:38 2009 +0200
@@ -1,6 +1,5 @@
#!/usr/bin/env bash
#
-# $Id$
# Author: Gerwin Klein, TU Muenchen
#
# DESCRIPTION: Run isabelle makeall from specified distribution and settings.
@@ -71,14 +70,19 @@
NICE="nice"
;;
+ macbroy2)
+ MFLAGS=""
+ NICE=""
+ ;;
+
macbroy5)
MFLAGS="-j 2"
NICE=""
;;
macbroy23)
- MFLAGS=""
- NICE=""
+ MFLAGS="-j 2"
+ NICE="nice"
;;
macbroy2[0-9])
--- a/Admin/isatest/isatest-makedist Sat Jun 13 10:01:01 2009 +0200
+++ b/Admin/isatest/isatest-makedist Sat Jun 13 16:32:38 2009 +0200
@@ -1,6 +1,5 @@
#!/usr/bin/env bash
#
-# $Id$
# Author: Gerwin Klein, TU Muenchen
#
# DESCRIPTION: Build distribution and run isatest-make for lots of platforms.
@@ -14,7 +13,6 @@
MAKEDIST=$HOME/bin/makedist
MAKEALL=$HOME/bin/isatest-makeall
TAR=tar
-CVS2CL="$HOME/bin/cvs2cl --follow-only TRUNK"
SSH="ssh -f"
@@ -46,7 +44,6 @@
rm -f $RUNNING/*.runnning
export DISTPREFIX
-export CVS2CL
DATE=$(date "+%Y-%m-%d")
DISTLOG=$LOGPREFIX/isatest-makedist-$DATE.log
@@ -109,9 +106,11 @@
sleep 15
$SSH atbroy101 "$MAKEALL $HOME/settings/at64-poly"
sleep 15
+$SSH macbroy2 "$MAKEALL $HOME/settings/at-mac-poly-5.1-para; $MAKEALL $HOME/settings/mac-poly-M8"
+sleep 15
$SSH macbroy5 "$MAKEALL $HOME/settings/mac-poly"
sleep 15
-$SSH macbroy6 "/usr/stud/isatest/bin/isatest-makeall $HOME/settings/at-mac-poly-5.1-para"
+$SSH macbroy6 "$MAKEALL $HOME/settings/mac-poly-M4"
sleep 15
$SSH atbroy51 "$HOME/admin/isatest/isatest-annomaly"
--- a/Admin/isatest/isatest-settings Sat Jun 13 10:01:01 2009 +0200
+++ b/Admin/isatest/isatest-settings Sat Jun 13 16:32:38 2009 +0200
@@ -1,5 +1,5 @@
# -*- shell-script -*- :mode=shellscript:
-# $Id$
+#
# Author: Gerwin Klein, NICTA
#
# DESCRIPTION: common settings for the isatest-* scripts
--- a/Admin/isatest/pmail Sat Jun 13 10:01:01 2009 +0200
+++ b/Admin/isatest/pmail Sat Jun 13 16:32:38 2009 +0200
@@ -1,6 +1,5 @@
#!/usr/bin/env bash
#
-# $Id$
# Author: Gerwin Klein, TU Muenchen
#
# DESCRIPTION: send email with text attachments.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Admin/isatest/settings/mac-poly-M4 Sat Jun 13 16:32:38 2009 +0200
@@ -0,0 +1,28 @@
+# -*- shell-script -*- :mode=shellscript:
+
+ POLYML_HOME="/home/polyml/polyml-svn"
+ ML_SYSTEM="polyml-experimental"
+ ML_PLATFORM="x86-darwin"
+ ML_HOME="$POLYML_HOME/$ML_PLATFORM"
+ ML_OPTIONS="--mutable 800 --immutable 2000"
+
+
+ISABELLE_HOME_USER=~/isabelle-mac-poly-M4
+
+# Where to look for isabelle tools (multiple dirs separated by ':').
+ISABELLE_TOOLS="$ISABELLE_HOME/lib/Tools"
+
+# Location for temporary files (should be on a local file system).
+ISABELLE_TMP_PREFIX="/tmp/isabelle-$USER"
+
+
+# Heap input locations. ML system identifier is included in lookup.
+ISABELLE_PATH="$ISABELLE_HOME_USER/heaps:$ISABELLE_HOME/heaps"
+
+# Heap output location. ML system identifier is appended automatically later on.
+ISABELLE_OUTPUT="$ISABELLE_HOME_USER/heaps"
+ISABELLE_BROWSER_INFO="$ISABELLE_HOME_USER/browser_info"
+
+ISABELLE_USEDIR_OPTIONS="-i false -d false -M 4"
+
+HOL_USEDIR_OPTIONS="-p 2 -Q false"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Admin/isatest/settings/mac-poly-M8 Sat Jun 13 16:32:38 2009 +0200
@@ -0,0 +1,28 @@
+# -*- shell-script -*- :mode=shellscript:
+
+ POLYML_HOME="/home/polyml/polyml-svn"
+ ML_SYSTEM="polyml-experimental"
+ ML_PLATFORM="x86-darwin"
+ ML_HOME="$POLYML_HOME/$ML_PLATFORM"
+ ML_OPTIONS="--mutable 800 --immutable 2000"
+
+
+ISABELLE_HOME_USER=~/isabelle-mac-poly-M8
+
+# Where to look for isabelle tools (multiple dirs separated by ':').
+ISABELLE_TOOLS="$ISABELLE_HOME/lib/Tools"
+
+# Location for temporary files (should be on a local file system).
+ISABELLE_TMP_PREFIX="/tmp/isabelle-$USER"
+
+
+# Heap input locations. ML system identifier is included in lookup.
+ISABELLE_PATH="$ISABELLE_HOME_USER/heaps:$ISABELLE_HOME/heaps"
+
+# Heap output location. ML system identifier is appended automatically later on.
+ISABELLE_OUTPUT="$ISABELLE_HOME_USER/heaps"
+ISABELLE_BROWSER_INFO="$ISABELLE_HOME_USER/browser_info"
+
+ISABELLE_USEDIR_OPTIONS="-i false -d false -M 8"
+
+HOL_USEDIR_OPTIONS="-p 2 -Q false"
--- a/NEWS Sat Jun 13 10:01:01 2009 +0200
+++ b/NEWS Sat Jun 13 16:32:38 2009 +0200
@@ -4,28 +4,38 @@
New in this Isabelle version
----------------------------
+*** General ***
+
+* Discontinued old form of "escaped symbols" such as \\<forall>. Only
+one backslash should be used, even in ML sources.
+
+
*** Pure ***
-* On instantiation of classes, remaining undefined class parameters are
-formally declared. INCOMPATIBILITY.
+* On instantiation of classes, remaining undefined class parameters
+are formally declared. INCOMPATIBILITY.
*** HOL ***
-* Class semiring_div requires superclass no_zero_divisors and proof of div_mult_mult1;
-theorems div_mult_mult1, div_mult_mult2, div_mult_mult1_if, div_mult_mult1 and
-div_mult_mult2 have been generalized to class semiring_div, subsuming former
-theorems zdiv_zmult_zmult1, zdiv_zmult_zmult1_if, zdiv_zmult_zmult1 and zdiv_zmult_zmult2.
-div_mult_mult1 is now [simp] by default. INCOMPATIBILITY.
-
-* Power operations on relations and functions are now one dedicate constant compow with
-infix syntax "^^". Power operations on multiplicative monoids retains syntax "^"
-and is now defined generic in class power. INCOMPATIBILITY.
-
-* ML antiquotation @{code_datatype} inserts definition of a datatype generated
-by the code generator; see Predicate.thy for an example.
-
-* New method "linarith" invokes existing linear arithmetic decision procedure only.
+* Class semiring_div requires superclass no_zero_divisors and proof of
+div_mult_mult1; theorems div_mult_mult1, div_mult_mult2,
+div_mult_mult1_if, div_mult_mult1 and div_mult_mult2 have been
+generalized to class semiring_div, subsuming former theorems
+zdiv_zmult_zmult1, zdiv_zmult_zmult1_if, zdiv_zmult_zmult1 and
+zdiv_zmult_zmult2. div_mult_mult1 is now [simp] by default.
+INCOMPATIBILITY.
+
+* Power operations on relations and functions are now one dedicate
+constant compow with infix syntax "^^". Power operations on
+multiplicative monoids retains syntax "^" and is now defined generic
+in class power. INCOMPATIBILITY.
+
+* ML antiquotation @{code_datatype} inserts definition of a datatype
+generated by the code generator; see Predicate.thy for an example.
+
+* New method "linarith" invokes existing linear arithmetic decision
+procedure only.
* Implementation of quickcheck using generic code generator; default generators
are provided for all suitable HOL types, records and datatypes.
--- a/doc-src/Codegen/Thy/document/Introduction.tex Sat Jun 13 10:01:01 2009 +0200
+++ b/doc-src/Codegen/Thy/document/Introduction.tex Sat Jun 13 16:32:38 2009 +0200
@@ -249,9 +249,9 @@
\hspace*{0pt}dequeue (AQueue [] []) = (Nothing,~AQueue [] []);\\
\hspace*{0pt}dequeue (AQueue xs (y :~ys)) = (Just y,~AQueue xs ys);\\
\hspace*{0pt}dequeue (AQueue (v :~va) []) =\\
-\hspace*{0pt} ~let {\char123}\\
+\hspace*{0pt} ~(let {\char123}\\
\hspace*{0pt} ~~~(y :~ys) = rev (v :~va);\\
-\hspace*{0pt} ~{\char125}~in (Just y,~AQueue [] ys);\\
+\hspace*{0pt} ~{\char125}~in (Just y,~AQueue [] ys) );\\
\hspace*{0pt}\\
\hspace*{0pt}enqueue ::~forall a.~a -> Queue a -> Queue a;\\
\hspace*{0pt}enqueue x (AQueue xs ys) = AQueue (x :~xs) ys;\\
--- a/doc-src/Codegen/Thy/document/Program.tex Sat Jun 13 10:01:01 2009 +0200
+++ b/doc-src/Codegen/Thy/document/Program.tex Sat Jun 13 16:32:38 2009 +0200
@@ -346,8 +346,8 @@
\hspace*{0pt}type 'a semigroup = {\char123}mult :~'a -> 'a -> 'a{\char125};\\
\hspace*{0pt}fun mult (A{\char95}:'a semigroup) = {\char35}mult A{\char95};\\
\hspace*{0pt}\\
-\hspace*{0pt}type 'a monoid = {\char123}Program{\char95}{\char95}semigroup{\char95}monoid :~'a semigroup,~neutral :~'a{\char125};\\
-\hspace*{0pt}fun semigroup{\char95}monoid (A{\char95}:'a monoid) = {\char35}Program{\char95}{\char95}semigroup{\char95}monoid A{\char95};\\
+\hspace*{0pt}type 'a monoid = {\char123}semigroup{\char95}monoid :~'a semigroup,~neutral :~'a{\char125};\\
+\hspace*{0pt}fun semigroup{\char95}monoid (A{\char95}:'a monoid) = {\char35}semigroup{\char95}monoid A{\char95};\\
\hspace*{0pt}fun neutral (A{\char95}:'a monoid) = {\char35}neutral A{\char95};\\
\hspace*{0pt}\\
\hspace*{0pt}fun pow A{\char95}~Zero{\char95}nat a = neutral A{\char95}\\
@@ -363,9 +363,8 @@
\hspace*{0pt}\\
\hspace*{0pt}val semigroup{\char95}nat = {\char123}mult = mult{\char95}nat{\char125}~:~nat semigroup;\\
\hspace*{0pt}\\
-\hspace*{0pt}val monoid{\char95}nat =\\
-\hspace*{0pt} ~{\char123}Program{\char95}{\char95}semigroup{\char95}monoid = semigroup{\char95}nat,~neutral = neutral{\char95}nat{\char125}~:\\
-\hspace*{0pt} ~nat monoid;\\
+\hspace*{0pt}val monoid{\char95}nat = {\char123}semigroup{\char95}monoid = semigroup{\char95}nat,~neutral = neutral{\char95}nat{\char125}\\
+\hspace*{0pt} ~:~nat monoid;\\
\hspace*{0pt}\\
\hspace*{0pt}fun bexp n = pow monoid{\char95}nat n (Suc (Suc Zero{\char95}nat));\\
\hspace*{0pt}\\
@@ -967,9 +966,9 @@
\noindent%
\hspace*{0pt}strict{\char95}dequeue ::~forall a.~Queue a -> (a,~Queue a);\\
\hspace*{0pt}strict{\char95}dequeue (AQueue xs []) =\\
-\hspace*{0pt} ~let {\char123}\\
+\hspace*{0pt} ~(let {\char123}\\
\hspace*{0pt} ~~~(y :~ys) = rev xs;\\
-\hspace*{0pt} ~{\char125}~in (y,~AQueue [] ys);\\
+\hspace*{0pt} ~{\char125}~in (y,~AQueue [] ys) );\\
\hspace*{0pt}strict{\char95}dequeue (AQueue xs (y :~ys)) = (y,~AQueue xs ys);%
\end{isamarkuptext}%
\isamarkuptrue%
--- a/doc-src/Codegen/Thy/examples/Example.hs Sat Jun 13 10:01:01 2009 +0200
+++ b/doc-src/Codegen/Thy/examples/Example.hs Sat Jun 13 16:32:38 2009 +0200
@@ -23,9 +23,9 @@
dequeue (AQueue [] []) = (Nothing, AQueue [] []);
dequeue (AQueue xs (y : ys)) = (Just y, AQueue xs ys);
dequeue (AQueue (v : va) []) =
- let {
+ (let {
(y : ys) = rev (v : va);
- } in (Just y, AQueue [] ys);
+ } in (Just y, AQueue [] ys) );
enqueue :: forall a. a -> Queue a -> Queue a;
enqueue x (AQueue xs ys) = AQueue (x : xs) ys;
--- a/doc-src/antiquote_setup.ML Sat Jun 13 10:01:01 2009 +0200
+++ b/doc-src/antiquote_setup.ML Sat Jun 13 16:32:38 2009 +0200
@@ -19,16 +19,16 @@
| "{" => "\\{"
| "|" => "$\\mid$"
| "}" => "\\}"
- | "\\<dash>" => "-"
+ | "\<dash>" => "-"
| c => c);
-fun clean_name "\\<dots>" = "dots"
+fun clean_name "\<dots>" = "dots"
| clean_name ".." = "ddot"
| clean_name "." = "dot"
| clean_name "_" = "underscore"
| clean_name "{" = "braceleft"
| clean_name "}" = "braceright"
- | clean_name s = s |> translate (fn "_" => "-" | "\\<dash>" => "-" | c => c);
+ | clean_name s = s |> translate (fn "_" => "-" | "\<dash>" => "-" | c => c);
(* verbatim text *)
--- a/src/HOL/Library/Convex_Euclidean_Space.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/Library/Convex_Euclidean_Space.thy Sat Jun 13 16:32:38 2009 +0200
@@ -39,10 +39,6 @@
lemma norm_not_0:"(x::real^'n::finite)\<noteq>0 \<Longrightarrow> norm x \<noteq> 0" by auto
-lemma vector_unminus_smult[simp]: "(-1::real) *s x = -x"
- unfolding vector_sneg_minus1 by simp
- (* TODO: move to Euclidean_Space.thy *)
-
lemma setsum_delta_notmem: assumes "x\<notin>s"
shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
"setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
@@ -51,23 +47,23 @@
apply(rule_tac [!] setsum_cong2) using assms by auto
lemma setsum_delta'': fixes s::"(real^'n) set" assumes "finite s"
- shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *s x) = (if y\<in>s then (f y) *s y else 0)"
+ shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
proof-
- have *:"\<And>x y. (if y = x then f x else (0::real)) *s x = (if x=y then (f x) *s x else 0)" by auto
- show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *s x"] by auto
+ have *:"\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)" by auto
+ show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
qed
lemma not_disjointI:"x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A \<inter> B \<noteq> {}" by blast
-lemma if_smult:"(if P then x else (y::real)) *s v = (if P then x *s v else y *s v)" by auto
+lemma if_smult:"(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)" by auto
lemma mem_interval_1: fixes x :: "real^1" shows
"(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
"(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
by(simp_all add: Cart_eq vector_less_def vector_less_eq_def dest_vec1_def all_1)
-lemma image_smult_interval:"(\<lambda>x. m *s (x::real^'n::finite)) ` {a..b} =
- (if {a..b} = {} then {} else if 0 \<le> m then {m *s a..m *s b} else {m *s b..m *s a})"
+lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::real^'n::finite)) ` {a..b} =
+ (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
using image_affinity_interval[of m 0 a b] by auto
lemma dest_vec1_inverval:
@@ -87,9 +83,11 @@
shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
using dest_vec1_sum[OF assms] by auto
-lemma dist_triangle_eq:"dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *s (y - z) = norm (y - z) *s (x - y)"
+lemma dist_triangle_eq:
+ fixes x y z :: "real ^ _"
+ shows "dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
proof- have *:"x - y + (y - z) = x - z" by auto
- show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
+ show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded smult_conv_scaleR *]
by(auto simp add:norm_minus_commute) qed
lemma norm_eqI:"x = y \<Longrightarrow> norm x = norm y" by auto
@@ -108,12 +106,14 @@
subsection {* Affine set and affine hull.*}
-definition "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v::real. u + v = 1 \<longrightarrow> (u *s x + v *s y) \<in> s)"
-
-lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *s x + u *s y \<in> s)"
+definition
+ affine :: "(real ^ 'n::finite) set \<Rightarrow> bool" where
+ "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v::real. u + v = 1 \<longrightarrow> (u *\<^sub>R x + v *\<^sub>R y) \<in> s)"
+
+lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
proof- have *:"\<And>u v ::real. u + v = 1 \<longleftrightarrow> v = 1 - u" by auto
{ fix x y assume "x\<in>s" "y\<in>s"
- hence "(\<forall>u v::real. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s) \<longleftrightarrow> (\<forall>u::real. (1 - u) *s x + u *s y \<in> s)" apply auto
+ hence "(\<forall>u v::real. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s) \<longleftrightarrow> (\<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)" apply auto
apply(erule_tac[!] x="1 - u" in allE) unfolding * by auto }
thus ?thesis unfolding affine_def by auto qed
@@ -121,7 +121,7 @@
unfolding affine_def by auto
lemma affine_sing[intro]: "affine {x}"
- unfolding affine_alt by (auto simp add: vector_sadd_rdistrib[THEN sym])
+ unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
lemma affine_UNIV[intro]: "affine UNIV"
unfolding affine_def by auto
@@ -149,30 +149,30 @@
subsection {* Some explicit formulations (from Lars Schewe). *}
-lemma affine: fixes V::"(real^'n) set"
- shows "affine V \<longleftrightarrow> (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *s x)) s \<in> V)"
+lemma affine: fixes V::"(real^'n::finite) set"
+ shows "affine V \<longleftrightarrow> (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
unfolding affine_def apply rule apply(rule, rule, rule) apply(erule conjE)+
defer apply(rule, rule, rule, rule, rule) proof-
fix x y u v assume as:"x \<in> V" "y \<in> V" "u + v = (1::real)"
- "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *s x) \<in> V"
- thus "u *s x + v *s y \<in> V" apply(cases "x=y")
+ "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
+ thus "u *\<^sub>R x + v *\<^sub>R y \<in> V" apply(cases "x=y")
using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]] and as(1-3)
- by(auto simp add: vector_sadd_rdistrib[THEN sym])
+ by(auto simp add: scaleR_left_distrib[THEN sym])
next
- fix s u assume as:"\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> V"
+ fix s u assume as:"\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
"finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
def n \<equiv> "card s"
have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
- thus "(\<Sum>x\<in>s. u x *s x) \<in> V" proof(auto simp only: disjE)
+ thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" proof(auto simp only: disjE)
assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
by(auto simp add: setsum_clauses(2))
next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
case (Suc n) fix s::"(real^'n) set" and u::"real^'n\<Rightarrow> real"
- assume IA:"\<And>u s. \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> V; finite s;
- s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n \<equiv> card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *s x) \<in> V" and
- as:"Suc n \<equiv> card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> V"
+ assume IA:"\<And>u s. \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
+ s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n \<equiv> card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" and
+ as:"Suc n \<equiv> card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
"finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
assume " \<not> (\<exists>x\<in>s. u x \<noteq> 1)" hence "setsum u s = real_of_nat (card s)" unfolding card_eq_setsum by auto
@@ -185,7 +185,7 @@
have **:"setsum u (s - {x}) = 1 - u x"
using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[THEN sym] as(7)] by auto
have ***:"inverse (1 - u x) * setsum u (s - {x}) = 1" unfolding ** using `u x \<noteq> 1` by auto
- have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *s xa) \<in> V" proof(cases "card (s - {x}) > 2")
+ have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V" proof(cases "card (s - {x}) > 2")
case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr)
assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp
thus False using True by auto qed auto
@@ -195,9 +195,9 @@
then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b" unfolding card_Suc_eq by auto
thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
using *** *(2) and `s \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
- thus "(\<Sum>x\<in>s. u x *s x) \<in> V" unfolding vector_smult_assoc[THEN sym] and setsum_cmul
+ thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
- using as(3)[THEN bspec[where x=x], THEN bspec[where x="(inverse (1 - u x)) *s (\<Sum>xa\<in>s - {x}. u xa *s xa)"],
+ using as(3)[THEN bspec[where x=x], THEN bspec[where x="(inverse (1 - u x)) *\<^sub>R (\<Sum>xa\<in>s - {x}. u xa *\<^sub>R xa)"],
THEN spec[where x="u x"], THEN spec[where x="1 - u x"]] and rev_subsetD[OF `x\<in>s` `s\<subseteq>V`] and `u x \<noteq> 1` by auto
qed auto
next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
@@ -206,44 +206,44 @@
qed
lemma affine_hull_explicit:
- "affine hull p = {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *s v) s = y}"
+ "affine hull p = {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq and mem_def[of _ affine]
apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
- fix x assume "x\<in>p" thus "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = x"
+ fix x assume "x\<in>p" thus "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
next
- fix t x s u assume as:"p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}" "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = x"
+ fix t x s u assume as:"p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}" "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
next
- show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = y}" unfolding affine_def
+ show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y}" unfolding affine_def
apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
fix u v ::real assume uv:"u + v = 1"
- fix x assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = x"
- then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *s v) = x" by auto
- fix y assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = y"
- then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *s v) = y" by auto
+ fix x assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+ then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x" by auto
+ fix y assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+ then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
- show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *s v) = u *s x + v *s y"
+ show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
apply(rule_tac x="sx \<union> sy" in exI)
apply(rule_tac x="\<lambda>a. (if a\<in>sx then u * ux a else 0) + (if a\<in>sy then v * uy a else 0)" in exI)
- unfolding vector_sadd_rdistrib setsum_addf if_smult vector_smult_lzero ** setsum_restrict_set[OF xy, THEN sym]
- unfolding vector_smult_assoc[THEN sym] setsum_cmul and setsum_right_distrib[THEN sym]
+ unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left ** setsum_restrict_set[OF xy, THEN sym]
+ unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
unfolding x y using x(1-3) y(1-3) uv by simp qed qed
lemma affine_hull_finite:
assumes "finite s"
- shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}"
+ shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
unfolding affine_hull_explicit and expand_set_eq and mem_Collect_eq apply (rule,rule)
apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
- fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = x"
- thus "\<exists>sa u. finite sa \<and> \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *s v) = x"
+ fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+ thus "\<exists>sa u. finite sa \<and> \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = x"
apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
next
fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
- assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *s v) = x"
- thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
- unfolding if_smult vector_smult_lzero and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
+ assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
+ thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
+ unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
subsection {* Stepping theorems and hence small special cases. *}
@@ -251,14 +251,14 @@
apply(rule hull_unique) unfolding mem_def by auto
lemma affine_hull_finite_step:
- shows "(\<exists>u::real^'n=>real. setsum u {} = w \<and> setsum (\<lambda>x. u x *s x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
- "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *s x) (insert a s) = y) \<longleftrightarrow>
- (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *s x) s = y - v *s a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
+ shows "(\<exists>u::real^'n=>real. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
+ "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
+ (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
proof-
show ?th1 by simp
assume ?as
{ assume ?lhs
- then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *s x) = y" by auto
+ then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
have ?rhs proof(cases "a\<in>s")
case True hence *:"insert a s = s" by auto
show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
@@ -266,41 +266,41 @@
case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto
qed } moreover
{ assume ?rhs
- then obtain v u where vu:"setsum u s = w - v" "(\<Sum>x\<in>s. u x *s x) = y - v *s a" by auto
- have *:"\<And>x M. (if x = a then v else M) *s x = (if x = a then v *s x else M *s x)" by auto
+ then obtain v u where vu:"setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
+ have *:"\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)" by auto
have ?lhs proof(cases "a\<in>s")
case True thus ?thesis
apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
unfolding setsum_clauses(2)[OF `?as`] apply simp
- unfolding vector_sadd_rdistrib and setsum_addf
- unfolding vu and * and vector_smult_lzero
+ unfolding scaleR_left_distrib and setsum_addf
+ unfolding vu and * and scaleR_zero_left
by (auto simp add: setsum_delta[OF `?as`])
next
case False
hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
- "\<And>x. x \<in> s \<Longrightarrow> u x *s x = (if x = a then v *s x else u x *s x)" by auto
+ "\<And>x. x \<in> s \<Longrightarrow> u x *\<^sub>R x = (if x = a then v *\<^sub>R x else u x *\<^sub>R x)" by auto
from False show ?thesis
apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
unfolding setsum_clauses(2)[OF `?as`] and * using vu
- using setsum_cong2[of s "\<lambda>x. u x *s x" "\<lambda>x. if x = a then v *s x else u x *s x", OF **(2)]
+ using setsum_cong2[of s "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF **(2)]
using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto
qed }
ultimately show "?lhs = ?rhs" by blast
qed
-lemma affine_hull_2: "affine hull {a,b::real^'n} = {u *s a + v *s b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
+lemma affine_hull_2:"affine hull {a,b::real^'n::finite} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
proof-
have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real^'n)" by auto
- have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *s v) = y}"
+ have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
using affine_hull_finite[of "{a,b}"] by auto
- also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *s b = y - v *s a}"
+ also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
by(simp add: affine_hull_finite_step(2)[of "{b}" a])
also have "\<dots> = ?rhs" unfolding * by auto
finally show ?thesis by auto
qed
-lemma affine_hull_3: "affine hull {a,b,c::real^'n} = { u *s a + v *s b + w *s c| u v w. u + v + w = 1}" (is "?lhs = ?rhs")
+lemma affine_hull_3: "affine hull {a,b,c::real^'n::finite} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}" (is "?lhs = ?rhs")
proof-
have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real^'n)" by auto
@@ -314,15 +314,20 @@
lemma affine_hull_insert_subset_span:
"affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
- unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq
+ unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq smult_conv_scaleR
apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
- fix x t u assume as:"finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *s v) = x"
+ fix x t u assume as:"finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
- thus "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *s v) = v)"
- apply(rule_tac x="x - a" in exI) apply rule defer apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
- apply(rule_tac x="\<lambda>x. u (x + a)" in exI) using as(1)
- apply(simp add: setsum_reindex[unfolded inj_on_def] setsum_subtractf setsum_diff1 setsum_vmul[THEN sym])
- unfolding as by simp_all qed
+ thus "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = v)"
+ apply(rule_tac x="x - a" in exI)
+ apply (rule conjI, simp)
+ apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
+ apply(rule_tac x="\<lambda>x. u (x + a)" in exI)
+ apply (rule conjI) using as(1) apply simp
+ apply (erule conjI)
+ using as(1)
+ apply (simp add: setsum_reindex[unfolded inj_on_def] scaleR_right_diff_distrib setsum_subtractf scaleR_left.setsum[THEN sym] setsum_diff1 scaleR_left_diff_distrib)
+ unfolding as by simp qed
lemma affine_hull_insert_span:
assumes "a \<notin> s"
@@ -331,17 +336,17 @@
apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
- then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *s v) = y" unfolding span_explicit by auto
+ then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *\<^sub>R v) = y" unfolding span_explicit smult_conv_scaleR by auto
def f \<equiv> "(\<lambda>x. x + a) ` t"
- have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *s (v - a)) = y - a" unfolding f_def using obt
+ have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a" unfolding f_def using obt
by(auto simp add: setsum_reindex[unfolded inj_on_def])
have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
- show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *s v) = y"
+ show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
apply(rule_tac x="insert a f" in exI)
apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
unfolding setsum_cases[OF f(1), of "{a}", unfolded singleton_iff] and *
- by (auto simp add: setsum_subtractf setsum_vmul field_simps) qed
+ by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps) qed
lemma affine_hull_span:
assumes "a \<in> s"
@@ -350,10 +355,10 @@
subsection {* Convexity. *}
-definition "convex (s::(real^'n) set) \<longleftrightarrow>
- (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. (u + v = 1) \<longrightarrow> (u *s x + v *s y) \<in> s)"
-
-lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *s x + u *s y) \<in> s)"
+definition "convex (s::(real^'n::finite) set) \<longleftrightarrow>
+ (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. (u + v = 1) \<longrightarrow> (u *\<^sub>R x + v *\<^sub>R y) \<in> s)"
+
+lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
proof- have *:"\<And>u v::real. u + v = 1 \<longleftrightarrow> u = 1 - v" by auto
show ?thesis unfolding convex_def apply auto
apply(erule_tac x=x in ballE) apply(erule_tac x=y in ballE) apply(erule_tac x="1 - u" in allE)
@@ -361,14 +366,14 @@
lemma mem_convex:
assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
- shows "((1 - u) *s a + u *s b) \<in> s"
+ shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
using assms unfolding convex_alt by auto
lemma convex_empty[intro]: "convex {}"
unfolding convex_def by simp
lemma convex_singleton[intro]: "convex {a}"
- unfolding convex_def by (auto simp add:vector_sadd_rdistrib[THEN sym])
+ unfolding convex_def by (auto simp add:scaleR_left_distrib[THEN sym])
lemma convex_UNIV[intro]: "convex UNIV"
unfolding convex_def by auto
@@ -379,28 +384,30 @@
lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
unfolding convex_def by auto
-lemma convex_halfspace_le: "convex {x. a \<bullet> x \<le> b}"
+lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
unfolding convex_def apply auto
- unfolding dot_radd dot_rmult by (metis real_convex_bound_le)
-
-lemma convex_halfspace_ge: "convex {x. a \<bullet> x \<ge> b}"
-proof- have *:"{x. a \<bullet> x \<ge> b} = {x. -a \<bullet> x \<le> -b}" by auto
+ unfolding inner_add inner_scaleR
+ by (metis real_convex_bound_le)
+
+lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
+proof- have *:"{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}" by auto
show ?thesis apply(unfold *) using convex_halfspace_le[of "-a" "-b"] by auto qed
-lemma convex_hyperplane: "convex {x. a \<bullet> x = b}"
+lemma convex_hyperplane: "convex {x. inner a x = b}"
proof-
- have *:"{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}" by auto
+ have *:"{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}" by auto
show ?thesis unfolding * apply(rule convex_Int)
using convex_halfspace_le convex_halfspace_ge by auto
qed
-lemma convex_halfspace_lt: "convex {x. a \<bullet> x < b}"
- unfolding convex_def by(auto simp add: real_convex_bound_lt dot_radd dot_rmult)
-
-lemma convex_halfspace_gt: "convex {x. a \<bullet> x > b}"
- using convex_halfspace_lt[of "-a" "-b"] by(auto simp add: dot_lneg neg_less_iff_less)
-
-lemma convex_positive_orthant: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
+lemma convex_halfspace_lt: "convex {x. inner a x < b}"
+ unfolding convex_def
+ by(auto simp add: real_convex_bound_lt inner_add)
+
+lemma convex_halfspace_gt: "convex {x. inner a x > b}"
+ using convex_halfspace_lt[of "-a" "-b"] by auto
+
+lemma convex_positive_orthant: "convex {x::real^'n::finite. (\<forall>i. 0 \<le> x$i)}"
unfolding convex_def apply auto apply(erule_tac x=i in allE)+
apply(rule add_nonneg_nonneg) by(auto simp add: mult_nonneg_nonneg)
@@ -408,18 +415,18 @@
lemma convex: "convex s \<longleftrightarrow>
(\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (setsum u {1..k} = 1)
- \<longrightarrow> setsum (\<lambda>i. u i *s x i) {1..k} \<in> s)"
+ \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
unfolding convex_def apply rule apply(rule allI)+ defer apply(rule ballI)+ apply(rule allI)+ proof(rule,rule,rule,rule)
- fix x y u v assume as:"\<forall>(k::nat) u x. (\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *s x i) \<in> s"
+ fix x y u v assume as:"\<forall>(k::nat) u x. (\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R x i) \<in> s"
"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
- show "u *s x + v *s y \<in> s" using as(1)[THEN spec[where x=2], THEN spec[where x="\<lambda>n. if n=1 then u else v"], THEN spec[where x="\<lambda>n. if n=1 then x else y"]] and as(2-)
+ show "u *\<^sub>R x + v *\<^sub>R y \<in> s" using as(1)[THEN spec[where x=2], THEN spec[where x="\<lambda>n. if n=1 then u else v"], THEN spec[where x="\<lambda>n. if n=1 then x else y"]] and as(2-)
by (auto simp add: setsum_head_Suc)
next
- fix k u x assume as:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s"
- show "(\<forall>i::nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *s x i) \<in> s" apply(rule,erule conjE) proof(induct k arbitrary: u)
+ fix k u x assume as:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
+ show "(\<forall>i::nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R x i) \<in> s" apply(rule,erule conjE) proof(induct k arbitrary: u)
case (Suc k) show ?case proof(cases "u (Suc k) = 1")
- case True hence "(\<Sum>i = Suc 0..k. u i *s x i) = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
- fix i assume i:"i \<in> {Suc 0..k}" "u i *s x i \<noteq> 0"
+ case True hence "(\<Sum>i = Suc 0..k. u i *\<^sub>R x i) = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
+ fix i assume i:"i \<in> {Suc 0..k}" "u i *\<^sub>R x i \<noteq> 0"
hence ui:"u i \<noteq> 0" by auto
hence "setsum (\<lambda>k. if k=i then u i else 0) {1 .. k} \<le> setsum u {1 .. k}" apply(rule_tac setsum_mono) using Suc(2) by auto
hence "setsum u {1 .. k} \<ge> u i" using i(1) by(auto simp add: setsum_delta)
@@ -429,32 +436,32 @@
next
have *:"setsum u {1..k} = 1 - u (Suc k)" using Suc(3)[unfolded setsum_cl_ivl_Suc] by auto
have **:"u (Suc k) \<le> 1" apply(rule ccontr) unfolding not_le using Suc(3) using setsum_nonneg[of "{1..k}" u] using Suc(2) by auto
- have ***:"\<And>i k. (u i / (1 - u (Suc k))) *s x i = (inverse (1 - u (Suc k))) *s (u i *s x i)" unfolding real_divide_def by auto
+ have ***:"\<And>i k. (u i / (1 - u (Suc k))) *\<^sub>R x i = (inverse (1 - u (Suc k))) *\<^sub>R (u i *\<^sub>R x i)" unfolding real_divide_def by (auto simp add: algebra_simps)
case False hence nn:"1 - u (Suc k) \<noteq> 0" by auto
- have "(\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *s x i) \<in> s" apply(rule Suc(1)) unfolding setsum_divide_distrib[THEN sym] and *
+ have "(\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *\<^sub>R x i) \<in> s" apply(rule Suc(1)) unfolding setsum_divide_distrib[THEN sym] and *
apply(rule_tac allI) apply(rule,rule) apply(rule divide_nonneg_pos) using nn Suc(2) ** by auto
- hence "(1 - u (Suc k)) *s (\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *s x i) + u (Suc k) *s x (Suc k) \<in> s"
+ hence "(1 - u (Suc k)) *\<^sub>R (\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *\<^sub>R x i) + u (Suc k) *\<^sub>R x (Suc k) \<in> s"
apply(rule as[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using Suc(2)[THEN spec[where x="Suc k"]] and ** by auto
- thus ?thesis unfolding setsum_cl_ivl_Suc and *** and setsum_cmul using nn by auto qed qed auto qed
-
-
-lemma convex_explicit: "convex (s::(real^'n) set) \<longleftrightarrow>
- (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *s x) t \<in> s)"
+ thus ?thesis unfolding setsum_cl_ivl_Suc and *** and scaleR_right.setsum [symmetric] using nn by auto qed qed auto qed
+
+
+lemma convex_explicit: "convex (s::(real^'n::finite) set) \<longleftrightarrow>
+ (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
unfolding convex_def apply(rule,rule,rule) apply(subst imp_conjL,rule) defer apply(rule,rule,rule,rule,rule,rule,rule) proof-
- fix x y u v assume as:"\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *s x) \<in> s" "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
- show "u *s x + v *s y \<in> s" proof(cases "x=y")
- case True show ?thesis unfolding True and vector_sadd_rdistrib[THEN sym] using as(3,6) by auto next
+ fix x y u v assume as:"\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
+ show "u *\<^sub>R x + v *\<^sub>R y \<in> s" proof(cases "x=y")
+ case True show ?thesis unfolding True and scaleR_left_distrib[THEN sym] using as(3,6) by auto next
case False thus ?thesis using as(1)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>z. if z=x then u else v"]] and as(2-) by auto qed
next
- fix t u assume asm:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *s x + v *s y \<in> s" "finite (t::(real^'n) set)"
+ fix t u assume asm:"\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s" "finite (t::(real^'n) set)"
(*"finite t" "t \<subseteq> s" "\<forall>x\<in>t. (0::real) \<le> u x" "setsum u t = 1"*)
- from this(2) have "\<forall>u. t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *s x) \<in> s" apply(induct_tac t rule:finite_induct)
+ from this(2) have "\<forall>u. t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" apply(induct_tac t rule:finite_induct)
prefer 3 apply (rule,rule) apply(erule conjE)+ proof-
- fix x f u assume ind:"\<forall>u. f \<subseteq> s \<and> (\<forall>x\<in>f. 0 \<le> u x) \<and> setsum u f = 1 \<longrightarrow> (\<Sum>x\<in>f. u x *s x) \<in> s"
+ fix x f u assume ind:"\<forall>u. f \<subseteq> s \<and> (\<forall>x\<in>f. 0 \<le> u x) \<and> setsum u f = 1 \<longrightarrow> (\<Sum>x\<in>f. u x *\<^sub>R x) \<in> s"
assume as:"finite f" "x \<notin> f" "insert x f \<subseteq> s" "\<forall>x\<in>insert x f. 0 \<le> u x" "setsum u (insert x f) = (1::real)"
- show "(\<Sum>x\<in>insert x f. u x *s x) \<in> s" proof(cases "u x = 1")
- case True hence "setsum (\<lambda>x. u x *s x) f = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
- fix y assume y:"y \<in> f" "u y *s y \<noteq> 0"
+ show "(\<Sum>x\<in>insert x f. u x *\<^sub>R x) \<in> s" proof(cases "u x = 1")
+ case True hence "setsum (\<lambda>x. u x *\<^sub>R x) f = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
+ fix y assume y:"y \<in> f" "u y *\<^sub>R y \<noteq> 0"
hence uy:"u y \<noteq> 0" by auto
hence "setsum (\<lambda>k. if k=y then u y else 0) f \<le> setsum u f" apply(rule_tac setsum_mono) using as(4) by auto
hence "setsum u f \<ge> u y" using y(1) and as(1) by(auto simp add: setsum_delta)
@@ -465,28 +472,28 @@
have *:"setsum u f = setsum u (insert x f) - u x" using as(2) unfolding setsum_clauses(2)[OF as(1)] by auto
have **:"u x \<le> 1" apply(rule ccontr) unfolding not_le using as(5)[unfolded setsum_clauses(2)[OF as(1)]] and as(2)
using setsum_nonneg[of f u] and as(4) by auto
- case False hence "inverse (1 - u x) *s (\<Sum>x\<in>f. u x *s x) \<in> s" unfolding setsum_cmul[THEN sym] and vector_smult_assoc
+ case False hence "inverse (1 - u x) *\<^sub>R (\<Sum>x\<in>f. u x *\<^sub>R x) \<in> s" unfolding scaleR_right.setsum and scaleR_scaleR
apply(rule_tac ind[THEN spec, THEN mp]) apply rule defer apply rule apply rule apply(rule mult_nonneg_nonneg)
unfolding setsum_right_distrib[THEN sym] and * using as and ** by auto
- hence "u x *s x + (1 - u x) *s ((inverse (1 - u x)) *s setsum (\<lambda>x. u x *s x) f) \<in>s"
+ hence "u x *\<^sub>R x + (1 - u x) *\<^sub>R ((inverse (1 - u x)) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) f) \<in>s"
apply(rule_tac asm(1)[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using as and ** False by auto
thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2) and False by auto qed
- qed auto thus "t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *s x) \<in> s" by auto
+ qed auto thus "t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" by auto
qed
lemma convex_finite: assumes "finite s"
shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
- \<longrightarrow> setsum (\<lambda>x. u x *s x) s \<in> s)"
+ \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
unfolding convex_explicit apply(rule, rule, rule) defer apply(rule,rule,rule)apply(erule conjE)+ proof-
- fix t u assume as:"\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *s x) \<in> s" " finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
+ fix t u assume as:"\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s" " finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
have *:"s \<inter> t = t" using as(3) by auto
- show "(\<Sum>x\<in>t. u x *s x) \<in> s" using as(1)[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]]
+ show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" using as(1)[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]]
unfolding if_smult and setsum_cases[OF assms] and * using as(2-) by auto
qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
subsection {* Cones. *}
-definition "cone (s::(real^'n) set) \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *s x) \<in> s)"
+definition "cone (s::(real^'n) set) \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
lemma cone_empty[intro, simp]: "cone {}"
unfolding cone_def by auto
@@ -509,43 +516,45 @@
subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}
-definition "affine_dependent (s::(real^'n) set) \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
+definition
+ affine_dependent :: "(real ^ 'n::finite) set \<Rightarrow> bool" where
+ "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
lemma affine_dependent_explicit:
"affine_dependent p \<longleftrightarrow>
(\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
- (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *s v) s = 0)"
+ (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
proof-
- fix x s u assume as:"x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = x"
+ fix x s u assume as:"x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
have "x\<notin>s" using as(1,4) by auto
- show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *s v) = 0"
+ show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as using as by auto
next
- fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *s v) = 0" "v \<in> s" "u v \<noteq> 0"
+ fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
have "s \<noteq> {v}" using as(3,6) by auto
- thus "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = x"
+ thus "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
apply(rule_tac x=v in bexI, rule_tac x="s - {v}" in exI, rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
- unfolding vector_smult_assoc[THEN sym] and setsum_cmul unfolding setsum_right_distrib[THEN sym] and setsum_diff1_ring[OF as(1,5)] using as by auto
+ unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] unfolding setsum_right_distrib[THEN sym] and setsum_diff1_ring[OF as(1,5)] using as by auto
qed
lemma affine_dependent_explicit_finite:
- assumes "finite (s::(real^'n) set)"
- shows "affine_dependent s \<longleftrightarrow> (\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *s v) s = 0)"
+ assumes "finite (s::(real^'n::finite) set)"
+ shows "affine_dependent s \<longleftrightarrow> (\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
(is "?lhs = ?rhs")
proof
- have *:"\<And>vt u v. (if vt then u v else 0) *s v = (if vt then (u v) *s v else (0::real^'n))" by auto
+ have *:"\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else (0::real^'n))" by auto
assume ?lhs
- then obtain t u v where "finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0" "(\<Sum>v\<in>t. u v *s v) = 0"
+ then obtain t u v where "finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0" "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
unfolding affine_dependent_explicit by auto
thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
unfolding Int_absorb2[OF `t\<subseteq>s`, unfolded Int_commute] by auto
next
assume ?rhs
- then obtain u v where "setsum u s = 0" "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *s v) = 0" by auto
+ then obtain u v where "setsum u s = 0" "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" by auto
thus ?lhs unfolding affine_dependent_explicit using assms by auto
qed
@@ -560,24 +569,24 @@
hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
{ fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
- { fix y have *:"(1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2) = (y - x) *s x1 - (y - x) *s x2"
- by(simp add: ring_simps vector_sadd_rdistrib vector_sub_rdistrib)
+ { fix y have *:"(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2) = (y - x) *\<^sub>R x1 - (y - x) *\<^sub>R x2"
+ by (simp add: algebra_simps)
assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
- hence "norm ((1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2)) < e"
- unfolding * and vector_ssub_ldistrib[THEN sym] and norm_mul
+ hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
+ unfolding * and scaleR_right_diff_distrib[THEN sym]
unfolding less_divide_eq using n by auto }
- hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *s x1 + x *s x2 - ((1 - y) *s x1 + y *s x2)) < e"
+ hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
apply auto unfolding zero_less_divide_iff using n by simp } note * = this
- have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *s x1 + x *s x2 \<notin> e1 \<and> (1 - x) *s x1 + x *s x2 \<notin> e2"
+ have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2"
apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
using * apply(simp add: dist_norm)
using as(1,2)[unfolded open_dist] apply simp
using as(1,2)[unfolded open_dist] apply simp
using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
using as(3) by auto
- then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *s x1 + x *s x2 \<notin> e1" "(1 - x) *s x1 + x *s x2 \<notin> e2" by auto
+ then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2" by auto
hence False using as(4)
using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
using x1(2) x2(2) by auto }
@@ -592,7 +601,7 @@
subsection {* Convex functions into the reals. *}
definition "convex_on s (f::real^'n \<Rightarrow> real) =
- (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *s x + v *s y) \<le> u * f x + v * f y)"
+ (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
unfolding convex_on_def by auto
@@ -603,11 +612,11 @@
proof-
{ fix x y assume "x\<in>s" "y\<in>s" moreover
fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
- ultimately have "f (u *s x + v *s y) + g (u *s x + v *s y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
+ ultimately have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
apply - apply(rule add_mono) by auto
- hence "f (u *s x + v *s y) + g (u *s x + v *s y) \<le> u * (f x + g x) + v * (f y + g y)" by (simp add: ring_simps) }
+ hence "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)" by (simp add: ring_simps) }
thus ?thesis unfolding convex_on_def by auto
qed
@@ -621,7 +630,7 @@
lemma convex_lower:
assumes "convex_on s f" "x\<in>s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
- shows "f (u *s x + v *s y) \<le> max (f x) (f y)"
+ shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
proof-
let ?m = "max (f x) (f y)"
have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)" apply(rule add_mono)
@@ -642,13 +651,13 @@
then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
using real_lbound_gt_zero[of 1 "e / dist x y"] using xy `e>0` and divide_pos_pos[of e "dist x y"] by auto
- hence "f ((1-u) *s x + u *s y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>s`
+ hence "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>s`
using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]] by auto
moreover
- have *:"x - ((1 - u) *s x + u *s y) = u *s (x - y)" by (simp add: vector_ssub_ldistrib vector_sub_rdistrib)
- have "(1 - u) *s x + u *s y \<in> ball x e" unfolding mem_ball dist_norm unfolding * and norm_mul and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
+ have *:"x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)" by (simp add: algebra_simps)
+ have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e" unfolding mem_ball dist_norm unfolding * and norm_scaleR and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
using u unfolding pos_less_divide_eq[OF xy] by auto
- hence "f x \<le> f ((1 - u) *s x + u *s y)" using assms(4) by auto
+ hence "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)" using assms(4) by auto
ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
qed
@@ -656,31 +665,32 @@
proof(auto simp add: convex_on_def dist_norm)
fix x y assume "x\<in>s" "y\<in>s"
fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
- have "a = u *s a + v *s a" unfolding vector_sadd_rdistrib[THEN sym] and `u+v=1` by simp
- hence *:"a - (u *s x + v *s y) = (u *s (a - x)) + (v *s (a - y))" by auto
- show "norm (a - (u *s x + v *s y)) \<le> u * norm (a - x) + v * norm (a - y)"
- unfolding * using norm_triangle_ineq[of "u *s (a - x)" "v *s (a - y)"] unfolding norm_mul
+ have "a = u *\<^sub>R a + v *\<^sub>R a" unfolding scaleR_left_distrib[THEN sym] and `u+v=1` by simp
+ hence *:"a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
+ by (auto simp add: algebra_simps)
+ show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
+ unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
using `0 \<le> u` `0 \<le> v` by auto
qed
subsection {* Arithmetic operations on sets preserve convexity. *}
-lemma convex_scaling: "convex s \<Longrightarrow> convex ((\<lambda>x. c *s x) ` s)"
+lemma convex_scaling: "convex s \<Longrightarrow> convex ((\<lambda>x. c *\<^sub>R x) ` s)"
unfolding convex_def and image_iff apply auto
- apply (rule_tac x="u *s x+v *s y" in bexI) by (auto simp add: field_simps)
+ apply (rule_tac x="u *\<^sub>R x+v *\<^sub>R y" in bexI) by (auto simp add: algebra_simps)
lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)"
unfolding convex_def and image_iff apply auto
- apply (rule_tac x="u *s x+v *s y" in bexI) by auto
+ apply (rule_tac x="u *\<^sub>R x+v *\<^sub>R y" in bexI) by auto
lemma convex_sums:
assumes "convex s" "convex t"
shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
-proof(auto simp add: convex_def image_iff)
+proof(auto simp add: convex_def image_iff scaleR_right_distrib)
fix xa xb ya yb assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
- show "\<exists>x y. u *s xa + u *s ya + (v *s xb + v *s yb) = x + y \<and> x \<in> s \<and> y \<in> t"
- apply(rule_tac x="u *s xa + v *s xb" in exI) apply(rule_tac x="u *s ya + v *s yb" in exI)
+ show "\<exists>x y. u *\<^sub>R xa + u *\<^sub>R ya + (v *\<^sub>R xb + v *\<^sub>R yb) = x + y \<and> x \<in> s \<and> y \<in> t"
+ apply(rule_tac x="u *\<^sub>R xa + v *\<^sub>R xb" in exI) apply(rule_tac x="u *\<^sub>R ya + v *\<^sub>R yb" in exI)
using assms(1)[unfolded convex_def, THEN bspec[where x=xa], THEN bspec[where x=xb]]
using assms(2)[unfolded convex_def, THEN bspec[where x=ya], THEN bspec[where x=yb]]
using uv xy by auto
@@ -700,17 +710,17 @@
proof- have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
thus ?thesis using convex_sums[OF convex_singleton[of a] assms] by auto qed
-lemma convex_affinity: assumes "convex (s::(real^'n) set)" shows "convex ((\<lambda>x. a + c *s x) ` s)"
-proof- have "(\<lambda>x. a + c *s x) ` s = op + a ` op *s c ` s" by auto
+lemma convex_affinity: assumes "convex (s::(real^'n::finite) set)" shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` s)"
+proof- have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s" by auto
thus ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed
lemma convex_linear_image: assumes c:"convex s" and l:"linear f" shows "convex(f ` s)"
proof(auto simp add: convex_def)
fix x y assume xy:"x \<in> s" "y \<in> s"
fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
- show "u *s f x + v *s f y \<in> f ` s" unfolding image_iff
- apply(rule_tac x="u *s x + v *s y" in bexI)
- unfolding linear_add[OF l] linear_cmul[OF l]
+ show "u *\<^sub>R f x + v *\<^sub>R f y \<in> f ` s" unfolding image_iff
+ apply(rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in bexI)
+ unfolding linear_add[OF l] linear_cmul[OF l, unfolded smult_conv_scaleR]
using c[unfolded convex_def] xy uv by auto
qed
@@ -720,18 +730,18 @@
proof(auto simp add: convex_def)
fix y z assume yz:"dist x y < e" "dist x z < e"
fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
- have "dist x (u *s y + v *s z) \<le> u * dist x y + v * dist x z" using uv yz
+ have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
- thus "dist x (u *s y + v *s z) < e" using real_convex_bound_lt[OF yz uv] by auto
+ thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using real_convex_bound_lt[OF yz uv] by auto
qed
lemma convex_cball: "convex(cball x e)"
proof(auto simp add: convex_def Ball_def mem_cball)
fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
- have "dist x (u *s y + v *s z) \<le> u * dist x y + v * dist x z" using uv yz
+ have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
- thus "dist x (u *s y + v *s z) \<le> e" using real_convex_bound_le[OF yz uv] by auto
+ thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using real_convex_bound_le[OF yz uv] by auto
qed
lemma connected_ball: "connected(ball (x::real^_) e)" (* FIXME: generalize *)
@@ -770,14 +780,14 @@
lemma convex_hull_insert:
assumes "s \<noteq> {}"
shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
- b \<in> (convex hull s) \<and> (x = u *s a + v *s b)}" (is "?xyz = ?hull")
+ b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
fix x assume x:"x = a \<or> x \<in> s"
thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer
apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
next
fix x assume "x\<in>?hull"
- then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *s a + v *s b" by auto
+ then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *\<^sub>R a + v *\<^sub>R b" by auto
have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
@@ -785,16 +795,16 @@
next
show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
- from as(4) obtain u1 v1 b1 where obt1:"u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *s a + v1 *s b1" by auto
- from as(5) obtain u2 v2 b2 where obt2:"u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *s a + v2 *s b2" by auto
- have *:"\<And>x s1 s2. x - s1 *s x - s2 *s x = ((1::real) - (s1 + s2)) *s x" by auto
- have "\<exists>b \<in> convex hull s. u *s x + v *s y = (u * u1) *s a + (v * u2) *s a + (b - (u * u1) *s b - (v * u2) *s b)"
+ from as(4) obtain u1 v1 b1 where obt1:"u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1" by auto
+ from as(5) obtain u2 v2 b2 where obt2:"u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2" by auto
+ have *:"\<And>(x::real^_) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
+ have "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
proof(cases "u * v1 + v * v2 = 0")
- have *:"\<And>x s1 s2. x - s1 *s x - s2 *s x = ((1::real) - (s1 + s2)) *s x" by auto
+ have *:"\<And>(x::real^_) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
case True hence **:"u * v1 = 0" "v * v2 = 0" apply- apply(rule_tac [!] ccontr)
using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by auto
hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
- thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: **)
+ thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
next
have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
@@ -803,9 +813,10 @@
apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
using as(1,2) obt1(1,2) obt2(1,2) by auto
thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
- apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *s b1 + ((v * v2) / (u * v1 + v * v2)) *s b2" in bexI) defer
+ apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI) defer
apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
- unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff by auto
+ unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff
+ by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
qed note * = this
have u1:"u1 \<le> 1" apply(rule ccontr) unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
have u2:"u2 \<le> 1" apply(rule ccontr) unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
@@ -813,9 +824,9 @@
apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] and as(3) using u1 u2 by auto
finally
- show "u *s x + v *s y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
+ show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
- using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add:field_simps)
+ using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
qed
qed
@@ -825,7 +836,7 @@
lemma convex_hull_indexed:
"convex hull s = {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
(setsum u {1..k} = 1) \<and>
- (setsum (\<lambda>i. u i *s x i) {1..k} = y)}" (is "?xyz = ?hull")
+ (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}" (is "?xyz = ?hull")
apply(rule hull_unique) unfolding mem_def[of _ convex] apply(rule) defer
apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
proof-
@@ -834,22 +845,22 @@
next
fix t assume as:"s \<subseteq> t" "convex t"
show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
- fix x k u y assume assm:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *s y i) = x"
+ fix x k u y assume assm:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
using assm(1,2) as(1) by auto qed
next
fix x y u v assume uv:"0\<le>u" "0\<le>v" "u+v=(1::real)" and xy:"x\<in>?hull" "y\<in>?hull"
- from xy obtain k1 u1 x1 where x:"\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *s x1 i) = x" by auto
- from xy obtain k2 u2 x2 where y:"\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *s x2 i) = y" by auto
- have *:"\<And>P x1 x2 s1 s2 i.(if P i then s1 else s2) *s (if P i then x1 else x2) = (if P i then s1 *s x1 else s2 *s x2)"
+ from xy obtain k1 u1 x1 where x:"\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x" by auto
+ from xy obtain k2 u2 x2 where y:"\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y" by auto
+ have *:"\<And>P x1 x2 s1 s2 i.(if P i then s1 else s2) *\<^sub>R (if P i then x1 else x2) = (if P i then s1 *\<^sub>R x1 else s2 *\<^sub>R x2)"
"{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto
- show "u *s x + v *s y \<in> ?hull" apply(rule)
+ show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" apply(rule)
apply(rule_tac x="k1 + k2" in exI, rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
apply(rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI) apply(rule,rule) defer apply(rule)
unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def
- unfolding vector_smult_assoc[THEN sym] setsum_cmul setsum_right_distrib[THEN sym] proof-
+ unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] proof-
fix i assume i:"i \<in> {1..k1+k2}"
show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and> (if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
proof(cases "i\<in>{1..k1}")
@@ -862,11 +873,11 @@
qed
lemma convex_hull_finite:
- assumes "finite (s::(real^'n)set)"
+ assumes "finite (s::(real^'n::finite)set)"
shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
- setsum u s = 1 \<and> setsum (\<lambda>x. u x *s x) s = y}" (is "?HULL = ?set")
+ setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
proof(rule hull_unique, auto simp add: mem_def[of _ convex] convex_def[of ?set])
- fix x assume "x\<in>s" thus " \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *s x) = x"
+ fix x assume "x\<in>s" thus " \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x"
apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto
next
@@ -878,14 +889,14 @@
by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2)) }
moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
- moreover have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *s x) = u *s (\<Sum>x\<in>s. ux x *s x) + v *s (\<Sum>x\<in>s. uy x *s x)"
- unfolding vector_sadd_rdistrib and setsum_addf and vector_smult_assoc[THEN sym] and setsum_cmul by auto
- ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<Sum>x\<in>s. uc x *s x) = u *s (\<Sum>x\<in>s. ux x *s x) + v *s (\<Sum>x\<in>s. uy x *s x)"
+ moreover have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
+ unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] by auto
+ ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<Sum>x\<in>s. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto
next
fix t assume t:"s \<subseteq> t" "convex t"
fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
- thus "(\<Sum>x\<in>s. u x *s x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
+ thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
using assms and t(1) by auto
qed
@@ -893,10 +904,10 @@
lemma convex_hull_explicit:
"convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
- (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}" (is "?lhs = ?rhs")
+ (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}" (is "?lhs = ?rhs")
proof-
{ fix x assume "x\<in>?lhs"
- then obtain k u y where obt:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *s y i) = x"
+ then obtain k u y where obt:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
unfolding convex_hull_indexed by auto
have fin:"finite {1..k}" by auto
@@ -908,16 +919,16 @@
moreover
have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"
unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
- moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *s v) = x"
- using setsum_image_gen[OF fin, of "\<lambda>i. u i *s y i" y, THEN sym]
- unfolding setsum_vmul[OF fin'] using obt(3) by auto
- ultimately have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = x"
+ moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
+ using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, THEN sym]
+ unfolding scaleR_left.setsum using obt(3) by auto
+ ultimately have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
apply(rule_tac x="y ` {1..k}" in exI)
apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
hence "x\<in>?rhs" by auto }
moreover
{ fix y assume "y\<in>?rhs"
- then obtain s u where obt:"finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = y" by auto
+ then obtain s u where obt:"finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
obtain f where f:"inj_on f {1..card s}" "f ` {1..card s} = s" using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
@@ -929,14 +940,14 @@
then obtain i where "i\<in>{1..card s}" "f i = y" using f using image_iff[of y f "{1..card s}"] by auto
hence "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}" apply auto using f(1)[unfolded inj_on_def] apply(erule_tac x=x in ballE) by auto
hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
- hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y" "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *s f x) = u y *s y" by auto }
-
- hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *s f i) = y"
- unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *s f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
- unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *s f x)" "\<lambda>v. u v *s v"]
+ hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y" "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y" by auto }
+
+ hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
+ unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
+ unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
using setsum_cong2 [of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u] unfolding obt(4,5) by auto
- ultimately have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and> (\<Sum>i::nat = 1..k. u i *s x i) = y"
+ ultimately have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and> (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
apply(rule_tac x="card s" in exI) apply(rule_tac x="u \<circ> f" in exI) apply(rule_tac x=f in exI) by fastsimp
hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto }
ultimately show ?thesis unfolding expand_set_eq by blast
@@ -946,24 +957,24 @@
lemma convex_hull_finite_step:
assumes "finite (s::(real^'n) set)"
- shows "(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *s x) (insert a s) = y)
- \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = w - v \<and> setsum (\<lambda>x. u x *s x) s = y - v *s a)" (is "?lhs = ?rhs")
+ shows "(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y)
+ \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?lhs = ?rhs")
proof(rule, case_tac[!] "a\<in>s")
assume "a\<in>s" hence *:"insert a s = s" by auto
assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
next
- assume ?lhs then obtain u where u:"\<forall>x\<in>insert a s. 0 \<le> u x" "setsum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *s x) = y" by auto
+ assume ?lhs then obtain u where u:"\<forall>x\<in>insert a s. 0 \<le> u x" "setsum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
assume "a\<notin>s" thus ?rhs apply(rule_tac x="u a" in exI) using u(1)[THEN bspec[where x=a]] apply simp
apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
next
assume "a\<in>s" hence *:"insert a s = s" by auto
have fin:"finite (insert a s)" using assms by auto
- assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *s x) = y - v *s a" by auto
- show ?lhs apply(rule_tac x="\<lambda>x. (if a = x then v else 0) + u x" in exI) unfolding vector_sadd_rdistrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
+ assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
+ show ?lhs apply(rule_tac x="\<lambda>x. (if a = x then v else 0) + u x" in exI) unfolding scaleR_left_distrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
next
- assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *s x) = y - v *s a" by auto
- moreover assume "a\<notin>s" moreover have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s" "(\<Sum>x\<in>s. (if a = x then v else u x) *s x) = (\<Sum>x\<in>s. u x *s x)"
+ assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
+ moreover assume "a\<notin>s" moreover have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s" "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
ultimately show ?lhs apply(rule_tac x="\<lambda>x. if a = x then v else u x" in exI) unfolding setsum_clauses(2)[OF assms] by auto
qed
@@ -971,20 +982,20 @@
subsection {* Hence some special cases. *}
lemma convex_hull_2:
- "convex hull {a,b} = {u *s a + v *s b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
+ "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
proof- have *:"\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b" by auto have **:"finite {b}" by auto
show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
-lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *s (b - a) | u. 0 \<le> u \<and> u \<le> 1}"
+lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u. 0 \<le> u \<and> u \<le> 1}"
unfolding convex_hull_2 unfolding Collect_def
proof(rule ext) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
- fix x show "(\<exists>v u. x = v *s a + u *s b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) = (\<exists>u. x = a + u *s (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
- unfolding * apply auto apply(rule_tac[!] x=u in exI) by auto qed
+ fix x show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) = (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
+ unfolding * apply auto apply(rule_tac[!] x=u in exI) by (auto simp add: algebra_simps) qed
lemma convex_hull_3:
- "convex hull {a::real^'n,b,c} = { u *s a + v *s b + w *s c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
+ "convex hull {a::real^'n::finite,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
proof-
have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
@@ -995,15 +1006,15 @@
apply(rule_tac x="1 - v - w" in exI) apply simp apply(rule_tac x=v in exI) apply simp apply(rule_tac x="\<lambda>x. w" in exI) by simp qed
lemma convex_hull_3_alt:
- "convex hull {a,b,c} = {a + u *s (b - a) + v *s (c - a) | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
+ "convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
- show ?thesis unfolding convex_hull_3 apply (auto simp add: *) apply(rule_tac x=v in exI) apply(rule_tac x=w in exI) apply simp
- apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by simp qed
+ show ?thesis unfolding convex_hull_3 apply (auto simp add: *) apply(rule_tac x=v in exI) apply(rule_tac x=w in exI) apply (simp add: algebra_simps)
+ apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by (simp add: algebra_simps) qed
subsection {* Relations among closure notions and corresponding hulls. *}
lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
- unfolding subspace_def affine_def by auto
+ unfolding subspace_def affine_def smult_conv_scaleR by auto
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
unfolding affine_def convex_def by auto
@@ -1031,8 +1042,8 @@
assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
shows "affine_dependent (insert a s)"
proof-
- from assms(1)[unfolded dependent_explicit] obtain S u v
- where obt:"finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *s v) = 0" by auto
+ from assms(1)[unfolded dependent_explicit smult_conv_scaleR] obtain S u v
+ where obt:"finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" by auto
def t \<equiv> "(\<lambda>x. x + a) ` S"
have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
@@ -1046,24 +1057,24 @@
unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
- moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *s x) = (\<Sum>x\<in>t. Q x *s x)"
+ moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
- have "(\<Sum>x\<in>t. u (x - a)) *s a = (\<Sum>v\<in>t. u (v - a) *s v)"
- unfolding setsum_vmul[OF fin(1)] unfolding t_def and setsum_reindex[OF inj] and o_def
- using obt(5) by (auto simp add: setsum_addf)
- hence "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *s v) = 0"
+ have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
+ unfolding scaleR_left.setsum unfolding t_def and setsum_reindex[OF inj] and o_def
+ using obt(5) by (auto simp add: setsum_addf scaleR_right_distrib)
+ hence "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: * vector_smult_lneg)
ultimately show ?thesis unfolding affine_dependent_explicit
apply(rule_tac x="insert a t" in exI) by auto
qed
lemma convex_cone:
- "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *s x) \<in> s)" (is "?lhs = ?rhs")
+ "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)" (is "?lhs = ?rhs")
proof-
{ fix x y assume "x\<in>s" "y\<in>s" and ?lhs
- hence "2 *s x \<in>s" "2 *s y \<in> s" unfolding cone_def by auto
+ hence "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" unfolding cone_def by auto
hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
- apply(erule_tac x="2*s x" in ballE) apply(erule_tac x="2*s y" in ballE)
+ apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto }
thus ?thesis unfolding convex_def cone_def by blast
qed
@@ -1104,20 +1115,20 @@
lemma convex_hull_caratheodory: fixes p::"(real^'n::finite) set"
shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and>
- (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *s v) s = y}"
+ (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
unfolding convex_hull_explicit expand_set_eq mem_Collect_eq
proof(rule,rule)
- fix y let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = y"
- assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = y"
+ fix y let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+ assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
then obtain N where "?P N" by auto
hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
- then obtain s u where obt:"finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = y" by auto
+ then obtain s u where obt:"finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
have "card s \<le> CARD('n) + 1" proof(rule ccontr, simp only: not_le)
assume "CARD('n) + 1 < card s"
hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
- then obtain w v where wv:"setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *s v) = 0"
+ then obtain w v where wv:"setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
using affine_dependent_explicit_finite[OF obt(1)] by auto
def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}" def t \<equiv> "Min i"
have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
@@ -1147,15 +1158,15 @@
have *:"\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'a::ring)" unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto
have "(\<Sum>v\<in>s. u v + t * w v) = 1"
unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
- moreover have "(\<Sum>v\<in>s. u v *s v + (t * w v) *s v) - (u a *s a + (t * w a) *s a) = y"
- unfolding setsum_addf obt(6) vector_smult_assoc[THEN sym] setsum_cmul wv(4)
+ moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
+ unfolding setsum_addf obt(6) scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] wv(4)
using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]]
by (simp add: vector_smult_lneg)
ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
- apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a by (auto simp add: *)
+ apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a by (auto simp add: * scaleR_left_distrib)
thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1
- \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *s v) = y" using obt by auto
+ \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y" using obt by auto
qed auto
lemma caratheodory:
@@ -1164,7 +1175,7 @@
unfolding expand_set_eq apply(rule, rule) unfolding mem_Collect_eq proof-
fix x assume "x \<in> convex hull p"
then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> CARD('n) + 1"
- "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *s v) = x"unfolding convex_hull_caratheodory by auto
+ "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"unfolding convex_hull_caratheodory by auto
thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> CARD('n) + 1 \<and> x \<in> convex hull s"
apply(rule_tac x=s in exI) using hull_subset[of s convex]
using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
@@ -1181,14 +1192,14 @@
shows "open(convex hull s)"
unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10)
proof(rule, rule) fix a
- assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *s v) = a"
- then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *s v) = a" by auto
+ assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
+ then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a" by auto
from assms[unfolded open_contains_cball] obtain b where b:"\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
have "b ` t\<noteq>{}" unfolding i_def using obt by auto def i \<equiv> "b ` t"
- show "\<exists>e>0. cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *s v) = y}"
+ show "\<exists>e>0. cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
proof-
show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
@@ -1204,41 +1215,65 @@
have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
unfolding setsum_reindex[OF *] o_def using obt(4) by auto
- moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *s v) = y"
+ moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
unfolding setsum_reindex[OF *] o_def using obt(4,5)
- by (simp add: setsum_addf setsum_subtractf setsum_vmul[OF obt(1), THEN sym])
- ultimately show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *s v) = y"
+ by (simp add: setsum_addf setsum_subtractf scaleR_left.setsum[THEN sym] scaleR_right_distrib)
+ ultimately show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
apply(rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI) apply(rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
using obt(1, 3) by auto
qed
qed
+lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
+unfolding open_vector_def all_1
+by (auto simp add: dest_vec1_def)
+
+lemma tendsto_dest_vec1 [tendsto_intros]:
+ "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
+ unfolding tendsto_def
+ apply clarify
+ apply (drule_tac x="dest_vec1 -` S" in spec)
+ apply (simp add: open_dest_vec1_vimage)
+ done
+
+lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
+ unfolding continuous_def by (rule tendsto_dest_vec1)
+
+(* TODO: move *)
+lemma compact_real_interval:
+ fixes a b :: real shows "compact {a..b}"
+proof -
+ have "continuous_on {vec1 a .. vec1 b} dest_vec1"
+ unfolding continuous_on
+ by (simp add: tendsto_dest_vec1 Lim_at_within Lim_ident_at)
+ moreover have "compact {vec1 a .. vec1 b}" by (rule compact_interval)
+ ultimately have "compact (dest_vec1 ` {vec1 a .. vec1 b})"
+ by (rule compact_continuous_image)
+ also have "dest_vec1 ` {vec1 a .. vec1 b} = {a..b}"
+ by (auto simp add: image_def Bex_def exists_vec1)
+ finally show ?thesis .
+qed
lemma compact_convex_combinations:
- fixes s t :: "(real ^ _) set"
+ fixes s t :: "(real ^ 'n::finite) set"
assumes "compact s" "compact t"
- shows "compact { (1 - u) *s x + u *s y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
+ shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
proof-
- let ?X = "{ pastecart u w | u w. u \<in> {vec1 0 .. vec1 1} \<and> w \<in> { pastecart x y |x y. x \<in> s \<and> y \<in> t} }"
- let ?h = "(\<lambda>z. (1 - dest_vec1(fstcart z)) *s fstcart(sndcart z) + dest_vec1(fstcart z) *s sndcart(sndcart z))"
- have *:"{ (1 - u) *s x + u *s y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
- apply(rule set_ext) unfolding image_iff mem_Collect_eq unfolding mem_interval_1 vec1_dest_vec1
- apply rule apply auto apply(rule_tac x="pastecart (vec1 u) (pastecart xa y)" in exI) apply simp
- apply(rule_tac x="vec1 u" in exI) apply(rule_tac x="pastecart xa y" in exI) by auto
- { fix u::"real^1" fix x y assume as:"0 \<le> dest_vec1 u" "dest_vec1 u \<le> 1" "x \<in> s" "y \<in> t"
- hence "continuous (at (pastecart u (pastecart x y)))
- (\<lambda>z. fstcart (sndcart z) - dest_vec1 (fstcart z) *s fstcart (sndcart z) +
- dest_vec1 (fstcart z) *s sndcart (sndcart z))"
- apply (auto intro!: continuous_add continuous_sub continuous_mul simp add: o_def vec1_dest_vec1)
- using linear_continuous_at linear_fstcart linear_sndcart linear_sndcart
- using linear_compose[unfolded o_def] by auto }
- hence "continuous_on {pastecart u w |u w. u \<in> {vec1 0..vec1 1} \<and> w \<in> {pastecart x y |x y. x \<in> s \<and> y \<in> t}}
- (\<lambda>z. (1 - dest_vec1 (fstcart z)) *s fstcart (sndcart z) + dest_vec1 (fstcart z) *s sndcart (sndcart z))"
- apply(rule_tac continuous_at_imp_continuous_on) unfolding mem_Collect_eq
- unfolding mem_interval_1 vec1_dest_vec1 by auto
- thus ?thesis unfolding * apply(rule compact_continuous_image)
- defer apply(rule compact_pastecart) defer apply(rule compact_pastecart)
- using compact_interval assms by auto
+ let ?X = "{0..1} \<times> s \<times> t"
+ let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
+ have *:"{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
+ apply(rule set_ext) unfolding image_iff mem_Collect_eq
+ apply rule apply auto
+ apply (rule_tac x=u in rev_bexI, simp)
+ apply (erule rev_bexI, erule rev_bexI, simp)
+ by auto
+ have "continuous_on ({0..1} \<times> s \<times> t)
+ (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
+ unfolding continuous_on by (rule ballI) (intro tendsto_intros)
+ thus ?thesis unfolding *
+ apply (rule compact_continuous_image)
+ apply (intro compact_Times compact_real_interval assms)
+ done
qed
lemma compact_convex_hull: fixes s::"(real^'n::finite) set"
@@ -1273,14 +1308,14 @@
qed thus ?thesis using assms by simp
next
case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
- { (1 - u) *s x + u *s y | x y u.
+ { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u.
0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> {x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> x \<in> convex hull t}}"
unfolding expand_set_eq and mem_Collect_eq proof(rule,rule)
- fix x assume "\<exists>u v c. x = (1 - c) *s u + c *s v \<and>
+ fix x assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
- then obtain u v c t where obt:"x = (1 - c) *s u + c *s v"
+ then obtain u v c t where obt:"x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
"0 \<le> c \<and> c \<le> 1" "u \<in> s" "finite t" "t \<subseteq> s" "card t \<le> n" "v \<in> convex hull t" by auto
- moreover have "(1 - c) *s u + c *s v \<in> convex hull insert u t"
+ moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
using obt(7) and hull_mono[of t "insert u t"] by auto
ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
@@ -1288,7 +1323,7 @@
next
fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
- let ?P = "\<exists>u v c. x = (1 - c) *s u + c *s v \<and>
+ let ?P = "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
show ?P proof(cases "card t = Suc n")
case False hence "card t \<le> n" using t(3) by auto
@@ -1301,7 +1336,7 @@
show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"] simp add: convex_hull_singleton)
next
- case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *s a + vx *s b"
+ case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
using t(4)[unfolded au convex_hull_insert[OF False]] by auto
have *:"1 - vx = ux" using obt(3) by auto
show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
@@ -1325,16 +1360,16 @@
fixes a b d :: "real ^ 'n::finite"
assumes "d \<noteq> 0"
shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
-proof(cases "a \<bullet> d - b \<bullet> d > 0")
- case True hence "0 < d \<bullet> d + (a \<bullet> d * 2 - b \<bullet> d * 2)"
+proof(cases "inner a d - inner b d > 0")
+ case True hence "0 < inner d d + (inner a d * 2 - inner b d * 2)"
apply(rule_tac add_pos_pos) using assms by auto
- thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and real_vector_norm_def and real_sqrt_less_iff
- by(simp add: dot_rsub dot_radd dot_lsub dot_ladd dot_sym field_simps)
+ thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
+ by (simp add: algebra_simps inner_commute)
next
- case False hence "0 < d \<bullet> d + (b \<bullet> d * 2 - a \<bullet> d * 2)"
+ case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)"
apply(rule_tac add_pos_nonneg) using assms by auto
- thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and real_vector_norm_def and real_sqrt_less_iff
- by(simp add: dot_rsub dot_radd dot_lsub dot_ladd dot_sym field_simps)
+ thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
+ by (simp add: algebra_simps inner_commute)
qed
lemma norm_increases_online:
@@ -1349,7 +1384,7 @@
show "\<forall>xa\<in>convex hull insert x s. xa \<notin> insert x s \<longrightarrow> (\<exists>y\<in>convex hull insert x s. norm (xa - a) < norm (y - a))"
proof(rule,rule,cases "s = {}")
case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
- obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "y = u *s x + v *s b"
+ obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "y = u *\<^sub>R x + v *\<^sub>R b"
using y(1)[unfolded convex_hull_insert[OF False]] by auto
show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
proof(cases "y\<in>convex hull s")
@@ -1368,24 +1403,24 @@
then obtain w where w:"w>0" "w<u" "w<v" using real_lbound_gt_zero[of u v] and obt(1,2) by auto
have "x\<noteq>b" proof(rule ccontr)
assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
- using obt(3) by(auto simp add: vector_sadd_rdistrib[THEN sym])
+ using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
thus False using obt(4) and False by simp qed
- hence *:"w *s (x - b) \<noteq> 0" using w(1) by auto
+ hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
show ?thesis using dist_increases_online[OF *, of a y]
proof(erule_tac disjE)
- assume "dist a y < dist a (y + w *s (x - b))"
- hence "norm (y - a) < norm ((u + w) *s x + (v - w) *s b - a)"
- unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: ring_simps)
- moreover have "(u + w) *s x + (v - w) *s b \<in> convex hull insert x s"
+ assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
+ hence "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
+ unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
+ moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
apply(rule_tac x="u + w" in exI) apply rule defer
apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
ultimately show ?thesis by auto
next
- assume "dist a y < dist a (y - w *s (x - b))"
- hence "norm (y - a) < norm ((u - w) *s x + (v + w) *s b - a)"
- unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: ring_simps)
- moreover have "(u - w) *s x + (v + w) *s b \<in> convex hull insert x s"
+ assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
+ hence "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
+ unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
+ moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
apply(rule_tac x="u - w" in exI) apply rule defer
apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
@@ -1469,38 +1504,59 @@
"closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
using closest_point_in_set[of s x] closest_point_self[of x s] by auto
+(* TODO: move *)
+lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
+ unfolding norm_eq_sqrt_inner by simp
+
+(* TODO: move *)
+lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
+ unfolding norm_eq_sqrt_inner by simp
+
lemma closer_points_lemma: fixes y::"real^'n::finite"
- assumes "y \<bullet> z > 0"
- shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *s z - y) < norm y"
-proof- have z:"z \<bullet> z > 0" unfolding dot_pos_lt using assms by auto
- thus ?thesis using assms apply(rule_tac x="(y \<bullet> z) / (z \<bullet> z)" in exI) apply(rule) defer proof(rule+)
- fix v assume "0<v" "v \<le> y \<bullet> z / (z \<bullet> z)"
- thus "norm (v *s z - y) < norm y" unfolding norm_lt using z and assms
- by (simp add: field_simps dot_sym mult_strict_left_mono[OF _ `0<v`])
+ assumes "inner y z > 0"
+ shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
+proof- have z:"inner z z > 0" unfolding inner_gt_zero_iff using assms by auto
+ thus ?thesis using assms apply(rule_tac x="inner y z / inner z z" in exI) apply(rule) defer proof(rule+)
+ fix v assume "0<v" "v \<le> inner y z / inner z z"
+ thus "norm (v *\<^sub>R z - y) < norm y" unfolding norm_lt using z and assms
+ by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
qed(rule divide_pos_pos, auto) qed
lemma closer_point_lemma:
fixes x y z :: "real ^ 'n::finite"
- assumes "(y - x) \<bullet> (z - x) > 0"
- shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *s (z - x)) y < dist x y"
-proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *s (z - x) - (y - x)) < norm (y - x)"
+ assumes "inner (y - x) (z - x) > 0"
+ shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
+proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
using closer_points_lemma[OF assms] by auto
show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
lemma any_closest_point_dot:
assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
- shows "(a - x) \<bullet> (y - x) \<le> 0"
-proof(rule ccontr) assume "\<not> (a - x) \<bullet> (y - x) \<le> 0"
- then obtain u where u:"u>0" "u\<le>1" "dist (x + u *s (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
- let ?z = "(1 - u) *s x + u *s y" have "?z \<in> s" using mem_convex[OF assms(1,3,4), of u] using u by auto
- thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute field_simps) qed
+ shows "inner (a - x) (y - x) \<le> 0"
+proof(rule ccontr) assume "\<not> inner (a - x) (y - x) \<le> 0"
+ then obtain u where u:"u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
+ let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y" have "?z \<in> s" using mem_convex[OF assms(1,3,4), of u] using u by auto
+ thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute algebra_simps) qed
+
+(* TODO: move *)
+lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> inner x x \<le> a\<twosuperior>"
+proof -
+ have "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> norm x \<le> a"
+ using norm_ge_zero [of x] by arith
+ also have "\<dots> \<longleftrightarrow> 0 \<le> a \<and> (norm x)\<twosuperior> \<le> a\<twosuperior>"
+ by (auto intro: power_mono dest: power2_le_imp_le)
+ also have "\<dots> \<longleftrightarrow> 0 \<le> a \<and> inner x x \<le> a\<twosuperior>"
+ unfolding power2_norm_eq_inner ..
+ finally show ?thesis .
+qed
lemma any_closest_point_unique:
assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
"\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
shows "x = y" using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
- unfolding norm_pths(1) and norm_le_square by auto
+ unfolding norm_pths(1) and norm_le_square
+ by (auto simp add: algebra_simps)
lemma closest_point_unique:
assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
@@ -1510,7 +1566,7 @@
lemma closest_point_dot:
assumes "convex s" "closed s" "x \<in> s"
- shows "(a - closest_point s a) \<bullet> (x - closest_point s a) \<le> 0"
+ shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
using closest_point_exists[OF assms(2)] and assms(3) by auto
@@ -1525,13 +1581,13 @@
assumes "convex s" "closed s" "s \<noteq> {}"
shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
proof-
- have "(x - closest_point s x) \<bullet> (closest_point s y - closest_point s x) \<le> 0"
- "(y - closest_point s y) \<bullet> (closest_point s x - closest_point s y) \<le> 0"
+ have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
+ "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
using closest_point_exists[OF assms(2-3)] by auto
thus ?thesis unfolding dist_norm and norm_le
- using dot_pos_le[of "(x - closest_point s x) - (y - closest_point s y)"]
- by (auto simp add: dot_sym dot_ladd dot_radd) qed
+ using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
+ by (simp add: inner_add inner_diff inner_commute) qed
lemma continuous_at_closest_point:
assumes "convex s" "closed s" "s \<noteq> {}"
@@ -1548,50 +1604,50 @@
lemma supporting_hyperplane_closed_point:
assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
- shows "\<exists>a b. \<exists>y\<in>s. a \<bullet> z < b \<and> (a \<bullet> y = b) \<and> (\<forall>x\<in>s. a \<bullet> x \<ge> b)"
+ shows "\<exists>a b. \<exists>y\<in>s. inner a z < b \<and> (inner a y = b) \<and> (\<forall>x\<in>s. inner a x \<ge> b)"
proof-
from distance_attains_inf[OF assms(2-3)] obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x" by auto
- show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="(y - z) \<bullet> y" in exI, rule_tac x=y in bexI)
+ show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) y" in exI, rule_tac x=y in bexI)
apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
- show "(y - z) \<bullet> z < (y - z) \<bullet> y" apply(subst diff_less_iff(1)[THEN sym])
- unfolding dot_rsub[THEN sym] and dot_pos_lt using `y\<in>s` `z\<notin>s` by auto
+ show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
+ unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y\<in>s` `z\<notin>s` by auto
next
- fix x assume "x\<in>s" have *:"\<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *s y + u *s x)"
+ fix x assume "x\<in>s" have *:"\<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
- assume "\<not> (y - z) \<bullet> y \<le> (y - z) \<bullet> x" then obtain v where
- "v>0" "v\<le>1" "dist (y + v *s (x - y)) z < dist y z" using closer_point_lemma[of z y x] by auto
- thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute field_simps)
+ assume "\<not> inner (y - z) y \<le> inner (y - z) x" then obtain v where
+ "v>0" "v\<le>1" "dist (y + v *\<^sub>R (x - y)) z < dist y z" using closer_point_lemma[of z y x] apply - by (auto simp add: inner_diff)
+ thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute algebra_simps)
qed auto
qed
lemma separating_hyperplane_closed_point:
assumes "convex s" "closed s" "z \<notin> s"
- shows "\<exists>a b. a \<bullet> z < b \<and> (\<forall>x\<in>s. a \<bullet> x > b)"
+ shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
proof(cases "s={}")
case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
- using less_le_trans[OF _ dot_pos_le[of z]] by auto
+ using less_le_trans[OF _ inner_ge_zero[of z]] by auto
next
case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
using distance_attains_inf[OF assms(2) False] by auto
- show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="(y - z) \<bullet> z + (norm(y - z))\<twosuperior> / 2" in exI)
+ show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) z + (norm(y - z))\<twosuperior> / 2" in exI)
apply rule defer apply rule proof-
fix x assume "x\<in>s"
- have "\<not> 0 < (z - y) \<bullet> (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
- assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *s (x - y)) z < dist y z"
- then obtain u where "u>0" "u\<le>1" "dist (y + u *s (x - y)) z < dist y z" by auto
- thus False using y[THEN bspec[where x="y + u *s (x - y)"]]
+ have "\<not> 0 < inner (z - y) (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
+ assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
+ then obtain u where "u>0" "u\<le>1" "dist (y + u *\<^sub>R (x - y)) z < dist y z" by auto
+ thus False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
- using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute field_simps) qed
+ using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps) qed
moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
- hence "0 < (y - z) \<bullet> (y - z)" unfolding norm_pow_2 by simp
- ultimately show "(y - z) \<bullet> z + (norm (y - z))\<twosuperior> / 2 < (y - z) \<bullet> x"
- unfolding norm_pow_2 and dlo_simps(3) by (auto simp add: field_simps dot_sym)
+ hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
+ ultimately show "inner (y - z) z + (norm (y - z))\<twosuperior> / 2 < inner (y - z) x"
+ unfolding power2_norm_eq_inner and not_less by (auto simp add: field_simps inner_commute inner_diff)
qed(insert `y\<in>s` `z\<notin>s`, auto)
qed
lemma separating_hyperplane_closed_0:
assumes "convex (s::(real^'n::finite) set)" "closed s" "0 \<notin> s"
- shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. a \<bullet> x > b)"
+ shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
proof(cases "s={}") guess a using UNIV_witness[where 'a='n] ..
case True have "norm ((basis a)::real^'n::finite) = 1"
using norm_basis and dimindex_ge_1 by auto
@@ -1603,41 +1659,41 @@
lemma separating_hyperplane_closed_compact:
assumes "convex (s::(real^'n::finite) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
- shows "\<exists>a b. (\<forall>x\<in>s. a \<bullet> x < b) \<and> (\<forall>x\<in>t. a \<bullet> x > b)"
+ shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
proof(cases "s={}")
case True
obtain b where b:"b>0" "\<forall>x\<in>t. norm x \<le> b" using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
obtain z::"real^'n" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
- then obtain a b where ab:"a \<bullet> z < b" "\<forall>x\<in>t. b < a \<bullet> x"
+ then obtain a b where ab:"inner a z < b" "\<forall>x\<in>t. b < inner a x"
using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
thus ?thesis using True by auto
next
case False then obtain y where "y\<in>s" by auto
- obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < a \<bullet> x"
+ obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
- hence ab:"\<forall>x\<in>s. \<forall>y\<in>t. b + a \<bullet> y < a \<bullet> x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by auto
- def k \<equiv> "rsup ((\<lambda>x. a \<bullet> x) ` t)"
+ hence ab:"\<forall>x\<in>s. \<forall>y\<in>t. b + inner a y < inner a x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by (auto simp add: inner_diff)
+ def k \<equiv> "rsup ((\<lambda>x. inner a x) ` t)"
show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
- apply(rule,rule) defer apply(rule) unfolding dot_lneg and neg_less_iff_less proof-
- from ab have "((\<lambda>x. a \<bullet> x) ` t) *<= (a \<bullet> y - b)"
+ apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
+ from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
- hence k:"isLub UNIV ((\<lambda>x. a \<bullet> x) ` t) k" unfolding k_def apply(rule_tac rsup) using assms(5) by auto
- fix x assume "x\<in>t" thus "a \<bullet> x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "a \<bullet> x"] by auto
+ hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac rsup) using assms(5) by auto
+ fix x assume "x\<in>t" thus "inner a x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "inner a x"] by auto
next
fix x assume "x\<in>s"
- hence "k \<le> a \<bullet> x - b" unfolding k_def apply(rule_tac rsup_le) using assms(5)
+ hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac rsup_le) using assms(5)
unfolding setle_def
using ab[THEN bspec[where x=x]] by auto
- thus "k + b / 2 < a \<bullet> x" using `0 < b` by auto
+ thus "k + b / 2 < inner a x" using `0 < b` by auto
qed
qed
lemma separating_hyperplane_compact_closed:
assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
- shows "\<exists>a b. (\<forall>x\<in>s. a \<bullet> x < b) \<and> (\<forall>x\<in>t. a \<bullet> x > b)"
-proof- obtain a b where "(\<forall>x\<in>t. a \<bullet> x < b) \<and> (\<forall>x\<in>s. b < a \<bullet> x)"
+ shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
+proof- obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
@@ -1645,33 +1701,33 @@
lemma separating_hyperplane_set_0:
assumes "convex s" "(0::real^'n::finite) \<notin> s"
- shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> a \<bullet> x)"
-proof- let ?k = "\<lambda>c. {x::real^'n. 0 \<le> c \<bullet> x}"
+ shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
+proof- let ?k = "\<lambda>c. {x::real^'n. 0 \<le> inner c x}"
have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
defer apply(rule,rule,erule conjE) proof-
fix f assume as:"f \<subseteq> ?k ` s" "finite f"
obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
- then obtain a b where ab:"a \<noteq> 0" "0 < b" "\<forall>x\<in>convex hull c. b < a \<bullet> x"
+ then obtain a b where ab:"a \<noteq> 0" "0 < b" "\<forall>x\<in>convex hull c. b < inner a x"
using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
- hence "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> y \<bullet> x)" apply(rule_tac x="inverse(norm a) *s a" in exI)
- using hull_subset[of c convex] unfolding subset_eq and dot_rmult
+ hence "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)" apply(rule_tac x="inverse(norm a) *\<^sub>R a" in exI)
+ using hull_subset[of c convex] unfolding subset_eq and inner_scaleR
apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
- by(auto simp add: dot_sym elim!: ballE)
+ by(auto simp add: inner_commute elim!: ballE)
thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
qed(insert closed_halfspace_ge, auto)
then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
- thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: dot_sym) qed
+ thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed
lemma separating_hyperplane_sets:
assumes "convex s" "convex (t::(real^'n::finite) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
- shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. a \<bullet> x \<le> b) \<and> (\<forall>x\<in>t. a \<bullet> x \<ge> b)"
+ shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. inner a x \<le> b) \<and> (\<forall>x\<in>t. inner a x \<ge> b)"
proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
- obtain a where "a\<noteq>0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> a \<bullet> x" using assms(3-5) by auto
- hence "\<forall>x\<in>t. \<forall>y\<in>s. a \<bullet> y \<le> a \<bullet> x" apply- apply(rule, rule) apply(erule_tac x="x - y" in ballE) by auto
- thus ?thesis apply(rule_tac x=a in exI, rule_tac x="rsup ((\<lambda>x. a \<bullet> x) ` s)" in exI) using `a\<noteq>0`
+ obtain a where "a\<noteq>0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> inner a x" using assms(3-5) by auto
+ hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x" apply- apply(rule, rule) apply(erule_tac x="x - y" in ballE) by (auto simp add: inner_diff)
+ thus ?thesis apply(rule_tac x=a in exI, rule_tac x="rsup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
apply(rule) apply(rule,rule) apply(rule rsup[THEN isLubD2]) prefer 4 apply(rule,rule rsup_le) unfolding setle_def
prefer 4 using assms(3-5) by blast+ qed
@@ -1680,7 +1736,7 @@
lemma convex_closure: assumes "convex s" shows "convex(closure s)"
unfolding convex_def Ball_def closure_sequential
apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
- apply(rule_tac x="\<lambda>n. u *s xb n + v *s xc n" in exI) apply(rule,rule)
+ apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
apply(rule assms[unfolded convex_def, rule_format]) prefer 6
apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
@@ -1688,13 +1744,13 @@
unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e"
- show "\<exists>e>0. ball ((1 - u) *s x + u *s y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
+ show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
apply rule unfolding subset_eq defer apply rule proof-
- fix z assume "z \<in> ball ((1 - u) *s x + u *s y) (min d e)"
- hence "(1- u) *s (z - u *s (y - x)) + u *s (z + (1 - u) *s (y - x)) \<in> s"
+ fix z assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
+ hence "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
apply(rule_tac assms[unfolded convex_alt, rule_format])
- using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: ring_simps)
- thus "z \<in> s" using u by (auto simp add: ring_simps) qed(insert u ed(3-4), auto) qed
+ using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: algebra_simps)
+ thus "z \<in> s" using u by (auto simp add: algebra_simps) qed(insert u ed(3-4), auto) qed
lemma convex_hull_eq_empty: "convex hull s = {} \<longleftrightarrow> s = {}"
using hull_subset[of s convex] convex_hull_empty by auto
@@ -1717,27 +1773,27 @@
apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
lemma convex_hull_scaling_lemma:
- "(convex hull ((\<lambda>x. c *s x) ` s)) \<subseteq> (\<lambda>x. c *s x) ` (convex hull s)"
+ "(convex hull ((\<lambda>x. c *\<^sub>R x) ` s)) \<subseteq> (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
apply(rule hull_minimal, rule image_mono, rule hull_subset)
unfolding mem_def by(rule convex_scaling, rule convex_convex_hull)
lemma convex_hull_scaling:
- "convex hull ((\<lambda>x. c *s x) ` s) = (\<lambda>x. c *s x) ` (convex hull s)"
+ "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
- unfolding image_image vector_smult_assoc by(auto simp add:image_constant_conv convex_hull_eq_empty)
+ unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv convex_hull_eq_empty)
lemma convex_hull_affinity:
- "convex hull ((\<lambda>x. a + c *s x) ` s) = (\<lambda>x. a + c *s x) ` (convex hull s)"
+ "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
unfolding image_image[THEN sym] convex_hull_scaling convex_hull_translation ..
subsection {* Convex set as intersection of halfspaces. *}
lemma convex_halfspace_intersection:
assumes "closed s" "convex s"
- shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. a \<bullet> x \<le> b})}"
+ shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
apply(rule set_ext, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof-
- fix x assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. a \<bullet> x \<le> b}) \<longrightarrow> x \<in> xa"
- hence "\<forall>a b. s \<subseteq> {x. a \<bullet> x \<le> b} \<longrightarrow> x \<in> {x. a \<bullet> x \<le> b}" by blast
+ fix x assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
+ hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
qed auto
@@ -1746,9 +1802,9 @@
lemma radon_ex_lemma:
assumes "finite c" "affine_dependent c"
- shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *s v) c = 0"
+ shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) c = 0"
proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
- thus ?thesis apply(rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) unfolding if_smult vector_smult_lzero
+ thus ?thesis apply(rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) unfolding if_smult scaleR_zero_left
and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
lemma radon_s_lemma:
@@ -1769,9 +1825,9 @@
lemma radon_partition:
assumes "finite c" "affine_dependent c"
shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
- obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0" "(\<Sum>v\<in>c. u v *s v) = 0" using radon_ex_lemma[OF assms] by auto
+ obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0" "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0" using radon_ex_lemma[OF assms] by auto
have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
- def z \<equiv> "(inverse (setsum u {x\<in>c. u x > 0})) *s setsum (\<lambda>x. u x *s x) {x\<in>c. u x > 0}"
+ def z \<equiv> "(inverse (setsum u {x\<in>c. u x > 0})) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
case False hence "u v < 0" by auto
thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
@@ -1783,23 +1839,23 @@
hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding real_less_def apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
- "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *s x) = (\<Sum>x\<in>c. u x *s x)"
+ "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
- "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *s x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *s x)"
+ "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)"
unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add: setsum_Un_zero[OF fin, THEN sym])
moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x"
apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
apply(rule_tac x="{v \<in> c. u v < 0}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
- using assms(1) unfolding vector_smult_assoc[THEN sym] setsum_cmul and z_def
+ using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x"
apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
apply(rule_tac x="{v \<in> c. 0 < u v}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
- using assms(1) unfolding vector_smult_assoc[THEN sym] setsum_cmul and z_def using *
+ using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
by(auto simp add: setsum_negf vector_smult_lneg mult_right.setsum[THEN sym])
ultimately show ?thesis apply(rule_tac x="{v\<in>c. u v \<le> 0}" in exI, rule_tac x="{v\<in>c. u v > 0}" in exI) by auto
qed
@@ -1865,10 +1921,10 @@
apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
proof- show "convex {x. f x \<in> convex hull f ` s}"
- unfolding convex_def by(auto simp add: linear_cmul[OF assms] linear_add[OF assms]
+ unfolding convex_def by(auto simp add: linear_cmul[OF assms, unfolded smult_conv_scaleR] linear_add[OF assms]
convex_convex_hull[unfolded convex_def, rule_format]) next
show "convex {x. x \<in> f ` (convex hull s)}" using convex_convex_hull[unfolded convex_def, of s]
- unfolding convex_def by (auto simp add: linear_cmul[OF assms, THEN sym] linear_add[OF assms, THEN sym])
+ unfolding convex_def by (auto simp add: linear_cmul[OF assms, THEN sym, unfolded smult_conv_scaleR] linear_add[OF assms, THEN sym])
qed auto
lemma in_convex_hull_linear_image:
@@ -1880,72 +1936,75 @@
lemma compact_frontier_line_lemma:
fixes s :: "(real ^ _) set"
assumes "compact s" "0 \<in> s" "x \<noteq> 0"
- obtains u where "0 \<le> u" "(u *s x) \<in> frontier s" "\<forall>v>u. (v *s x) \<notin> s"
+ obtains u where "0 \<le> u" "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
proof-
obtain b where b:"b>0" "\<forall>x\<in>s. norm x \<le> b" using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
- let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *s x)}"
- have A:"?A = (\<lambda>u. dest_vec1 u *s x) ` {0 .. vec1 (b / norm x)}"
- unfolding image_image[of "\<lambda>u. u *s x" "\<lambda>x. dest_vec1 x", THEN sym]
+ let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
+ have A:"?A = (\<lambda>u. dest_vec1 u *\<^sub>R x) ` {0 .. vec1 (b / norm x)}"
+ unfolding image_image[of "\<lambda>u. u *\<^sub>R x" "\<lambda>x. dest_vec1 x", THEN sym]
unfolding dest_vec1_inverval vec1_dest_vec1 by auto
have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
- apply(rule, rule continuous_vmul) unfolding o_def vec1_dest_vec1 apply(rule continuous_at_id) by(rule compact_interval)
- moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *s x} \<inter> s \<noteq> {}" apply(rule not_disjointI[OF _ assms(2)])
+ apply(rule, rule continuous_vmul)
+ apply (rule continuous_dest_vec1)
+ apply(rule continuous_at_id) by(rule compact_interval)
+ moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *\<^sub>R x} \<inter> s \<noteq> {}" apply(rule not_disjointI[OF _ assms(2)])
unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
- ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *s x"
+ ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
"y\<in>?A" "y\<in>s" "\<forall>z\<in>?A \<inter> s. dist 0 z \<le> dist 0 y" using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0] by auto
have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
- { fix v assume as:"v > u" "v *s x \<in> s"
+ { fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)]
- using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] and norm_mul by auto
- hence "norm (v *s x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer
+ using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
+ hence "norm (v *\<^sub>R x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer
apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI)
using as(1) `u\<ge>0` by(auto simp add:field_simps)
- hence False unfolding obt(3) unfolding norm_mul using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
+ hence False unfolding obt(3) using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
} note u_max = this
- have "u *s x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *s x" in bexI) unfolding obt(3)[THEN sym]
- prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *s x" in exI) apply(rule, rule) proof-
- fix e assume "0 < e" and as:"(u + e / 2 / norm x) *s x \<in> s"
+ have "u *\<^sub>R x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *\<^sub>R x" in bexI) unfolding obt(3)[THEN sym]
+ prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
+ fix e assume "0 < e" and as:"(u + e / 2 / norm x) *\<^sub>R x \<in> s"
hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
thus False using u_max[OF _ as] by auto
- qed(insert `y\<in>s`, auto simp add: dist_norm obt(3))
+ qed(insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
thus ?thesis apply(rule_tac that[of u]) apply(rule obt(1), assumption)
apply(rule,rule,rule ccontr) apply(rule u_max) by auto qed
lemma starlike_compact_projective:
assumes "compact s" "cball (0::real^'n::finite) 1 \<subseteq> s "
- "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *s x) \<in> (s - frontier s )"
+ "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *\<^sub>R x) \<in> (s - frontier s )"
shows "s homeomorphic (cball (0::real^'n::finite) 1)"
proof-
have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
- def pi \<equiv> "\<lambda>x::real^'n. inverse (norm x) *s x"
+ def pi \<equiv> "\<lambda>x::real^'n. inverse (norm x) *\<^sub>R x"
have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
- apply rule unfolding pi_def apply(rule continuous_mul) unfolding o_def
- apply(rule continuous_at_inv[unfolded o_def]) unfolding continuous_at_vec1_range[unfolded o_def]
- apply(rule,rule) apply(rule_tac x=e in exI) apply(rule,assumption,rule,rule)
- proof- fix e x y assume "0 < e" "norm (y - x::real^'n) < e"
- thus "\<bar>norm y - norm x\<bar> < e" using norm_triangle_ineq3[of y x] by auto
- qed(auto intro!:continuous_at_id)
+ apply rule unfolding pi_def
+ apply (rule continuous_mul)
+ apply (rule continuous_at_inv[unfolded o_def])
+ apply (rule continuous_at_norm)
+ apply simp
+ apply (rule continuous_at_id)
+ done
def sphere \<equiv> "{x::real^'n. norm x = 1}"
- have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *s x) = pi x" unfolding pi_def sphere_def by auto
+ have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *\<^sub>R x) = pi x" unfolding pi_def sphere_def by auto
have "0\<in>s" using assms(2) and centre_in_cball[of 0 1] by auto
- have front_smul:"\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *s x \<in> s \<longleftrightarrow> u \<le> 1" proof(rule,rule,rule)
+ have front_smul:"\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" proof(rule,rule,rule)
fix x u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
hence "x\<noteq>0" using `0\<notin>frontier s` by auto
- obtain v where v:"0 \<le> v" "v *s x \<in> frontier s" "\<forall>w>v. w *s x \<notin> s"
+ obtain v where v:"0 \<le> v" "v *\<^sub>R x \<in> frontier s" "\<forall>w>v. w *\<^sub>R x \<notin> s"
using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
- assume "v>1" thus False using assms(3)[THEN bspec[where x="v *s x"], THEN spec[where x="inverse v"]]
+ assume "v>1" thus False using assms(3)[THEN bspec[where x="v *\<^sub>R x"], THEN spec[where x="inverse v"]]
using v and x and fs unfolding inverse_less_1_iff by auto qed
- show "u *s x \<in> s \<longleftrightarrow> u \<le> 1" apply rule using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
- assume "u\<le>1" thus "u *s x \<in> s" apply(cases "u=1")
+ show "u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" apply rule using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
+ assume "u\<le>1" thus "u *\<^sub>R x \<in> s" apply(cases "u=1")
using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]] using `0\<le>u` and x and fs by auto qed auto qed
have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
@@ -1955,14 +2014,14 @@
proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
- then obtain u where "0 \<le> u" "u *s x \<in> frontier s" "\<forall>v>u. v *s x \<notin> s"
+ then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
- thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *s x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
+ thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *\<^sub>R x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
hence xys:"x\<in>s" "y\<in>s" using fs by auto
from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto
- from nor have x:"x = norm x *s ((inverse (norm y)) *s y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto
- from nor have y:"y = norm y *s ((inverse (norm x)) *s x)" unfolding as(3)[unfolded pi_def] by auto
+ from nor have x:"x = norm x *\<^sub>R ((inverse (norm y)) *\<^sub>R y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto
+ from nor have y:"y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)" unfolding as(3)[unfolded pi_def] by auto
have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
@@ -1978,7 +2037,7 @@
apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
{ fix x assume as:"x \<in> cball (0::real^'n) 1"
- have "norm x *s surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1")
+ have "norm x *\<^sub>R surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1")
case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
apply(rule_tac fs[unfolded subset_eq, rule_format])
@@ -1987,35 +2046,35 @@
unfolding surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
{ fix x assume "x\<in>s"
- hence "x \<in> (\<lambda>x. norm x *s surf (pi x)) ` cball 0 1" proof(cases "x=0")
+ hence "x \<in> (\<lambda>x. norm x *\<^sub>R surf (pi x)) ` cball 0 1" proof(cases "x=0")
case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
next let ?a = "inverse (norm (surf (pi x)))"
case False hence invn:"inverse (norm x) \<noteq> 0" by auto
from False have pix:"pi x\<in>sphere" using pi(1) by auto
hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
- hence **:"norm x *s (?a *s surf (pi x)) = x" apply(rule_tac vector_mul_lcancel_imp[OF invn]) unfolding pi_def by auto
+ hence **:"norm x *\<^sub>R (?a *\<^sub>R surf (pi x)) = x" apply(rule_tac scaleR_left_imp_eq[OF invn]) unfolding pi_def using invn by auto
hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
- hence "norm x = norm ((?a * norm x) *s surf (pi x))"
- unfolding norm_mul abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
- moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *s surf (pi x))"
+ hence "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
+ unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
+ moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))"
unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
- hence "dist 0 (inverse (norm (surf (pi x))) *s x) \<le> 1" unfolding dist_norm
+ hence "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1" unfolding dist_norm
using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
using False `x\<in>s` by(auto simp add:field_simps)
- ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *s x" in bexI)
- apply(subst injpi[THEN sym]) unfolding norm_mul abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
+ ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *\<^sub>R x" in bexI)
+ apply(subst injpi[THEN sym]) unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
unfolding pi(2)[OF `?a > 0`] by auto
qed } note hom2 = this
- show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *s surf (pi x)"])
+ show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *\<^sub>R surf (pi x)"])
apply(rule compact_cball) defer apply(rule set_ext, rule, erule imageE, drule hom)
prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
fix x::"real^'n" assume as:"x \<in> cball 0 1"
- thus "continuous (at x) (\<lambda>x. norm x *s surf (pi x))" proof(cases "x=0")
- case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_vec1_norm)
+ thus "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))" proof(cases "x=0")
+ case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
next guess a using UNIV_witness[where 'a = 'n] ..
obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
@@ -2023,7 +2082,7 @@
unfolding Ball_def mem_cball dist_norm by (auto simp add: norm_basis[unfolded One_nat_def])
case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
- unfolding norm_0 vector_smult_lzero dist_norm diff_0_right norm_mul abs_norm_cancel proof-
+ unfolding norm_0 scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel proof-
fix e and x::"real^'n" assume as:"norm x < e / B" "0 < norm x" "0<e"
hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
hence "norm (surf (pi x)) \<le> B" using B fs by auto
@@ -2038,8 +2097,8 @@
hence "surf (pi x) \<in> frontier s" using surf(5) by auto
thus False using `0\<notin>frontier s` unfolding as by simp qed
} note surf_0 = this
- show "inj_on (\<lambda>x. norm x *s surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
- fix x y assume as:"x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *s surf (pi x) = norm y *s surf (pi y)"
+ show "inj_on (\<lambda>x. norm x *\<^sub>R surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
+ fix x y assume as:"x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *\<^sub>R surf (pi x) = norm y *\<^sub>R surf (pi y)"
thus "x=y" proof(cases "x=0 \<or> y=0")
case True thus ?thesis using as by(auto elim: surf_0) next
case False
@@ -2056,18 +2115,18 @@
shows "s homeomorphic (cball (0::real^'n) 1)"
apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
fix x u assume as:"x \<in> s" "0 \<le> u" "u < (1::real)"
- hence "u *s x \<in> interior s" unfolding interior_def mem_Collect_eq
- apply(rule_tac x="ball (u *s x) (1 - u)" in exI) apply(rule, rule open_ball)
+ hence "u *\<^sub>R x \<in> interior s" unfolding interior_def mem_Collect_eq
+ apply(rule_tac x="ball (u *\<^sub>R x) (1 - u)" in exI) apply(rule, rule open_ball)
unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
- fix y assume "dist (u *s x) y < 1 - u"
- hence "inverse (1 - u) *s (y - u *s x) \<in> s"
+ fix y assume "dist (u *\<^sub>R x) y < 1 - u"
+ hence "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s"
using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
- unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_mul
+ unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_scaleR
apply (rule mult_left_le_imp_le[of "1 - u"])
unfolding class_semiring.mul_a using `u<1` by auto
- thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *s (y - u *s x)" x "1 - u" u]
- using as unfolding vector_smult_assoc by auto qed auto
- thus "u *s x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
+ thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *\<^sub>R (y - u *\<^sub>R x)" x "1 - u" u]
+ using as unfolding scaleR_scaleR by auto qed auto
+ thus "u *\<^sub>R x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
lemma homeomorphic_convex_compact_cball: fixes e::real and s::"(real^'n::finite) set"
assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
@@ -2075,16 +2134,16 @@
proof- obtain a where "a\<in>interior s" using assms(3) by auto
then obtain d where "d>0" and d:"cball a d \<subseteq> s" unfolding mem_interior_cball by auto
let ?d = "inverse d" and ?n = "0::real^'n"
- have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *s (x - a)) ` s"
- apply(rule, rule_tac x="d *s x + a" in image_eqI) defer
+ have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
+ apply(rule, rule_tac x="d *\<^sub>R x + a" in image_eqI) defer
apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
by(auto simp add: mult_right_le_one_le)
- hence "(\<lambda>x. inverse d *s (x - a)) ` s homeomorphic cball ?n 1"
- using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *s -a + ?d *s x) ` s", OF convex_affinity compact_affinity]
- using assms(1,2) by(auto simp add: uminus_add_conv_diff)
+ hence "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
+ using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s", OF convex_affinity compact_affinity]
+ using assms(1,2) by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib)
thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
- apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *s -a"]])
- using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff) qed
+ apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
+ using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib) qed
lemma homeomorphic_convex_compact: fixes s::"(real^'n::finite) set" and t::"(real^'n) set"
assumes "convex s" "compact s" "interior s \<noteq> {}"
@@ -2101,8 +2160,8 @@
lemma convex_epigraph:
"convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
unfolding convex_def convex_on_def unfolding Ball_def forall_pastecart epigraph_def
- unfolding mem_Collect_eq fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul fstcart_add fstcart_cmul
- unfolding Ball_def[symmetric] unfolding dest_vec1_add dest_vec1_cmul
+ unfolding mem_Collect_eq fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul [where 'a=real, unfolded smult_conv_scaleR] fstcart_add fstcart_cmul [where 'a=real, unfolded smult_conv_scaleR]
+ unfolding Ball_def[symmetric] unfolding dest_vec1_add dest_vec1_cmul [where 'a=real, unfolded smult_conv_scaleR]
apply(subst forall_dest_vec1[THEN sym])+ by(meson real_le_refl real_le_trans add_mono mult_left_mono)
lemma convex_epigraphI: assumes "convex_on s f" "convex s"
@@ -2131,11 +2190,11 @@
lemma convex_on:
assumes "convex s"
shows "convex_on s f \<longleftrightarrow> (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow>
- f (setsum (\<lambda>i. u i *s x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
+ f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
unfolding sndcart_setsum[OF finite_atLeastAtMost] fstcart_setsum[OF finite_atLeastAtMost] dest_vec1_setsum[OF finite_atLeastAtMost]
- unfolding fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul fstcart_add fstcart_cmul
- unfolding dest_vec1_add dest_vec1_cmul apply(subst forall_of_pastecart)+ apply(subst forall_of_dest_vec1)+ apply rule
+ unfolding fstcart_pastecart sndcart_pastecart sndcart_add sndcart_cmul [where 'a=real, unfolded smult_conv_scaleR] fstcart_add fstcart_cmul [where 'a=real, unfolded smult_conv_scaleR]
+ unfolding dest_vec1_add dest_vec1_cmul [where 'a=real, unfolded smult_conv_scaleR] apply(subst forall_of_pastecart)+ apply(subst forall_of_dest_vec1)+ apply rule
using assms[unfolded convex] apply simp apply(rule,rule,rule)
apply(erule_tac x=k in allE, erule_tac x=u in allE, erule_tac x=x in allE) apply rule apply rule apply rule defer
apply(rule_tac j="\<Sum>i = 1..k. u i * f (x i)" in real_le_trans)
@@ -2157,7 +2216,7 @@
hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: ring_simps)
hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
- ultimately show "u *s x + v *s y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
+ ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
using as(3-) dimindex_ge_1 apply- by(auto simp add: vector_component) qed
lemma is_interval_connected:
@@ -2179,10 +2238,10 @@
apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
fix a b x assume as:"connected s" "a \<in> s" "b \<in> s" "dest_vec1 a \<le> dest_vec1 x" "dest_vec1 x \<le> dest_vec1 b" "x\<notin>s"
hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
- let ?halfl = "{z. basis 1 \<bullet> z < dest_vec1 x} " and ?halfr = "{z. basis 1 \<bullet> z > dest_vec1 x} "
+ let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
{ fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
- using as(6) `y\<in>s` by (auto simp add: basis_component field_simps dest_vec1_eq[unfolded dest_vec1_def One_nat_def] dest_vec1_def) }
- moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: basis_component field_simps dest_vec1_def)
+ using as(6) `y\<in>s` by (auto simp add: inner_vector_def dest_vec1_eq [unfolded dest_vec1_def] dest_vec1_def) }
+ moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def dest_vec1_def)
hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}" using as(2-3) by auto
ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI)
@@ -2232,7 +2291,7 @@
assumes "convex_on (convex hull s) f" "\<forall>x\<in>s. f x \<le> b"
shows "\<forall>x\<in> convex hull s. f x \<le> b" proof
fix x assume "x\<in>convex hull s"
- then obtain k u v where obt:"\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *s v i) = x"
+ then obtain k u v where obt:"\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
unfolding convex_hull_indexed mem_Collect_eq by auto
have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b" using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
@@ -2270,7 +2329,7 @@
thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
by(auto simp add: Cart_lambda_beta)
next let ?y = "\<lambda>j. if x$j = 0 then 0 else (x$j - x$i) / (1 - x$i)"
- case False hence *:"x = x$i *s (\<chi> j. if x$j = 0 then 0 else 1) + (1 - x$i) *s (\<chi> j. ?y j)" unfolding Cart_eq
+ case False hence *:"x = x$i *\<^sub>R (\<chi> j. if x$j = 0 then 0 else 1) + (1 - x$i) *\<^sub>R (\<chi> j. ?y j)" unfolding Cart_eq
by(auto simp add: Cart_lambda_beta vector_add_component vector_smult_component vector_minus_component field_simps)
{ fix j have "x$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $ j - x $ i) / (1 - x $ i)" "(x $ j - x $ i) / (1 - x $ i) \<le> 1"
apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
@@ -2304,7 +2363,7 @@
lemma cube_convex_hull:
assumes "0 < d" obtains s::"(real^'n::finite) set" where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s" proof-
let ?d = "(\<chi> i. d)::real^'n"
- have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *s y) ` {0 .. 1}" apply(rule set_ext, rule)
+ have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. 1}" apply(rule set_ext, rule)
unfolding image_iff defer apply(erule bexE) proof-
fix y assume as:"y\<in>{x - ?d .. x + ?d}"
{ fix i::'n have "x $ i \<le> d + y $ i" "y $ i \<le> d + x $ i" using as[unfolded mem_interval, THEN spec[where x=i]]
@@ -2314,26 +2373,26 @@
using assms by(auto simp add: field_simps right_inverse)
hence "inverse d * (x $ i * 2) \<le> 2 + inverse d * (y $ i * 2)"
"inverse d * (y $ i * 2) \<le> 2 + inverse d * (x $ i * 2)" by(auto simp add:field_simps) }
- hence "inverse (2 * d) *s (y - (x - ?d)) \<in> {0..1}" unfolding mem_interval using assms
+ hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..1}" unfolding mem_interval using assms
by(auto simp add: Cart_eq vector_component_simps field_simps)
- thus "\<exists>z\<in>{0..1}. y = x - ?d + (2 * d) *s z" apply- apply(rule_tac x="inverse (2 * d) *s (y - (x - ?d))" in bexI)
+ thus "\<exists>z\<in>{0..1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI)
using assms by(auto simp add: Cart_eq vector_less_eq_def Cart_lambda_beta)
next
- fix y z assume as:"z\<in>{0..1}" "y = x - ?d + (2*d) *s z"
+ fix y z assume as:"z\<in>{0..1}" "y = x - ?d + (2*d) *\<^sub>R z"
have "\<And>i. 0 \<le> d * z $ i \<and> d * z $ i \<le> d" using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
using assms by(auto simp add: vector_component_simps Cart_eq)
thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
apply(erule_tac x=i in allE) using assms by(auto simp add: vector_component_simps Cart_eq) qed
obtain s where "finite s" "{0..1::real^'n} = convex hull s" using unit_cube_convex_hull by auto
- thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *s y)` s"]) unfolding * and convex_hull_affinity by auto qed
+ thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"]) unfolding * and convex_hull_affinity by auto qed
subsection {* Bounded convex function on open set is continuous. *}
lemma convex_on_bounded_continuous:
assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
- shows "continuous_on s (vec1 o f)"
- apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_vec1_range proof(rule,rule,rule)
+ shows "continuous_on s f"
+ apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_real_range proof(rule,rule,rule)
fix x e assume "x\<in>s" "(0::real) < e"
def B \<equiv> "abs b + 1"
have B:"0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
@@ -2347,12 +2406,12 @@
have "2 < t" "0<t" unfolding t_def using as False and `k>0` by(auto simp add:field_simps)
have "y\<in>s" apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
apply(rule order_trans[of _ "2 * norm (x - y)"]) using as by(auto simp add: field_simps norm_minus_commute)
- { def w \<equiv> "x + t *s (y - x)"
+ { def w \<equiv> "x + t *\<^sub>R (y - x)"
have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
- unfolding t_def using `k>0` by(auto simp add: norm_mul simp del: vector_ssub_ldistrib)
- have "(1 / t) *s x + - x + ((t - 1) / t) *s x = (1 / t - 1 + (t - 1) / t) *s x" by auto
- also have "\<dots> = 0" using `t>0` by(auto simp add:field_simps simp del:vector_sadd_rdistrib)
- finally have w:"(1 / t) *s w + ((t - 1) / t) *s x = y" unfolding w_def using False and `t>0` by auto
+ unfolding t_def using `k>0` by auto
+ have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x" by (auto simp add: algebra_simps)
+ also have "\<dots> = 0" using `t>0` by(auto simp add:field_simps)
+ finally have w:"(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
have "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps)
hence "(f w - f x) / t < e"
using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`] using `t>0` by(auto simp add:field_simps)
@@ -2360,12 +2419,12 @@
using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
using `0<t` `2<t` and `x\<in>s` `w\<in>s` by(auto simp add:field_simps) }
moreover
- { def w \<equiv> "x - t *s (y - x)"
+ { def w \<equiv> "x - t *\<^sub>R (y - x)"
have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
- unfolding t_def using `k>0` by(auto simp add: norm_mul simp del: vector_ssub_ldistrib)
- have "(1 / (1 + t)) *s x + (t / (1 + t)) *s x = (1 / (1 + t) + t / (1 + t)) *s x" by auto
- also have "\<dots>=x" using `t>0` by (auto simp add:field_simps simp del:vector_sadd_rdistrib)
- finally have w:"(1 / (1+t)) *s w + (t / (1 + t)) *s y = x" unfolding w_def using False and `t>0` by auto
+ unfolding t_def using `k>0` by auto
+ have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x" by (auto simp add: algebra_simps)
+ also have "\<dots>=x" using `t>0` by (auto simp add:field_simps)
+ finally have w:"(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
have "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps)
hence *:"(f w - f y) / t < e" using B(2)[OF `w\<in>s`] and B(2)[OF `y\<in>s`] using `t>0` by(auto simp add:field_simps)
have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y"
@@ -2384,10 +2443,10 @@
assumes "convex_on (cball x e) f" "\<forall>y \<in> cball x e. f y \<le> b"
shows "\<forall>y \<in> cball x e. abs(f y) \<le> b + 2 * abs(f x)"
apply(rule) proof(cases "0 \<le> e") case True
- fix y assume y:"y\<in>cball x e" def z \<equiv> "2 *s x - y"
- have *:"x - (2 *s x - y) = y - x" by vector
+ fix y assume y:"y\<in>cball x e" def z \<equiv> "2 *\<^sub>R x - y"
+ have *:"x - (2 *\<^sub>R x - y) = y - x" by vector
have z:"z\<in>cball x e" using y unfolding z_def mem_cball dist_norm * by(auto simp add: norm_minus_commute)
- have "(1 / 2) *s y + (1 / 2) *s z = x" unfolding z_def by auto
+ have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x" unfolding z_def by (auto simp add: algebra_simps)
thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z] by(auto simp add:field_simps)
next case False fix y assume "y\<in>cball x e"
@@ -2398,7 +2457,7 @@
lemma convex_on_continuous:
assumes "open (s::(real^'n::finite) set)" "convex_on s f"
- shows "continuous_on s (vec1 \<circ> f)"
+ shows "continuous_on s f"
unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
note dimge1 = dimindex_ge_1[where 'a='n]
fix x assume "x\<in>s"
@@ -2428,19 +2487,25 @@
using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add: vector_component) }
thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm
by(auto simp add: vector_component_simps) qed
- hence "continuous_on (ball x d) (vec1 \<circ> f)" apply(rule_tac convex_on_bounded_continuous)
+ hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball) by auto
- thus "continuous (at x) (vec1 \<circ> f)" unfolding continuous_on_eq_continuous_at[OF open_ball] using `d>0` by auto qed
+ thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball] using `d>0` by auto qed
subsection {* Line segments, starlike sets etc. *)
(* Use the same overloading tricks as for intervals, so that *)
(* segment[a,b] is closed and segment(a,b) is open relative to affine hull. *}
-definition "midpoint a b = (inverse (2::real)) *s (a + b)"
-
-definition "open_segment a b = {(1 - u) *s a + u *s b | u::real. 0 < u \<and> u < 1}"
-
-definition "closed_segment a b = {(1 - u) *s a + u *s b | u::real. 0 \<le> u \<and> u \<le> 1}"
+definition
+ midpoint :: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> real ^ 'n" where
+ "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
+
+definition
+ open_segment :: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> (real ^ 'n) set" where
+ "open_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 < u \<and> u < 1}"
+
+definition
+ closed_segment :: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> (real ^ 'n) set" where
+ "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 \<le> u \<and> u \<le> 1}"
definition "between = (\<lambda> (a,b). closed_segment a b)"
@@ -2449,9 +2514,9 @@
definition "starlike s \<longleftrightarrow> (\<exists>a\<in>s. \<forall>x\<in>s. closed_segment a x \<subseteq> s)"
lemma midpoint_refl: "midpoint x x = x"
- unfolding midpoint_def unfolding vector_add_ldistrib unfolding vector_sadd_rdistrib[THEN sym] by auto
-
-lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by auto
+ unfolding midpoint_def unfolding scaleR_right_distrib unfolding scaleR_left_distrib[THEN sym] by auto
+
+lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
lemma dist_midpoint:
"dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
@@ -2459,8 +2524,9 @@
"dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
"dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
proof-
- have *: "\<And>x y::real^'n::finite. 2 *s x = - y \<Longrightarrow> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
- have **:"\<And>x y::real^'n::finite. 2 *s x = y \<Longrightarrow> norm x = (norm y) / 2" by auto
+ have *: "\<And>x y::real^'n::finite. 2 *\<^sub>R x = - y \<Longrightarrow> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
+ have **:"\<And>x y::real^'n::finite. 2 *\<^sub>R x = y \<Longrightarrow> norm x = (norm y) / 2" by auto
+ note scaleR_right_distrib [simp]
show ?t1 unfolding midpoint_def dist_norm apply (rule **) by(auto,vector)
show ?t2 unfolding midpoint_def dist_norm apply (rule *) by(auto,vector)
show ?t3 unfolding midpoint_def dist_norm apply (rule *) by(auto,vector)
@@ -2507,7 +2573,7 @@
using segment_furthest_le[OF assms, of b]
by (auto simp add:norm_minus_commute)
-lemma segment_refl:"closed_segment a a = {a}" unfolding segment by auto
+lemma segment_refl:"closed_segment a a = {a}" unfolding segment by (auto simp add: algebra_simps)
lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
unfolding between_def mem_def by auto
@@ -2517,32 +2583,32 @@
case True thus ?thesis unfolding between_def split_conv mem_def[of x, symmetric]
by(auto simp add:segment_refl dist_commute) next
case False hence Fal:"norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0" by auto
- have *:"\<And>u. a - ((1 - u) *s a + u *s b) = u *s (a - b)" by auto
+ have *:"\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)" by (auto simp add: algebra_simps)
show ?thesis unfolding between_def split_conv mem_def[of x, symmetric] closed_segment_def mem_Collect_eq
apply rule apply(erule exE, (erule conjE)+) apply(subst dist_triangle_eq) proof-
- fix u assume as:"x = (1 - u) *s a + u *s b" "0 \<le> u" "u \<le> 1"
- hence *:"a - x = u *s (a - b)" "x - b = (1 - u) *s (a - b)"
- unfolding as(1) by(auto simp add:field_simps)
- show "norm (a - x) *s (x - b) = norm (x - b) *s (a - x)"
- unfolding norm_minus_commute[of x a] * norm_mul Cart_eq using as(2,3)
+ fix u assume as:"x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
+ hence *:"a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
+ unfolding as(1) by(auto simp add:algebra_simps)
+ show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
+ unfolding norm_minus_commute[of x a] * Cart_eq using as(2,3)
by(auto simp add: vector_component_simps field_simps)
next assume as:"dist a b = dist a x + dist x b"
have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2] unfolding as[unfolded dist_norm] norm_ge_zero by auto
- thus "\<exists>u. x = (1 - u) *s a + u *s b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
+ thus "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
unfolding dist_norm Cart_eq apply- apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4 proof rule
- fix i::'n have "((1 - norm (a - x) / norm (a - b)) *s a + (norm (a - x) / norm (a - b)) *s b) $ i =
+ fix i::'n have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $ i =
((norm (a - b) - norm (a - x)) * (a $ i) + norm (a - x) * (b $ i)) / norm (a - b)"
using Fal by(auto simp add:vector_component_simps field_simps)
also have "\<dots> = x$i" apply(rule divide_eq_imp[OF Fal])
unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq Cart_eq,rule_format, of i]
by(auto simp add:field_simps vector_component_simps)
- finally show "x $ i = ((1 - norm (a - x) / norm (a - b)) *s a + (norm (a - x) / norm (a - b)) *s b) $ i" by auto
+ finally show "x $ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $ i" by auto
qed(insert Fal2, auto) qed qed
lemma between_midpoint: fixes a::"real^'n::finite" shows
"between (a,b) (midpoint a b)" (is ?t1)
"between (b,a) (midpoint a b)" (is ?t2)
-proof- have *:"\<And>x y z. x = (1/2::real) *s z \<Longrightarrow> y = (1/2) *s z \<Longrightarrow> norm z = norm x + norm y" by auto
+proof- have *:"\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y" by auto
show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
by(auto simp add:field_simps Cart_eq vector_component_simps) qed
@@ -2554,16 +2620,16 @@
lemma mem_interior_convex_shrink:
assumes "convex s" "c \<in> interior s" "x \<in> s" "0 < e" "e \<le> 1"
- shows "x - e *s (x - c) \<in> interior s"
+ shows "x - e *\<^sub>R (x - c) \<in> interior s"
proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
show ?thesis unfolding mem_interior apply(rule_tac x="e*d" in exI)
apply(rule) defer unfolding subset_eq Ball_def mem_ball proof(rule,rule)
- fix y assume as:"dist (x - e *s (x - c)) y < e * d"
- have *:"y = (1 - (1 - e)) *s ((1 / e) *s y - ((1 - e) / e) *s x) + (1 - e) *s x" using `e>0` by auto
- have "dist c ((1 / e) *s y - ((1 - e) / e) *s x) = abs(1/e) * norm (e *s c - y + (1 - e) *s x)"
- unfolding dist_norm unfolding norm_mul[THEN sym] apply(rule norm_eqI) using `e>0`
+ fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d"
+ have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
+ have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
+ unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule norm_eqI) using `e>0`
by(auto simp add:vector_component_simps Cart_eq field_simps)
- also have "\<dots> = abs(1/e) * norm (x - e *s (x - c) - y)" by(auto intro!:norm_eqI)
+ also have "\<dots> = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:norm_eqI simp add: algebra_simps)
also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
by(auto simp add:pos_divide_less_eq[OF `e>0`] real_mult_commute)
finally show "y \<in> s" apply(subst *) apply(rule assms(1)[unfolded convex_alt,rule_format])
@@ -2572,7 +2638,7 @@
lemma mem_interior_closure_convex_shrink:
assumes "convex s" "c \<in> interior s" "x \<in> closure s" "0 < e" "e \<le> 1"
- shows "x - e *s (x - c) \<in> interior s"
+ shows "x - e *\<^sub>R (x - c) \<in> interior s"
proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d" proof(cases "x\<in>s")
case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
@@ -2587,11 +2653,11 @@
using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
then obtain y where "y\<in>s" and y:"norm (y - x) * (1 - e) < e * d" by auto
- def z \<equiv> "c + ((1 - e) / e) *s (x - y)"
- have *:"x - e *s (x - c) = y - e *s (y - z)" unfolding z_def using `e>0` by auto
+ def z \<equiv> "c + ((1 - e) / e) *\<^sub>R (x - y)"
+ have *:"x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)" unfolding z_def using `e>0` by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
have "z\<in>interior s" apply(rule subset_interior[OF d,unfolded subset_eq,rule_format])
unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
- by(auto simp del:vector_ssub_ldistrib simp add:field_simps norm_minus_commute)
+ by(auto simp add:field_simps norm_minus_commute)
thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink)
using assms(1,4-5) `y\<in>s` by auto qed
@@ -2599,7 +2665,7 @@
lemma simplex:
assumes "finite s" "0 \<notin> s"
- shows "convex hull (insert 0 s) = { y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s \<le> 1 \<and> setsum (\<lambda>x. u x *s x) s = y)}"
+ shows "convex hull (insert 0 s) = { y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s \<le> 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y)}"
unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]] apply(rule set_ext, rule) unfolding mem_Collect_eq
apply(erule_tac[!] exE) apply(erule_tac[!] conjE)+ unfolding setsum_clauses(2)[OF assms(1)]
apply(rule_tac x=u in exI) defer apply(rule_tac x="\<lambda>x. if x = 0 then 1 - setsum u s else u x" in exI) using assms(2)
@@ -2614,16 +2680,16 @@
note sumbas = this setsum_reindex[OF basis_inj, unfolded o_def]
show ?thesis unfolding simplex[OF finite_stdbasis `0\<notin>?p`] apply(rule set_ext) unfolding mem_Collect_eq apply rule
apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
- fix x::"real^'n" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x" "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *s x) = x"
- have *:"\<forall>i. u (basis i) = x$i" using as(3) unfolding sumbas and basis_expansion_unique by auto
+ fix x::"real^'n" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x" "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *\<^sub>R x) = x"
+ have *:"\<forall>i. u (basis i) = x$i" using as(3) unfolding sumbas and basis_expansion_unique [where 'a=real, unfolded smult_conv_scaleR] by auto
hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $ x) ?D" unfolding sumbas by(rule_tac setsum_cong, auto)
show " (\<forall>i. 0 \<le> x $ i) \<and> setsum (op $ x) ?D \<le> 1" apply - proof(rule,rule)
fix i::'n show "0 \<le> x$i" unfolding *[rule_format,of i,THEN sym] apply(rule_tac as(1)[rule_format]) by auto
qed(insert as(2)[unfolded **], auto)
next fix x::"real^'n" assume as:"\<forall>i. 0 \<le> x $ i" "setsum (op $ x) ?D \<le> 1"
- show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and> setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *s x) = x"
- apply(rule_tac x="\<lambda>y. y \<bullet> x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE) using as(1) apply(erule_tac x=i in allE)
- unfolding sumbas using as(2) and basis_expansion_unique by(auto simp add:dot_basis) qed qed
+ show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and> setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *\<^sub>R x) = x"
+ apply(rule_tac x="\<lambda>y. inner y x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE) using as(1) apply(erule_tac x=i in allE)
+ unfolding sumbas using as(2) and basis_expansion_unique [where 'a=real, unfolded smult_conv_scaleR] by(auto simp add:inner_basis) qed qed
lemma interior_std_simplex:
"interior (convex hull (insert 0 { basis i| i. i\<in>UNIV})) =
@@ -2632,14 +2698,14 @@
unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
fix x::"real^'n" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x. 0 \<le> xa $ x) \<and> setsum (op $ xa) UNIV \<le> 1"
show "(\<forall>xa. 0 < x $ xa) \<and> setsum (op $ x) UNIV < 1" apply(rule,rule) proof-
- fix i::'n show "0 < x $ i" using as[THEN spec[where x="x - (e / 2) *s basis i"]] and `e>0`
+ fix i::'n show "0 < x $ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
unfolding dist_norm by(auto simp add: norm_basis vector_component_simps basis_component elim:allE[where x=i])
next guess a using UNIV_witness[where 'a='n] ..
- have **:"dist x (x + (e / 2) *s basis a) < e" using `e>0` and norm_basis[of a]
+ have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e" using `e>0` and norm_basis[of a]
unfolding dist_norm by(auto simp add: vector_component_simps basis_component intro!: mult_strict_left_mono_comm)
- have "\<And>i. (x + (e / 2) *s basis a) $ i = x$i + (if i = a then e/2 else 0)" by(auto simp add:vector_component_simps)
- hence *:"setsum (op $ (x + (e / 2) *s basis a)) UNIV = setsum (\<lambda>i. x$i + (if a = i then e/2 else 0)) UNIV" by(rule_tac setsum_cong, auto)
- have "setsum (op $ x) UNIV < setsum (op $ (x + (e / 2) *s basis a)) UNIV" unfolding * setsum_addf
+ have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $ i = x$i + (if i = a then e/2 else 0)" by(auto simp add:vector_component_simps)
+ hence *:"setsum (op $ (x + (e / 2) *\<^sub>R basis a)) UNIV = setsum (\<lambda>i. x$i + (if a = i then e/2 else 0)) UNIV" by(rule_tac setsum_cong, auto)
+ have "setsum (op $ x) UNIV < setsum (op $ (x + (e / 2) *\<^sub>R basis a)) UNIV" unfolding * setsum_addf
using `0<e` dimindex_ge_1 by(auto simp add: setsum_delta')
also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
finally show "setsum (op $ x) UNIV < 1" by auto qed
@@ -2665,8 +2731,8 @@
lemma interior_std_simplex_nonempty: obtains a::"real^'n::finite" where
"a \<in> interior(convex hull (insert 0 {basis i | i . i \<in> UNIV}))" proof-
- let ?D = "UNIV::'n set" let ?a = "setsum (\<lambda>b. inverse (2 * real CARD('n)) *s b) {(basis i) | i. i \<in> ?D}"
- have *:"{basis i | i. i \<in> ?D} = basis ` ?D" by auto
+ let ?D = "UNIV::'n set" let ?a = "setsum (\<lambda>b::real^'n. inverse (2 * real CARD('n)) *\<^sub>R b) {(basis i) | i. i \<in> ?D}"
+ have *:"{basis i :: real ^ 'n | i. i \<in> ?D} = basis ` ?D" by auto
{ fix i have "?a $ i = inverse (2 * real CARD('n))"
unfolding setsum_component vector_smult_component and * and setsum_reindex[OF basis_inj] and o_def
apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real CARD('n)) else 0) ?D"]) apply(rule setsum_cong2)
@@ -2691,7 +2757,7 @@
definition "reversepath (g::real^1 \<Rightarrow> real^'n) = (\<lambda>x. g(1 - x))"
definition joinpaths:: "(real^1 \<Rightarrow> real^'n) \<Rightarrow> (real^1 \<Rightarrow> real^'n) \<Rightarrow> (real^1 \<Rightarrow> real^'n)" (infixr "+++" 75)
- where "joinpaths g1 g2 = (\<lambda>x. if dest_vec1 x \<le> ((1 / 2)::real) then g1 (2 *s x) else g2(2 *s x - 1))"
+ where "joinpaths g1 g2 = (\<lambda>x. if dest_vec1 x \<le> ((1 / 2)::real) then g1 (2 *\<^sub>R x) else g2(2 *\<^sub>R x - 1))"
definition "simple_path (g::real^1 \<Rightarrow> real^'n) \<longleftrightarrow>
(\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
@@ -2736,7 +2802,7 @@
unfolding pathstart_def joinpaths_def pathfinish_def by auto
lemma pathfinish_join[simp]:"pathfinish(g1 +++ g2) = pathfinish g2" proof-
- have "2 *s 1 - 1 = (1::real^1)" unfolding Cart_eq by(auto simp add:vector_component_simps)
+ have "2 *\<^sub>R 1 - 1 = (1::real^1)" unfolding Cart_eq by(auto simp add:vector_component_simps)
thus ?thesis unfolding pathstart_def joinpaths_def pathfinish_def
unfolding vec_1[THEN sym] dest_vec1_vec by auto qed
@@ -2759,9 +2825,9 @@
lemma path_join[simp]: assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
unfolding path_def pathfinish_def pathstart_def apply rule defer apply(erule conjE) proof-
assume as:"continuous_on {0..1} (g1 +++ g2)"
- have *:"g1 = (\<lambda>x. g1 (2 *s x)) \<circ> (\<lambda>x. (1/2) *s x)"
- "g2 = (\<lambda>x. g2 (2 *s x - 1)) \<circ> (\<lambda>x. (1/2) *s (x + 1))" unfolding o_def by auto
- have "op *s (1 / 2) ` {0::real^1..1} \<subseteq> {0..1}" "(\<lambda>x. (1 / 2) *s (x + 1)) ` {(0::real^1)..1} \<subseteq> {0..1}"
+ have *:"g1 = (\<lambda>x. g1 (2 *\<^sub>R x)) \<circ> (\<lambda>x. (1/2) *\<^sub>R x)"
+ "g2 = (\<lambda>x. g2 (2 *\<^sub>R x - 1)) \<circ> (\<lambda>x. (1/2) *\<^sub>R (x + 1))" unfolding o_def by auto
+ have "op *\<^sub>R (1 / 2) ` {0::real^1..1} \<subseteq> {0..1}" "(\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {(0::real^1)..1} \<subseteq> {0..1}"
unfolding image_smult_interval by (auto, auto simp add:vector_less_eq_def vector_component_simps elim!:ballE)
thus "continuous_on {0..1} g1 \<and> continuous_on {0..1} g2" apply -apply rule
apply(subst *) defer apply(subst *) apply (rule_tac[!] continuous_on_compose)
@@ -2769,35 +2835,35 @@
apply (rule continuous_on_cmul, rule continuous_on_id) apply(rule_tac[!] continuous_on_eq[of _ "g1 +++ g2"]) defer prefer 3
apply(rule_tac[1-2] continuous_on_subset[of "{0 .. 1}"]) apply(rule as, assumption, rule as, assumption)
apply(rule) defer apply rule proof-
- fix x assume "x \<in> op *s (1 / 2) ` {0::real^1..1}"
+ fix x assume "x \<in> op *\<^sub>R (1 / 2) ` {0::real^1..1}"
hence "dest_vec1 x \<le> 1 / 2" unfolding image_iff by(auto simp add: vector_component_simps)
- thus "(g1 +++ g2) x = g1 (2 *s x)" unfolding joinpaths_def by auto next
- fix x assume "x \<in> (\<lambda>x. (1 / 2) *s (x + 1)) ` {0::real^1..1}"
+ thus "(g1 +++ g2) x = g1 (2 *\<^sub>R x)" unfolding joinpaths_def by auto next
+ fix x assume "x \<in> (\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {0::real^1..1}"
hence "dest_vec1 x \<ge> 1 / 2" unfolding image_iff by(auto simp add: vector_component_simps)
- thus "(g1 +++ g2) x = g2 (2 *s x - 1)" proof(cases "dest_vec1 x = 1 / 2")
- case True hence "x = (1/2) *s 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
+ thus "(g1 +++ g2) x = g2 (2 *\<^sub>R x - 1)" proof(cases "dest_vec1 x = 1 / 2")
+ case True hence "x = (1/2) *\<^sub>R 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
thus ?thesis unfolding joinpaths_def using assms[unfolded pathstart_def pathfinish_def] by auto
qed (auto simp add:le_less joinpaths_def) qed
next assume as:"continuous_on {0..1} g1" "continuous_on {0..1} g2"
- have *:"{0 .. 1::real^1} = {0.. (1/2)*s 1} \<union> {(1/2) *s 1 .. 1}" by(auto simp add: vector_component_simps)
- have **:"op *s 2 ` {0..(1 / 2) *s 1} = {0..1::real^1}" apply(rule set_ext, rule) unfolding image_iff
- defer apply(rule_tac x="(1/2)*s x" in bexI) by(auto simp add: vector_component_simps)
- have ***:"(\<lambda>x. 2 *s x - 1) ` {(1 / 2) *s 1..1} = {0..1::real^1}"
+ have *:"{0 .. 1::real^1} = {0.. (1/2)*\<^sub>R 1} \<union> {(1/2) *\<^sub>R 1 .. 1}" by(auto simp add: vector_component_simps)
+ have **:"op *\<^sub>R 2 ` {0..(1 / 2) *\<^sub>R 1} = {0..1::real^1}" apply(rule set_ext, rule) unfolding image_iff
+ defer apply(rule_tac x="(1/2)*\<^sub>R x" in bexI) by(auto simp add: vector_component_simps)
+ have ***:"(\<lambda>x. 2 *\<^sub>R x - 1) ` {(1 / 2) *\<^sub>R 1..1} = {0..1::real^1}"
unfolding image_affinity_interval[of _ "- 1", unfolded diff_def[symmetric]] and interval_eq_empty_1
by(auto simp add: vector_component_simps)
- have ****:"\<And>x::real^1. x $ 1 * 2 = 1 \<longleftrightarrow> x = (1/2) *s 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
+ have ****:"\<And>x::real^1. x $ 1 * 2 = 1 \<longleftrightarrow> x = (1/2) *\<^sub>R 1" unfolding Cart_eq by(auto simp add: forall_1 vector_component_simps)
show "continuous_on {0..1} (g1 +++ g2)" unfolding * apply(rule continuous_on_union) apply(rule closed_interval)+ proof-
- show "continuous_on {0..(1 / 2) *s 1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "\<lambda>x. g1 (2 *s x)"]) defer
+ show "continuous_on {0..(1 / 2) *\<^sub>R 1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "\<lambda>x. g1 (2 *\<^sub>R x)"]) defer
unfolding o_def[THEN sym] apply(rule continuous_on_compose) apply(rule continuous_on_cmul, rule continuous_on_id)
unfolding ** apply(rule as(1)) unfolding joinpaths_def by(auto simp add: vector_component_simps) next
- show "continuous_on {(1/2)*s1..1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "g2 \<circ> (\<lambda>x. 2 *s x - 1)"]) defer
+ show "continuous_on {(1/2)*\<^sub>R1..1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "g2 \<circ> (\<lambda>x. 2 *\<^sub>R x - 1)"]) defer
apply(rule continuous_on_compose) apply(rule continuous_on_sub, rule continuous_on_cmul, rule continuous_on_id, rule continuous_on_const)
unfolding *** o_def joinpaths_def apply(rule as(2)) using assms[unfolded pathstart_def pathfinish_def]
by(auto simp add: vector_component_simps ****) qed qed
lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)" proof
fix x assume "x \<in> path_image (g1 +++ g2)"
- then obtain y where y:"y\<in>{0..1}" "x = (if dest_vec1 y \<le> 1 / 2 then g1 (2 *s y) else g2 (2 *s y - 1))"
+ then obtain y where y:"y\<in>{0..1}" "x = (if dest_vec1 y \<le> 1 / 2 then g1 (2 *\<^sub>R y) else g2 (2 *\<^sub>R y - 1))"
unfolding path_image_def image_iff joinpaths_def by auto
thus "x \<in> path_image g1 \<union> path_image g2" apply(cases "dest_vec1 y \<le> 1/2")
apply(rule_tac UnI1) defer apply(rule_tac UnI2) unfolding y(2) path_image_def using y(1)
@@ -2814,12 +2880,12 @@
fix x assume "x \<in> path_image g1"
then obtain y where y:"y\<in>{0..1}" "x = g1 y" unfolding path_image_def image_iff by auto
thus "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
- apply(rule_tac x="(1/2) *s y" in bexI) by(auto simp add: vector_component_simps) next
+ apply(rule_tac x="(1/2) *\<^sub>R y" in bexI) by(auto simp add: vector_component_simps) next
fix x assume "x \<in> path_image g2"
then obtain y where y:"y\<in>{0..1}" "x = g2 y" unfolding path_image_def image_iff by auto
moreover have *:"y $ 1 = 0 \<Longrightarrow> y = 0" unfolding Cart_eq by auto
ultimately show "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff
- apply(rule_tac x="(1/2) *s (y + 1)" in bexI) using assms(3)[unfolded pathfinish_def pathstart_def]
+ apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI) using assms(3)[unfolded pathfinish_def pathstart_def]
by(auto simp add: vector_component_simps) qed
lemma not_in_path_image_join:
@@ -2831,6 +2897,10 @@
apply(erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
unfolding mem_interval_1 by(auto simp add:vector_component_simps)
+lemma dest_vec1_scaleR [simp]:
+ "dest_vec1 (scaleR a x) = scaleR a (dest_vec1 x)"
+unfolding dest_vec1_def by simp
+
lemma simple_path_join_loop:
assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
"(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
@@ -2840,40 +2910,40 @@
fix x y::"real^1" assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y"
show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" proof(case_tac "x$1 \<le> 1/2",case_tac[!] "y$1 \<le> 1/2", unfold not_le)
assume as:"x $ 1 \<le> 1 / 2" "y $ 1 \<le> 1 / 2"
- hence "g1 (2 *s x) = g1 (2 *s y)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
- moreover have "2 *s x \<in> {0..1}" "2 *s y \<in> {0..1}" using xy(1,2) as
+ hence "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
+ moreover have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" using xy(1,2) as
unfolding mem_interval_1 dest_vec1_def by(auto simp add:vector_component_simps)
- ultimately show ?thesis using inj(1)[of "2*s x" "2*s y"] by auto
+ ultimately show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto
next assume as:"x $ 1 > 1 / 2" "y $ 1 > 1 / 2"
- hence "g2 (2 *s x - 1) = g2 (2 *s y - 1)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
- moreover have "2 *s x - 1 \<in> {0..1}" "2 *s y - 1 \<in> {0..1}" using xy(1,2) as
+ hence "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)" using xy(3) unfolding joinpaths_def dest_vec1_def by auto
+ moreover have "2 *\<^sub>R x - 1 \<in> {0..1}" "2 *\<^sub>R y - 1 \<in> {0..1}" using xy(1,2) as
unfolding mem_interval_1 dest_vec1_def by(auto simp add:vector_component_simps)
- ultimately show ?thesis using inj(2)[of "2*s x - 1" "2*s y - 1"] by auto
+ ultimately show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto
next assume as:"x $ 1 \<le> 1 / 2" "y $ 1 > 1 / 2"
hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
moreover have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def
- using inj(2)[of "2 *s y - 1" 0] and xy(2)[unfolded mem_interval_1]
+ using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2)[unfolded mem_interval_1]
apply(rule_tac ccontr) by(auto simp add:vector_component_simps field_simps Cart_eq)
ultimately have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto
hence "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1)[unfolded mem_interval_1]
- using inj(1)[of "2 *s x" 0] by(auto simp add:vector_component_simps)
+ using inj(1)[of "2 *\<^sub>R x" 0] by(auto simp add:vector_component_simps)
moreover have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]
unfolding joinpaths_def pathfinish_def using as(2) and xy(2)[unfolded mem_interval_1]
- using inj(2)[of "2 *s y - 1" 1] by (auto simp add:vector_component_simps Cart_eq)
+ using inj(2)[of "2 *\<^sub>R y - 1" 1] by (auto simp add:vector_component_simps Cart_eq)
ultimately show ?thesis by auto
next assume as:"x $ 1 > 1 / 2" "y $ 1 \<le> 1 / 2"
hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
moreover have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def
- using inj(2)[of "2 *s x - 1" 0] and xy(1)[unfolded mem_interval_1]
+ using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1)[unfolded mem_interval_1]
apply(rule_tac ccontr) by(auto simp add:vector_component_simps field_simps Cart_eq)
ultimately have *:"?g y = pathstart g1" using assms(4) unfolding xy(3) by auto
hence "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2)[unfolded mem_interval_1]
- using inj(1)[of "2 *s y" 0] by(auto simp add:vector_component_simps)
+ using inj(1)[of "2 *\<^sub>R y" 0] by(auto simp add:vector_component_simps)
moreover have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]
unfolding joinpaths_def pathfinish_def using as(1) and xy(1)[unfolded mem_interval_1]
- using inj(2)[of "2 *s x - 1" 1] by(auto simp add:vector_component_simps Cart_eq)
+ using inj(2)[of "2 *\<^sub>R x - 1" 1] by(auto simp add:vector_component_simps Cart_eq)
ultimately show ?thesis by auto qed qed
lemma injective_path_join:
@@ -2884,22 +2954,22 @@
note inj = assms(1,2)[unfolded injective_path_def, rule_format]
fix x y assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"
show "x = y" proof(cases "x$1 \<le> 1/2", case_tac[!] "y$1 \<le> 1/2", unfold not_le)
- assume "x $ 1 \<le> 1 / 2" "y $ 1 \<le> 1 / 2" thus ?thesis using inj(1)[of "2*s x" "2*s y"] and xy
+ assume "x $ 1 \<le> 1 / 2" "y $ 1 \<le> 1 / 2" thus ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy
unfolding mem_interval_1 joinpaths_def by(auto simp add:vector_component_simps)
- next assume "x $ 1 > 1 / 2" "y $ 1 > 1 / 2" thus ?thesis using inj(2)[of "2*s x - 1" "2*s y - 1"] and xy
+ next assume "x $ 1 > 1 / 2" "y $ 1 > 1 / 2" thus ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy
unfolding mem_interval_1 joinpaths_def by(auto simp add:vector_component_simps)
next assume as:"x $ 1 \<le> 1 / 2" "y $ 1 > 1 / 2"
hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def
using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
hence "?g x = pathfinish g1" "?g y = pathstart g2" using assms(4) unfolding assms(3) xy(3) by auto
- thus ?thesis using as and inj(1)[of "2 *s x" 1] inj(2)[of "2 *s y - 1" 0] and xy(1,2)
+ thus ?thesis using as and inj(1)[of "2 *\<^sub>R x" 1] inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(1,2)
unfolding pathstart_def pathfinish_def joinpaths_def mem_interval_1
by(auto simp add:vector_component_simps Cart_eq forall_1)
next assume as:"x $ 1 > 1 / 2" "y $ 1 \<le> 1 / 2"
hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def
using xy(1,2)[unfolded mem_interval_1] by(auto simp add:vector_component_simps intro!: imageI)
hence "?g x = pathstart g2" "?g y = pathfinish g1" using assms(4) unfolding assms(3) xy(3) by auto
- thus ?thesis using as and inj(2)[of "2 *s x - 1" 0] inj(1)[of "2 *s y" 1] and xy(1,2)
+ thus ?thesis using as and inj(2)[of "2 *\<^sub>R x - 1" 0] inj(1)[of "2 *\<^sub>R y" 1] and xy(1,2)
unfolding pathstart_def pathfinish_def joinpaths_def mem_interval_1
by(auto simp add:vector_component_simps forall_1 Cart_eq) qed qed
@@ -2966,7 +3036,7 @@
definition
linepath :: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> real ^ 1 \<Rightarrow> real ^ 'n" where
- "linepath a b = (\<lambda>x. (1 - dest_vec1 x) *s a + dest_vec1 x *s b)"
+ "linepath a b = (\<lambda>x. (1 - dest_vec1 x) *\<^sub>R a + dest_vec1 x *\<^sub>R b)"
lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a"
unfolding pathstart_def linepath_def by auto
@@ -2975,7 +3045,8 @@
unfolding pathfinish_def linepath_def by auto
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
- unfolding linepath_def by(auto simp add: vec1_dest_vec1 o_def intro!: continuous_intros)
+ unfolding linepath_def
+ by (intro continuous_intros continuous_dest_vec1)
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on)
@@ -2985,7 +3056,7 @@
lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)"
unfolding path_image_def segment linepath_def apply (rule set_ext, rule) defer
- unfolding mem_Collect_eq image_iff apply(erule exE) apply(rule_tac x="u *s 1" in bexI)
+ unfolding mem_Collect_eq image_iff apply(erule exE) apply(rule_tac x="u *\<^sub>R 1" in bexI)
by(auto simp add:vector_component_simps)
lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a"
@@ -2993,10 +3064,10 @@
lemma injective_path_linepath: assumes "a \<noteq> b" shows "injective_path(linepath a b)" proof-
{ obtain i where i:"a$i \<noteq> b$i" using assms[unfolded Cart_eq] by auto
- fix x y::"real^1" assume "x $ 1 *s b + y $ 1 *s a = x $ 1 *s a + y $ 1 *s b"
+ fix x y::"real^1" assume "x $ 1 *\<^sub>R b + y $ 1 *\<^sub>R a = x $ 1 *\<^sub>R a + y $ 1 *\<^sub>R b"
hence "x$1 * (b$i - a$i) = y$1 * (b$i - a$i)" unfolding Cart_eq by(auto simp add:field_simps vector_component_simps)
hence "x = y" unfolding mult_cancel_right Cart_eq using i(1) by(auto simp add:field_simps) }
- thus ?thesis unfolding injective_path_def linepath_def by(auto simp add:vector_component_simps field_simps) qed
+ thus ?thesis unfolding injective_path_def linepath_def by(auto simp add:vector_component_simps algebra_simps) qed
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)" by(auto intro!: injective_imp_simple_path injective_path_linepath)
@@ -3136,7 +3207,7 @@
lemma path_connected_singleton: "path_connected {a}"
unfolding path_connected_def apply(rule,rule)
- apply(rule_tac x="linepath a a" in exI) by(auto simp add:segment)
+ apply(rule_tac x="linepath a a" in exI) by(auto simp add:segment scaleR_left_diff_distrib)
lemma path_connected_Un: assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}"
shows "path_connected (s \<union> t)" unfolding path_connected_component proof(rule,rule)
@@ -3153,18 +3224,18 @@
obtain \<psi> where \<psi>:"bij_betw \<psi> {1..CARD('n)} (UNIV::'n set)" using ex_bij_betw_nat_finite_1[OF finite_UNIV] by auto
let ?U = "UNIV::(real^'n) set" let ?u = "?U - {0}"
let ?basis = "\<lambda>k. basis (\<psi> k)"
- let ?A = "\<lambda>k. {x::real^'n. \<exists>i\<in>{1..k}. (basis (\<psi> i)) \<bullet> x \<noteq> 0}"
+ let ?A = "\<lambda>k. {x::real^'n. \<exists>i\<in>{1..k}. inner (basis (\<psi> i)) x \<noteq> 0}"
have "\<forall>k\<in>{2..CARD('n)}. path_connected (?A k)" proof
- have *:"\<And>k. ?A (Suc k) = {x. ?basis (Suc k) \<bullet> x < 0} \<union> {x. ?basis (Suc k) \<bullet> x > 0} \<union> ?A k" apply(rule set_ext,rule) defer
+ have *:"\<And>k. ?A (Suc k) = {x. inner (?basis (Suc k)) x < 0} \<union> {x. inner (?basis (Suc k)) x > 0} \<union> ?A k" apply(rule set_ext,rule) defer
apply(erule UnE)+ unfolding mem_Collect_eq apply(rule_tac[1-2] x="Suc k" in bexI)
by(auto elim!: ballE simp add: not_less le_Suc_eq)
fix k assume "k \<in> {2..CARD('n)}" thus "path_connected (?A k)" proof(induct k)
case (Suc k) show ?case proof(cases "k = 1")
case False from Suc have d:"k \<in> {1..CARD('n)}" "Suc k \<in> {1..CARD('n)}" by auto
hence "\<psi> k \<noteq> \<psi> (Suc k)" using \<psi>[unfolded bij_betw_def inj_on_def, THEN conjunct1, THEN bspec[where x=k]] by auto
- hence **:"?basis k + ?basis (Suc k) \<in> {x. 0 < ?basis (Suc k) \<bullet> x} \<inter> (?A k)"
- "?basis k - ?basis (Suc k) \<in> {x. 0 > ?basis (Suc k) \<bullet> x} \<inter> ({x. 0 < ?basis (Suc k) \<bullet> x} \<union> (?A k))" using d
- by(auto simp add: dot_basis vector_component_simps intro!:bexI[where x=k])
+ hence **:"?basis k + ?basis (Suc k) \<in> {x. 0 < inner (?basis (Suc k)) x} \<inter> (?A k)"
+ "?basis k - ?basis (Suc k) \<in> {x. 0 > inner (?basis (Suc k)) x} \<inter> ({x. 0 < inner (?basis (Suc k)) x} \<union> (?A k))" using d
+ by(auto simp add: inner_basis vector_component_simps intro!:bexI[where x=k])
show ?thesis unfolding * Un_assoc apply(rule path_connected_Un) defer apply(rule path_connected_Un)
prefer 5 apply(rule_tac[1-2] convex_imp_path_connected, rule convex_halfspace_lt, rule convex_halfspace_gt)
apply(rule Suc(1)) apply(rule_tac[2-3] ccontr) using d ** False by auto
@@ -3177,18 +3248,18 @@
apply(rule_tac[5] x=" ?basis 1 + ?basis 2" in nequals0I)
apply(rule_tac[6] x="-?basis 1 + ?basis 2" in nequals0I)
apply(rule_tac[7] x="-?basis 1 - ?basis 2" in nequals0I)
- using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add:vector_component_simps dot_basis)
+ using d unfolding *** by(auto intro!: convex_halfspace_gt convex_halfspace_lt, auto simp add:vector_component_simps inner_basis)
qed qed auto qed note lem = this
- have ***:"\<And>x::real^'n. (\<exists>i\<in>{1..CARD('n)}. basis (\<psi> i) \<bullet> x \<noteq> 0) \<longleftrightarrow> (\<exists>i. basis i \<bullet> x \<noteq> 0)"
+ have ***:"\<And>x::real^'n. (\<exists>i\<in>{1..CARD('n)}. inner (basis (\<psi> i)) x \<noteq> 0) \<longleftrightarrow> (\<exists>i. inner (basis i) x \<noteq> 0)"
apply rule apply(erule bexE) apply(rule_tac x="\<psi> i" in exI) defer apply(erule exE) proof-
- fix x::"real^'n" and i assume as:"basis i \<bullet> x \<noteq> 0"
+ fix x::"real^'n" and i assume as:"inner (basis i) x \<noteq> 0"
have "i\<in>\<psi> ` {1..CARD('n)}" using \<psi>[unfolded bij_betw_def, THEN conjunct2] by auto
then obtain j where "j\<in>{1..CARD('n)}" "\<psi> j = i" by auto
- thus "\<exists>i\<in>{1..CARD('n)}. basis (\<psi> i) \<bullet> x \<noteq> 0" apply(rule_tac x=j in bexI) using as by auto qed auto
+ thus "\<exists>i\<in>{1..CARD('n)}. inner (basis (\<psi> i)) x \<noteq> 0" apply(rule_tac x=j in bexI) using as by auto qed auto
have *:"?U - {a} = (\<lambda>x. x + a) ` {x. x \<noteq> 0}" apply(rule set_ext) unfolding image_iff
apply rule apply(rule_tac x="x - a" in bexI) by auto
- have **:"\<And>x::real^'n. x\<noteq>0 \<longleftrightarrow> (\<exists>i. basis i \<bullet> x \<noteq> 0)" unfolding Cart_eq by(auto simp add: dot_basis)
+ have **:"\<And>x::real^'n. x\<noteq>0 \<longleftrightarrow> (\<exists>i. inner (basis i) x \<noteq> 0)" unfolding Cart_eq by(auto simp add: inner_basis)
show ?thesis unfolding * apply(rule path_connected_continuous_image) apply(rule continuous_on_intros)+
unfolding ** apply(rule lem[THEN bspec[where x="CARD('n)"], unfolded ***]) using assms by auto qed
@@ -3197,14 +3268,14 @@
case False hence "{x::real^'n. norm(x - a) = r} = {}" using True by auto
thus ?thesis using path_connected_empty by auto
qed(auto intro!:path_connected_singleton) next
- case False hence *:"{x::real^'n. norm(x - a) = r} = (\<lambda>x. a + r *s x) ` {x. norm x = 1}" unfolding not_le apply -apply(rule set_ext,rule)
- unfolding image_iff apply(rule_tac x="(1/r) *s (x - a)" in bexI) unfolding mem_Collect_eq norm_mul by auto
+ case False hence *:"{x::real^'n. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" unfolding not_le apply -apply(rule set_ext,rule)
+ unfolding image_iff apply(rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI) unfolding mem_Collect_eq norm_scaleR by (auto simp add: scaleR_right_diff_distrib)
have ***:"\<And>xa. (if xa = 0 then 0 else 1) \<noteq> 1 \<Longrightarrow> xa = 0" apply(rule ccontr) by auto
- have **:"{x::real^'n. norm x = 1} = (\<lambda>x. (1/norm x) *s x) ` (UNIV - {0})" apply(rule set_ext,rule)
- unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq norm_mul by(auto intro!: ***)
- have "continuous_on (UNIV - {0}) (vec1 \<circ> (\<lambda>x::real^'n. 1 / norm x))" unfolding o_def continuous_on_eq_continuous_within
+ have **:"{x::real^'n. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})" apply(rule set_ext,rule)
+ unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq by(auto intro!: ***)
+ have "continuous_on (UNIV - {0}) (\<lambda>x::real^'n. 1 / norm x)" unfolding o_def continuous_on_eq_continuous_within
apply(rule, rule continuous_at_within_inv[unfolded o_def inverse_eq_divide]) apply(rule continuous_at_within)
- apply(rule continuous_at_vec1_norm[unfolded o_def]) by auto
+ apply(rule continuous_at_norm[unfolded o_def]) by auto
thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
by(auto intro!: path_connected_continuous_image continuous_on_intros continuous_on_mul) qed
--- a/src/HOL/Library/Euclidean_Space.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/Library/Euclidean_Space.thy Sat Jun 13 16:32:38 2009 +0200
@@ -369,19 +369,33 @@
end
-lemma tendsto_Cart_nth:
- fixes f :: "'a \<Rightarrow> 'b::topological_space ^ 'n::finite"
+lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
+unfolding open_vector_def by auto
+
+lemma open_vimage_Cart_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
+unfolding open_vector_def
+apply clarify
+apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
+done
+
+lemma closed_vimage_Cart_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
+unfolding closed_open vimage_Compl [symmetric]
+by (rule open_vimage_Cart_nth)
+
+lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
+proof -
+ have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto
+ thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
+ by (simp add: closed_INT closed_vimage_Cart_nth)
+qed
+
+lemma tendsto_Cart_nth [tendsto_intros]:
assumes "((\<lambda>x. f x) ---> a) net"
shows "((\<lambda>x. f x $ i) ---> a $ i) net"
proof (rule topological_tendstoI)
- fix S :: "'b set" assume "open S" "a $ i \<in> S"
+ fix S assume "open S" "a $ i \<in> S"
then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"
- unfolding open_vector_def
- apply simp_all
- apply clarify
- apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI)
- apply simp
- done
+ by (simp_all add: open_vimage_Cart_nth)
with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y $ i) -` S) net"
by (rule topological_tendstoD)
then show "eventually (\<lambda>x. f x $ i \<in> S) net"
@@ -736,7 +750,7 @@
instantiation "^" :: (real_normed_vector, finite) real_normed_vector
begin
-definition vector_norm_def:
+definition norm_vector_def:
"norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) UNIV"
definition vector_sgn_def:
@@ -745,30 +759,30 @@
instance proof
fix a :: real and x y :: "'a ^ 'b"
show "0 \<le> norm x"
- unfolding vector_norm_def
+ unfolding norm_vector_def
by (rule setL2_nonneg)
show "norm x = 0 \<longleftrightarrow> x = 0"
- unfolding vector_norm_def
+ unfolding norm_vector_def
by (simp add: setL2_eq_0_iff Cart_eq)
show "norm (x + y) \<le> norm x + norm y"
- unfolding vector_norm_def
+ unfolding norm_vector_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
apply (simp add: setL2_mono norm_triangle_ineq)
done
show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
- unfolding vector_norm_def
- by (simp add: norm_scaleR setL2_right_distrib)
+ unfolding norm_vector_def
+ by (simp add: setL2_right_distrib)
show "sgn x = scaleR (inverse (norm x)) x"
by (rule vector_sgn_def)
show "dist x y = norm (x - y)"
- unfolding dist_vector_def vector_norm_def
+ unfolding dist_vector_def norm_vector_def
by (simp add: dist_norm)
qed
end
lemma norm_nth_le: "norm (x $ i) \<le> norm x"
-unfolding vector_norm_def
+unfolding norm_vector_def
by (rule member_le_setL2) simp_all
interpretation Cart_nth: bounded_linear "\<lambda>x. x $ i"
@@ -785,28 +799,28 @@
instantiation "^" :: (real_inner, finite) real_inner
begin
-definition vector_inner_def:
+definition inner_vector_def:
"inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
instance proof
fix r :: real and x y z :: "'a ^ 'b"
show "inner x y = inner y x"
- unfolding vector_inner_def
+ unfolding inner_vector_def
by (simp add: inner_commute)
show "inner (x + y) z = inner x z + inner y z"
- unfolding vector_inner_def
- by (simp add: inner_left_distrib setsum_addf)
+ unfolding inner_vector_def
+ by (simp add: inner_add_left setsum_addf)
show "inner (scaleR r x) y = r * inner x y"
- unfolding vector_inner_def
- by (simp add: inner_scaleR_left setsum_right_distrib)
+ unfolding inner_vector_def
+ by (simp add: setsum_right_distrib)
show "0 \<le> inner x x"
- unfolding vector_inner_def
+ unfolding inner_vector_def
by (simp add: setsum_nonneg)
show "inner x x = 0 \<longleftrightarrow> x = 0"
- unfolding vector_inner_def
+ unfolding inner_vector_def
by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
show "norm x = sqrt (inner x x)"
- unfolding vector_inner_def vector_norm_def setL2_def
+ unfolding inner_vector_def norm_vector_def setL2_def
by (simp add: power2_norm_eq_inner)
qed
@@ -864,7 +878,7 @@
done
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
- by (simp add: vector_norm_def UNIV_1)
+ by (simp add: norm_vector_def UNIV_1)
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
by (simp add: norm_vector_1)
@@ -983,12 +997,12 @@
by (rule norm_zero)
lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
- by (simp add: vector_norm_def vector_component setL2_right_distrib
+ by (simp add: norm_vector_def vector_component setL2_right_distrib
abs_mult cong: strong_setL2_cong)
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
- by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
+ by (simp add: norm_vector_def dot_def setL2_def power2_eq_square)
lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
- by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
+ by (simp add: norm_vector_def setL2_def dot_def power2_eq_square)
lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
by (simp add: real_vector_norm_def)
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n::finite)" by (metis norm_eq_zero)
@@ -1064,7 +1078,7 @@
qed
lemma component_le_norm: "\<bar>x$i\<bar> <= norm (x::real ^ 'n::finite)"
- apply (simp add: vector_norm_def)
+ apply (simp add: norm_vector_def)
apply (rule member_le_setL2, simp_all)
done
@@ -1077,7 +1091,7 @@
by (metis component_le_norm basic_trans_rules(21))
lemma norm_le_l1: "norm (x:: real ^'n::finite) <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
- by (simp add: vector_norm_def setL2_le_setsum)
+ by (simp add: norm_vector_def setL2_le_setsum)
lemma real_abs_norm: "\<bar>norm x\<bar> = norm (x :: real ^ _)"
by (rule abs_norm_cancel)
@@ -1522,6 +1536,13 @@
shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n::finite) = (x$i :: 'a::semiring_1)"
by (auto simp add: dot_def basis_def cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
+lemma inner_basis:
+ fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n::finite"
+ shows "inner (basis i) x = inner 1 (x $ i)"
+ and "inner x (basis i) = inner (x $ i) 1"
+ unfolding inner_vector_def basis_def
+ by (auto simp add: cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
+
lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
by (auto simp add: Cart_eq)
@@ -2917,7 +2938,7 @@
done
lemma norm_pastecart: "norm (pastecart x y) <= norm x + norm y"
- unfolding vector_norm_def setL2_def setsum_UNIV_sum
+ unfolding norm_vector_def setL2_def setsum_UNIV_sum
by (simp add: sqrt_add_le_add_sqrt setsum_nonneg)
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
--- a/src/HOL/Library/Inner_Product.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/Library/Inner_Product.thy Sat Jun 13 16:32:38 2009 +0200
@@ -27,28 +27,28 @@
class real_inner = real_vector + sgn_div_norm + dist_norm + open_dist +
fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"
assumes inner_commute: "inner x y = inner y x"
- and inner_left_distrib: "inner (x + y) z = inner x z + inner y z"
- and inner_scaleR_left: "inner (scaleR r x) y = r * (inner x y)"
+ and inner_add_left: "inner (x + y) z = inner x z + inner y z"
+ and inner_scaleR_left [simp]: "inner (scaleR r x) y = r * (inner x y)"
and inner_ge_zero [simp]: "0 \<le> inner x x"
and inner_eq_zero_iff [simp]: "inner x x = 0 \<longleftrightarrow> x = 0"
and norm_eq_sqrt_inner: "norm x = sqrt (inner x x)"
begin
lemma inner_zero_left [simp]: "inner 0 x = 0"
- using inner_left_distrib [of 0 0 x] by simp
+ using inner_add_left [of 0 0 x] by simp
lemma inner_minus_left [simp]: "inner (- x) y = - inner x y"
- using inner_left_distrib [of x "- x" y] by simp
+ using inner_add_left [of x "- x" y] by simp
lemma inner_diff_left: "inner (x - y) z = inner x z - inner y z"
- by (simp add: diff_minus inner_left_distrib)
+ by (simp add: diff_minus inner_add_left)
text {* Transfer distributivity rules to right argument. *}
-lemma inner_right_distrib: "inner x (y + z) = inner x y + inner x z"
- using inner_left_distrib [of y z x] by (simp only: inner_commute)
+lemma inner_add_right: "inner x (y + z) = inner x y + inner x z"
+ using inner_add_left [of y z x] by (simp only: inner_commute)
-lemma inner_scaleR_right: "inner x (scaleR r y) = r * (inner x y)"
+lemma inner_scaleR_right [simp]: "inner x (scaleR r y) = r * (inner x y)"
using inner_scaleR_left [of r y x] by (simp only: inner_commute)
lemma inner_zero_right [simp]: "inner x 0 = 0"
@@ -60,9 +60,14 @@
lemma inner_diff_right: "inner x (y - z) = inner x y - inner x z"
using inner_diff_left [of y z x] by (simp only: inner_commute)
+lemmas inner_add [algebra_simps] = inner_add_left inner_add_right
+lemmas inner_diff [algebra_simps] = inner_diff_left inner_diff_right
+lemmas inner_scaleR = inner_scaleR_left inner_scaleR_right
+
+text {* Legacy theorem names *}
+lemmas inner_left_distrib = inner_add_left
+lemmas inner_right_distrib = inner_add_right
lemmas inner_distrib = inner_left_distrib inner_right_distrib
-lemmas inner_diff = inner_diff_left inner_diff_right
-lemmas inner_scaleR = inner_scaleR_left inner_scaleR_right
lemma inner_gt_zero_iff [simp]: "0 < inner x x \<longleftrightarrow> x \<noteq> 0"
by (simp add: order_less_le)
@@ -81,7 +86,7 @@
have "0 \<le> inner (x - scaleR ?r y) (x - scaleR ?r y)"
by (rule inner_ge_zero)
also have "\<dots> = inner x x - inner y x * ?r"
- by (simp add: inner_diff inner_scaleR)
+ by (simp add: inner_diff)
also have "\<dots> = inner x x - (inner x y)\<twosuperior> / inner y y"
by (simp add: power2_eq_square inner_commute)
finally have "0 \<le> inner x x - (inner x y)\<twosuperior> / inner y y" .
@@ -116,7 +121,7 @@
by (rule order_trans [OF abs_ge_self Cauchy_Schwarz_ineq2])
thus "(norm (x + y))\<twosuperior> \<le> (norm x + norm y)\<twosuperior>"
unfolding power2_sum power2_norm_eq_inner
- by (simp add: inner_distrib inner_commute)
+ by (simp add: inner_add inner_commute)
show "0 \<le> norm x + norm y"
unfolding norm_eq_sqrt_inner
by (simp add: add_nonneg_nonneg)
@@ -125,7 +130,7 @@
by (simp add: real_sqrt_mult_distrib)
then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
unfolding norm_eq_sqrt_inner
- by (simp add: inner_scaleR power2_eq_square mult_assoc)
+ by (simp add: power2_eq_square mult_assoc)
qed
end
@@ -149,9 +154,9 @@
proof
fix x y z :: 'a and r :: real
show "inner (x + y) z = inner x z + inner y z"
- by (rule inner_left_distrib)
+ by (rule inner_add_left)
show "inner x (y + z) = inner x y + inner x z"
- by (rule inner_right_distrib)
+ by (rule inner_add_right)
show "inner (scaleR r x) y = scaleR r (inner x y)"
unfolding real_scaleR_def by (rule inner_scaleR_left)
show "inner x (scaleR r y) = scaleR r (inner x y)"
@@ -244,7 +249,7 @@
\<Longrightarrow> GDERIV (\<lambda>x. g (f x)) x :> scaleR dg df"
unfolding gderiv_def deriv_fderiv
apply (drule (1) FDERIV_compose)
- apply (simp add: inner_scaleR_right mult_ac)
+ apply (simp add: mult_ac)
done
lemma FDERIV_subst: "\<lbrakk>FDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> FDERIV f x :> d"
@@ -286,7 +291,7 @@
unfolding gderiv_def
apply (rule FDERIV_subst)
apply (erule (1) FDERIV_mult)
- apply (simp add: inner_distrib inner_scaleR mult_ac)
+ apply (simp add: inner_add mult_ac)
done
lemma GDERIV_inverse:
@@ -302,7 +307,7 @@
have 1: "FDERIV (\<lambda>x. inner x x) x :> (\<lambda>h. inner x h + inner h x)"
by (intro inner.FDERIV FDERIV_ident)
have 2: "(\<lambda>h. inner x h + inner h x) = (\<lambda>h. inner h (scaleR 2 x))"
- by (simp add: expand_fun_eq inner_scaleR inner_commute)
+ by (simp add: expand_fun_eq inner_commute)
have "0 < inner x x" using `x \<noteq> 0` by simp
then have 3: "DERIV sqrt (inner x x) :> (inverse (sqrt (inner x x)) / 2)"
by (rule DERIV_real_sqrt)
--- a/src/HOL/Library/Product_Vector.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/Library/Product_Vector.thy Sat Jun 13 16:32:38 2009 +0200
@@ -72,6 +72,37 @@
end
+lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
+unfolding open_prod_def by auto
+
+lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
+by auto
+
+lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
+by auto
+
+lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
+by (simp add: fst_vimage_eq_Times open_Times)
+
+lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
+by (simp add: snd_vimage_eq_Times open_Times)
+
+lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
+unfolding closed_open vimage_Compl [symmetric]
+by (rule open_vimage_fst)
+
+lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
+unfolding closed_open vimage_Compl [symmetric]
+by (rule open_vimage_snd)
+
+lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
+proof -
+ have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
+ thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
+ by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
+qed
+
+
subsection {* Product is a metric space *}
instantiation
@@ -87,25 +118,21 @@
instance proof
fix x y :: "'a \<times> 'b"
show "dist x y = 0 \<longleftrightarrow> x = y"
- unfolding dist_prod_def
- by (simp add: expand_prod_eq)
+ unfolding dist_prod_def expand_prod_eq by simp
next
fix x y z :: "'a \<times> 'b"
show "dist x y \<le> dist x z + dist y z"
unfolding dist_prod_def
- apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
- apply (rule real_sqrt_le_mono)
- apply (rule order_trans [OF add_mono])
- apply (rule power_mono [OF dist_triangle2 [of _ _ "fst z"] zero_le_dist])
- apply (rule power_mono [OF dist_triangle2 [of _ _ "snd z"] zero_le_dist])
- apply (simp only: real_sum_squared_expand)
- done
+ by (intro order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]
+ real_sqrt_le_mono add_mono power_mono dist_triangle2 zero_le_dist)
next
(* FIXME: long proof! *)
(* Maybe it would be easier to define topological spaces *)
(* in terms of neighborhoods instead of open sets? *)
fix S :: "('a \<times> 'b) set"
show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
+ proof
+ assume "open S" thus "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
unfolding open_prod_def open_dist
apply safe
apply (drule (1) bspec)
@@ -121,7 +148,11 @@
apply (erule le_less_trans [OF real_sqrt_sum_squares_ge1])
apply (drule spec, erule mp)
apply (erule le_less_trans [OF real_sqrt_sum_squares_ge2])
-
+ done
+ next
+ assume "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" thus "open S"
+ unfolding open_prod_def open_dist
+ apply safe
apply (drule (1) bspec)
apply clarify
apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
@@ -132,14 +163,14 @@
apply clarify
apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
apply clarify
- apply (rule le_less_trans [OF dist_triangle])
- apply (erule less_le_trans [OF add_strict_right_mono], simp)
+ apply (simp add: less_diff_eq)
+ apply (erule le_less_trans [OF dist_triangle])
apply (rule conjI)
apply clarify
apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
apply clarify
- apply (rule le_less_trans [OF dist_triangle])
- apply (erule less_le_trans [OF add_strict_right_mono], simp)
+ apply (simp add: less_diff_eq)
+ apply (erule le_less_trans [OF dist_triangle])
apply (rule conjI)
apply simp
apply (clarify, rename_tac c d)
@@ -149,6 +180,7 @@
apply (rule_tac x="e / sqrt 2" in exI, simp add: divide_pos_pos)
apply (simp add: power_divide)
done
+ qed
qed
end
@@ -161,7 +193,7 @@
lemma dist_snd_le: "dist (snd x) (snd y) \<le> dist x y"
unfolding dist_prod_def by simp
-lemma tendsto_fst:
+lemma tendsto_fst [tendsto_intros]:
assumes "(f ---> a) net"
shows "((\<lambda>x. fst (f x)) ---> fst a) net"
proof (rule topological_tendstoI)
@@ -180,7 +212,7 @@
by simp
qed
-lemma tendsto_snd:
+lemma tendsto_snd [tendsto_intros]:
assumes "(f ---> a) net"
shows "((\<lambda>x. snd (f x)) ---> snd a) net"
proof (rule topological_tendstoI)
@@ -199,7 +231,7 @@
by simp
qed
-lemma tendsto_Pair:
+lemma tendsto_Pair [tendsto_intros]:
assumes "(f ---> a) net" and "(g ---> b) net"
shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
proof (rule topological_tendstoI)
@@ -315,7 +347,7 @@
done
show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
unfolding norm_prod_def
- apply (simp add: norm_scaleR power_mult_distrib)
+ apply (simp add: power_mult_distrib)
apply (simp add: right_distrib [symmetric])
apply (simp add: real_sqrt_mult_distrib)
done
@@ -349,10 +381,10 @@
by (simp add: inner_commute)
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_prod_def
- by (simp add: inner_left_distrib)
+ by (simp add: inner_add_left)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_prod_def
- by (simp add: inner_scaleR_left right_distrib)
+ by (simp add: right_distrib)
show "0 \<le> inner x x"
unfolding inner_prod_def
by (intro add_nonneg_nonneg inner_ge_zero)
--- a/src/HOL/Library/Topology_Euclidean_Space.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/Library/Topology_Euclidean_Space.thy Sat Jun 13 16:32:38 2009 +0200
@@ -6,7 +6,7 @@
header {* Elementary topology in Euclidean space. *}
theory Topology_Euclidean_Space
-imports SEQ Euclidean_Space
+imports SEQ Euclidean_Space Product_Vector
begin
declare fstcart_pastecart[simp] sndcart_pastecart[simp]
@@ -748,7 +748,7 @@
{ fix x
have "x \<in> interior S \<longleftrightarrow> x \<in> UNIV - (closure (UNIV - S))"
unfolding interior_def closure_def islimpt_def
- by blast
+ by blast (* FIXME: VERY slow! *)
}
thus ?thesis
by blast
@@ -1031,7 +1031,7 @@
unfolding trivial_limit_def Rep_net_at_infinity
apply (clarsimp simp add: expand_set_eq)
apply (drule_tac x="scaleR r (sgn 1)" in spec)
- apply (simp add: norm_scaleR norm_sgn)
+ apply (simp add: norm_sgn)
done
lemma trivial_limit_sequentially: "\<not> trivial_limit sequentially"
@@ -1245,17 +1245,16 @@
unfolding linear_conv_bounded_linear
by (rule bounded_linear.tendsto)
+lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
+ unfolding tendsto_def Limits.eventually_at_topological by fast
+
lemma Lim_const: "((\<lambda>x. a) ---> a) net"
by (rule tendsto_const)
lemma Lim_cmul:
fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
- shows "(f ---> l) net ==> ((\<lambda>x. c *s f x) ---> c *s l) net"
- apply (rule Lim_linear[where f = f])
- apply simp
- apply (rule linear_compose_cmul)
- apply (rule linear_id[unfolded id_def])
- done
+ shows "(f ---> l) net ==> ((\<lambda>x. c *\<^sub>R f x) ---> c *\<^sub>R l) net"
+ by (intro tendsto_intros)
lemma Lim_neg:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
@@ -1277,38 +1276,34 @@
lemma Lim_null_norm:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. vec1(norm(f x))) ---> 0) net"
- by (simp add: Lim dist_norm norm_vec1)
+ shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
+ by (simp add: Lim dist_norm)
lemma Lim_null_comparison:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "eventually (\<lambda>x. norm(f x) <= g x) net" "((\<lambda>x. vec1(g x)) ---> 0) net"
+ assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
shows "(f ---> 0) net"
proof(simp add: tendsto_iff, rule+)
fix e::real assume "0<e"
{ fix x
- assume "norm (f x) \<le> g x" "dist (vec1 (g x)) 0 < e"
- hence "dist (f x) 0 < e" unfolding vec_def using dist_vec1[of "g x" "0"]
- by (vector dist_norm norm_vec1 real_vector_norm_def dot_def vec1_def)
+ assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
+ hence "dist (f x) 0 < e" by (simp add: dist_norm)
}
thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
- using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (vec1 (g x)) 0 < e" net]
- using eventually_mono[of "(\<lambda>x. norm (f x) \<le> g x \<and> dist (vec1 (g x)) 0 < e)" "(\<lambda>x. dist (f x) 0 < e)" net]
+ using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
+ using eventually_mono[of "(\<lambda>x. norm (f x) \<le> g x \<and> dist (g x) 0 < e)" "(\<lambda>x. dist (f x) 0 < e)" net]
using assms `e>0` unfolding tendsto_iff by auto
qed
-lemma Lim_component: "(f ---> l) net
- ==> ((\<lambda>a. vec1((f a :: real ^'n::finite)$i)) ---> vec1(l$i)) net"
+lemma Lim_component:
+ fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n::finite"
+ shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $i) ---> l$i) net"
unfolding tendsto_iff
- apply (simp add: dist_norm vec1_sub[symmetric] norm_vec1 vector_minus_component[symmetric] del: vector_minus_component)
- apply (auto simp del: vector_minus_component)
- apply (erule_tac x=e in allE)
- apply clarify
- apply (erule eventually_rev_mono)
- apply (auto simp del: vector_minus_component)
- apply (rule order_le_less_trans)
- apply (rule component_le_norm)
- by auto
+ apply (clarify)
+ apply (drule spec, drule (1) mp)
+ apply (erule eventually_elim1)
+ apply (erule le_less_trans [OF dist_nth_le])
+ done
lemma Lim_transform_bound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
@@ -1504,12 +1499,6 @@
netlimit :: "'a::metric_space net \<Rightarrow> 'a" where
"netlimit net = (SOME a. \<forall>r>0. eventually (\<lambda>x. dist x a < r) net)"
-lemma dist_triangle3:
- fixes x y :: "'a::metric_space"
- shows "dist x y \<le> dist a x + dist a y"
-using dist_triangle2 [of x y a]
-by (simp add: dist_commute)
-
lemma netlimit_within:
assumes "\<not> trivial_limit (at a within S)"
shows "netlimit (at a within S) = a"
@@ -1694,14 +1683,14 @@
apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
by metis arith
-lemma seq_harmonic: "((\<lambda>n. vec1(inverse (real n))) ---> 0) sequentially"
+lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
proof-
{ fix e::real assume "e>0"
hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
- by (metis dlo_simps(4) le_imp_inverse_le linorder_not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
+ by (metis not_le le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
}
- thus ?thesis unfolding Lim_sequentially dist_norm apply simp unfolding norm_vec1 by auto
+ thus ?thesis unfolding Lim_sequentially dist_norm by simp
qed
text{* More properties of closed balls. *}
@@ -1768,7 +1757,7 @@
also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
unfolding scaleR_minus_left scaleR_one
- by (auto simp add: norm_minus_commute norm_scaleR)
+ by (auto simp add: norm_minus_commute)
also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
unfolding real_add_mult_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
@@ -1780,7 +1769,7 @@
have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
moreover
- have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel norm_scaleR
+ have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
unfolding dist_norm by auto
ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
@@ -1819,11 +1808,11 @@
unfolding z_def by (simp add: algebra_simps)
have "dist z y < r"
unfolding z_def k_def using `0 < r`
- by (simp add: dist_norm norm_scaleR min_def)
+ by (simp add: dist_norm min_def)
hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
have "dist x z < dist x y"
unfolding z_def2 dist_norm
- apply (simp add: norm_scaleR norm_minus_commute)
+ apply (simp add: norm_minus_commute)
apply (simp only: dist_norm [symmetric])
apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
apply (rule mult_strict_right_mono)
@@ -1856,9 +1845,10 @@
apply (simp add: zero_less_dist_iff)
done
+(* In a trivial vector space, this fails for e = 0. *)
lemma interior_cball:
- fixes x :: "real ^ _" (* FIXME: generalize *)
- shows "interior(cball x e) = ball x e"
+ fixes x :: "'a::{real_normed_vector, perfect_space}"
+ shows "interior (cball x e) = ball x e"
proof(cases "e\<ge>0")
case False note cs = this
from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
@@ -1873,9 +1863,9 @@
{ fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
- then obtain xa where xa:"dist y xa = d / 2" using vector_choose_dist[of "d/2" y] by auto
- hence xa_y:"xa \<noteq> y" using dist_nz[of y xa] using `d>0` by auto
- have "xa\<in>S" using d[THEN spec[where x=xa]] using xa apply(auto simp add: dist_commute) unfolding dist_nz[THEN sym] using xa_y by auto
+ then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
+ using perfect_choose_dist [of d] by auto
+ have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
hence xa_cball:"xa \<in> cball x e" using as(1) by auto
hence "y \<in> ball x e" proof(cases "x = y")
@@ -1884,18 +1874,19 @@
thus "y \<in> ball x e" using `x = y ` by simp
next
case False
- have "dist (y + (d / 2 / dist y x) *s (y - x)) y < d" unfolding dist_norm
+ have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
- hence *:"y + (d / 2 / dist y x) *s (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
+ hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
have "y - x \<noteq> 0" using `x \<noteq> y` by auto
hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
- have "dist (y + (d / 2 / dist y x) *s (y - x)) x = norm (y + (d / (2 * norm (y - x))) *s y - (d / (2 * norm (y - x))) *s x - x)"
- by (auto simp add: dist_norm vector_ssub_ldistrib add_diff_eq)
- also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *s (y - x))"
- by (auto simp add: vector_sadd_rdistrib vector_smult_lid ring_simps vector_sadd_rdistrib vector_ssub_ldistrib)
- also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)" using ** by auto
+ have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
+ by (auto simp add: dist_norm algebra_simps)
+ also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
+ by (auto simp add: algebra_simps)
+ also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
+ using ** by auto
also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
@@ -1905,14 +1896,14 @@
qed
lemma frontier_ball:
- fixes a :: "real ^ _" (* FIXME: generalize *)
+ fixes a :: "'a::real_normed_vector"
shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
apply (simp add: frontier_def closure_ball interior_open open_ball order_less_imp_le)
apply (simp add: expand_set_eq)
by arith
lemma frontier_cball:
- fixes a :: "real ^ _" (* FIXME: generalize *)
+ fixes a :: "'a::{real_normed_vector, perfect_space}"
shows "frontier(cball a e) = {x. dist a x = e}"
apply (simp add: frontier_def interior_cball closed_cball closure_closed order_less_imp_le)
apply (simp add: expand_set_eq)
@@ -1924,20 +1915,20 @@
lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
lemma cball_eq_sing:
- fixes x :: "real ^ _" (* FIXME: generalize *)
+ fixes x :: "'a::perfect_space"
shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
-proof-
- { assume as:"\<forall>xa. (dist x xa \<le> e) = (xa = x)"
- hence "e \<ge> 0" apply (erule_tac x=x in allE) by auto
- then obtain y where y:"dist x y = e" using vector_choose_dist[of e] by auto
- hence "e = 0" using as apply(erule_tac x=y in allE) by auto
- }
- thus ?thesis unfolding expand_set_eq mem_cball by (auto simp add: dist_nz)
-qed
+proof (rule linorder_cases)
+ assume e: "0 < e"
+ obtain a where "a \<noteq> x" "dist a x < e"
+ using perfect_choose_dist [OF e] by auto
+ hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
+ with e show ?thesis by (auto simp add: expand_set_eq)
+qed auto
lemma cball_sing:
- fixes x :: "real ^ _" (* FIXME: generalize *)
- shows "e = 0 ==> cball x e = {x}" by (simp add: cball_eq_sing)
+ fixes x :: "'a::metric_space"
+ shows "e = 0 ==> cball x e = {x}"
+ by (auto simp add: expand_set_eq)
text{* For points in the interior, localization of limits makes no difference. *}
@@ -2082,7 +2073,7 @@
fix b::real assume b: "b >0"
have b1: "b +1 \<ge> 0" using b by simp
with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
- by (simp add: norm_scaleR norm_sgn)
+ by (simp add: norm_sgn)
then show "\<exists>x::'a. b < norm x" ..
qed
@@ -2104,9 +2095,11 @@
lemma bounded_scaling:
fixes S :: "(real ^ 'n::finite) set"
- shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *s x) ` S)"
+ shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
apply (rule bounded_linear_image, assumption)
- by (rule linear_compose_cmul, rule linear_id[unfolded id_def])
+ apply (simp only: linear_conv_bounded_linear)
+ apply (rule scaleR.bounded_linear_right)
+ done
lemma bounded_translation:
fixes S :: "'a::real_normed_vector set"
@@ -2123,26 +2116,26 @@
text{* Some theorems on sups and infs using the notion "bounded". *}
-lemma bounded_vec1:
+lemma bounded_real:
fixes S :: "real set"
- shows "bounded(vec1 ` S) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
- by (simp add: bounded_iff forall_vec1 norm_vec1 vec1_in_image_vec1)
-
-lemma bounded_has_rsup: assumes "bounded(vec1 ` S)" "S \<noteq> {}"
+ shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
+ by (simp add: bounded_iff)
+
+lemma bounded_has_rsup: assumes "bounded S" "S \<noteq> {}"
shows "\<forall>x\<in>S. x <= rsup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> rsup S <= b"
proof
fix x assume "x\<in>S"
- from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_vec1 by auto
+ from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
hence *:"S *<= a" using setleI[of S a] by (metis abs_le_interval_iff mem_def)
- thus "x \<le> rsup S" using rsup[OF `S\<noteq>{}`] using assms(1)[unfolded bounded_vec1] using isLubD2[of UNIV S "rsup S" x] using `x\<in>S` by auto
+ thus "x \<le> rsup S" using rsup[OF `S\<noteq>{}`] using assms(1)[unfolded bounded_real] using isLubD2[of UNIV S "rsup S" x] using `x\<in>S` by auto
next
show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> rsup S \<le> b" using assms
using rsup[of S, unfolded isLub_def isUb_def leastP_def setle_def setge_def]
- apply (auto simp add: bounded_vec1)
+ apply (auto simp add: bounded_real)
by (auto simp add: isLub_def isUb_def leastP_def setle_def setge_def)
qed
-lemma rsup_insert: assumes "bounded (vec1 ` S)"
+lemma rsup_insert: assumes "bounded S"
shows "rsup(insert x S) = (if S = {} then x else max x (rsup S))"
proof(cases "S={}")
case True thus ?thesis using rsup_finite_in[of "{x}"] by auto
@@ -2168,17 +2161,17 @@
by simp
lemma bounded_has_rinf:
- assumes "bounded(vec1 ` S)" "S \<noteq> {}"
+ assumes "bounded S" "S \<noteq> {}"
shows "\<forall>x\<in>S. x >= rinf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> rinf S >= b"
proof
fix x assume "x\<in>S"
- from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_vec1 by auto
+ from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
hence *:"- a <=* S" using setgeI[of S "-a"] unfolding abs_le_interval_iff by auto
thus "x \<ge> rinf S" using rinf[OF `S\<noteq>{}`] using isGlbD2[of UNIV S "rinf S" x] using `x\<in>S` by auto
next
show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> rinf S \<ge> b" using assms
using rinf[of S, unfolded isGlb_def isLb_def greatestP_def setle_def setge_def]
- apply (auto simp add: bounded_vec1)
+ apply (auto simp add: bounded_real)
by (auto simp add: isGlb_def isLb_def greatestP_def setle_def setge_def)
qed
@@ -2189,7 +2182,7 @@
apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
done
-lemma rinf_insert: assumes "bounded (vec1 ` S)"
+lemma rinf_insert: assumes "bounded S"
shows "rinf(insert x S) = (if S = {} then x else min x (rinf S))" (is "?lhs = ?rhs")
proof(cases "S={}")
case True thus ?thesis using rinf_finite_in[of "{x}"] by auto
@@ -2217,8 +2210,8 @@
definition
compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
"compact S \<longleftrightarrow>
- (\<forall>f. (\<forall>n::nat. f n \<in> S) \<longrightarrow>
- (\<exists>l\<in>S. \<exists>r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((f o r) ---> l) sequentially))"
+ (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
+ (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
text {*
A metric space (or topological vector space) is said to have the
@@ -2227,44 +2220,43 @@
class heine_borel =
assumes bounded_imp_convergent_subsequence:
- "bounded s \<Longrightarrow> \<forall>n::nat. f n \<in> s
- \<Longrightarrow> \<exists>l r. (\<forall>m n. m < n --> r m < r n) \<and> ((f \<circ> r) ---> l) sequentially"
+ "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
+ \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
lemma bounded_closed_imp_compact:
fixes s::"'a::heine_borel set"
assumes "bounded s" and "closed s" shows "compact s"
proof (unfold compact_def, clarify)
fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
- obtain l r where r: "\<forall>m n. m < n \<longrightarrow> r m < r n" and l: "((f \<circ> r) ---> l) sequentially"
+ obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
have "l \<in> s" using `closed s` fr l
unfolding closed_sequential_limits by blast
- show "\<exists>l\<in>s. \<exists>r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((f \<circ> r) ---> l) sequentially"
+ show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
using `l \<in> s` r l by blast
qed
-lemma monotone_bigger: fixes r::"nat\<Rightarrow>nat"
- assumes "\<forall>m n::nat. m < n --> r m < r n"
- shows "n \<le> r n"
+lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
proof(induct n)
show "0 \<le> r 0" by auto
next
fix n assume "n \<le> r n"
- moreover have "r n < r (Suc n)" using assms by auto
+ moreover have "r n < r (Suc n)"
+ using assms [unfolded subseq_def] by auto
ultimately show "Suc n \<le> r (Suc n)" by auto
qed
-lemma eventually_subsequence:
- assumes r: "\<forall>m n. m < n \<longrightarrow> r m < r n"
+lemma eventually_subseq:
+ assumes r: "subseq r"
shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
unfolding eventually_sequentially
-by (metis monotone_bigger [OF r] le_trans)
-
-lemma lim_subsequence:
- fixes l :: "'a::metric_space" (* TODO: generalize *)
- shows "\<forall>m n. m < n \<longrightarrow> r m < r n \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
-unfolding Lim_sequentially by (simp, metis monotone_bigger le_trans)
+by (metis subseq_bigger [OF r] le_trans)
+
+lemma lim_subseq:
+ "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
+unfolding tendsto_def eventually_sequentially o_def
+by (metis subseq_bigger le_trans)
lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
unfolding Ex1_def
@@ -2280,9 +2272,8 @@
apply (simp)
done
-
lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
- assumes "\<forall>m n. m \<le> n --> s m \<le> s n" and "\<forall>n. abs(s n) \<le> b"
+ assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
proof-
have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
@@ -2292,27 +2283,27 @@
obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
with n have "s N \<le> t - e" using `e>0` by auto
- hence "s n \<le> t - e" using assms(1)[THEN spec[where x=n], THEN spec[where x=N]] using `n\<le>N` by auto }
+ hence "s n \<le> t - e" using assms(1)[unfolded incseq_def, THEN spec[where x=n], THEN spec[where x=N]] using `n\<le>N` by auto }
hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
thus ?thesis by blast
qed
lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
- assumes "\<forall>n. abs(s n) \<le> b" and "(\<forall>m n. m \<le> n --> s m \<le> s n) \<or> (\<forall>m n. m \<le> n --> s n \<le> s m)"
+ assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
+ unfolding monoseq_def incseq_def
apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
lemma compact_real_lemma:
assumes "\<forall>n::nat. abs(s n) \<le> b"
- shows "\<exists>(l::real) r. (\<forall>m n::nat. m < n --> r m < r n) \<and> ((s \<circ> r) ---> l) sequentially"
+ shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
proof-
- obtain r where r:"\<forall>m n::nat. m < n \<longrightarrow> r m < r n"
- "(\<forall>m n. m \<le> n \<longrightarrow> s (r m) \<le> s (r n)) \<or> (\<forall>m n. m \<le> n \<longrightarrow> s (r n) \<le> s (r m))"
- using seq_monosub[of s] by (auto simp add: subseq_def monoseq_def)
- thus ?thesis using convergent_bounded_monotone[of "s o r" b] and assms
+ obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
+ using seq_monosub[of s] by auto
+ thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
unfolding tendsto_iff dist_norm eventually_sequentially by auto
qed
@@ -2323,9 +2314,9 @@
then obtain b where b: "\<forall>n. abs (f n) \<le> b"
unfolding bounded_iff by auto
obtain l :: real and r :: "nat \<Rightarrow> nat" where
- r: "\<forall>m n. m < n \<longrightarrow> r m < r n" and l: "((f \<circ> r) ---> l) sequentially"
+ r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
using compact_real_lemma [OF b] by auto
- thus "\<exists>l r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((f \<circ> r) ---> l) sequentially"
+ thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
by auto
qed
@@ -2342,28 +2333,29 @@
fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n::finite"
assumes "bounded s" and "\<forall>n. f n \<in> s"
shows "\<forall>d.
- \<exists>l r. (\<forall>n m::nat. m < n --> r m < r n) \<and>
+ \<exists>l r. subseq r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
proof
fix d::"'n set" have "finite d" by simp
- thus "\<exists>l::'a ^ 'n. \<exists>r. (\<forall>n m::nat. m < n --> r m < r n) \<and>
+ thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
- proof(induct d) case empty thus ?case by auto
+ proof(induct d) case empty thus ?case unfolding subseq_def by auto
next case (insert k d)
have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component)
- obtain l1::"'a^'n" and r1 where r1:"\<forall>n m::nat. m < n \<longrightarrow> r1 m < r1 n" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
+ obtain l1::"'a^'n" and r1 where r1:"subseq r1" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
using insert(3) by auto
have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp
- obtain l2 r2 where r2:"\<forall>m n::nat. m < n \<longrightarrow> r2 m < r2 n" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
+ obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
- def r \<equiv> "r1 \<circ> r2" have r:"\<forall>m n. m < n \<longrightarrow> r m < r n" unfolding r_def o_def using r1 and r2 by auto
+ def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
+ using r1 and r2 unfolding r_def o_def subseq_def by auto
moreover
def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
{ fix e::real assume "e>0"
from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast
from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD)
from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
- by (rule eventually_subsequence)
+ by (rule eventually_subseq)
have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
}
@@ -2375,7 +2367,7 @@
proof
fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
- then obtain l r where r: "\<forall>n m::nat. m < n --> r m < r n"
+ then obtain l r where r: "subseq r"
and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
using compact_lemma [OF s f] by blast
let ?d = "UNIV::'b set"
@@ -2396,7 +2388,55 @@
by (rule eventually_elim1)
}
hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
- with r show "\<exists>l r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((f \<circ> r) ---> l) sequentially" by auto
+ with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
+qed
+
+lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
+unfolding bounded_def
+apply clarify
+apply (rule_tac x="a" in exI)
+apply (rule_tac x="e" in exI)
+apply clarsimp
+apply (drule (1) bspec)
+apply (simp add: dist_Pair_Pair)
+apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
+done
+
+lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
+unfolding bounded_def
+apply clarify
+apply (rule_tac x="b" in exI)
+apply (rule_tac x="e" in exI)
+apply clarsimp
+apply (drule (1) bspec)
+apply (simp add: dist_Pair_Pair)
+apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
+done
+
+instance "*" :: (heine_borel, heine_borel) heine_borel
+proof
+ fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
+ assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
+ from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
+ from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
+ obtain l1 r1 where r1: "subseq r1"
+ and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
+ using bounded_imp_convergent_subsequence [OF s1 f1]
+ unfolding o_def by fast
+ from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
+ from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
+ obtain l2 r2 where r2: "subseq r2"
+ and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
+ using bounded_imp_convergent_subsequence [OF s2 f2]
+ unfolding o_def by fast
+ have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
+ using lim_subseq [OF r2 l1] unfolding o_def .
+ have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
+ using tendsto_Pair [OF l1' l2] unfolding o_def by simp
+ have r: "subseq (r1 \<circ> r2)"
+ using r1 r2 unfolding subseq_def by simp
+ show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
+ using l r by fast
qed
subsection{* Completeness. *}
@@ -2461,14 +2501,9 @@
lemma compact_imp_complete: assumes "compact s" shows "complete s"
proof-
{ fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
- from as(1) obtain l r where lr: "l\<in>s" "(\<forall>m n. m < n \<longrightarrow> r m < r n)" "((f \<circ> r) ---> l) sequentially" using assms unfolding compact_def by blast
-
- { fix n :: nat have lr':"n \<le> r n"
- proof (induct n)
- show "0 \<le> r 0" using lr(2) by blast
- next fix na assume "na \<le> r na" moreover have "na < Suc na \<longrightarrow> r na < r (Suc na)" using lr(2) by blast
- ultimately show "Suc na \<le> r (Suc na)" by auto
- qed } note lr' = this
+ from as(1) obtain l r where lr: "l\<in>s" "subseq r" "((f \<circ> r) ---> l) sequentially" using assms unfolding compact_def by blast
+
+ note lr' = subseq_bigger [OF lr(2)]
{ fix e::real assume "e>0"
from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using `e>0` apply (erule_tac x="e/2" in allE) by auto
@@ -2575,12 +2610,12 @@
thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
qed }
hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
- then obtain l r where "l\<in>s" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded compact_def, THEN spec[where x=x]] by auto
+ then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded compact_def, THEN spec[where x=x]] by auto
from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using `e>0` by auto
show False
using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
- using r[THEN spec[where x=N], THEN spec[where x="N+1"]]
+ using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
qed
@@ -2599,7 +2634,7 @@
then obtain f where f:"\<forall>n::nat. f n \<in> s \<and> (\<forall>xa\<in>t. \<not> ball (f n) (inverse (real (n + 1))) \<subseteq> xa)"
using choice[of "\<lambda>n::nat. \<lambda>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)"] by auto
- then obtain l r where l:"l\<in>s" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and lr:"((f \<circ> r) ---> l) sequentially"
+ then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
@@ -2612,7 +2647,7 @@
obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
- using monotone_bigger[OF r, of "N1 + N2"] by auto
+ using subseq_bigger[OF r, of "N1 + N2"] by auto
def x \<equiv> "(f (r (N1 + N2)))"
have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
@@ -2926,7 +2961,7 @@
by blast
lemma compact_sing [simp]: "compact {a}"
- unfolding compact_def o_def
+ unfolding compact_def o_def subseq_def
by (auto simp add: tendsto_const)
lemma compact_cball[simp]:
@@ -2987,7 +3022,7 @@
from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto
from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
- then obtain l r where lr:"l\<in>s 0" "\<forall>m n. m < n \<longrightarrow> r m < r n" "((x \<circ> r) ---> l) sequentially"
+ then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
{ fix n::nat
@@ -2995,7 +3030,7 @@
with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
hence "dist ((x \<circ> r) (max N n)) l < e" by auto
moreover
- have "r (max N n) \<ge> n" using lr(2) using monotone_bigger[of r "max N n"] by auto
+ have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
hence "(x \<circ> r) (max N n) \<in> s n"
using x apply(erule_tac x=n in allE)
using x apply(erule_tac x="r (max N n)" in allE)
@@ -3415,7 +3450,7 @@
lemma continuous_cmul:
fixes f :: "'a::metric_space \<Rightarrow> real ^ 'n::finite"
- shows "continuous net f ==> continuous net (\<lambda>x. c *s f x)"
+ shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
by (auto simp add: continuous_def Lim_cmul)
lemma continuous_neg:
@@ -3441,7 +3476,7 @@
lemma continuous_on_cmul:
fixes f :: "'a::metric_space \<Rightarrow> real ^ _"
- shows "continuous_on s f ==> continuous_on s (\<lambda>x. c *s (f x))"
+ shows "continuous_on s f ==> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
unfolding continuous_on_eq_continuous_within using continuous_cmul by blast
lemma continuous_on_neg:
@@ -3470,12 +3505,12 @@
lemma uniformly_continuous_on_cmul:
fixes f :: "'a::real_normed_vector \<Rightarrow> real ^ _"
assumes "uniformly_continuous_on s f"
- shows "uniformly_continuous_on s (\<lambda>x. c *s f(x))"
+ shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
proof-
{ fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
- hence "((\<lambda>n. c *s f (x n) - c *s f (y n)) ---> 0) sequentially"
+ hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
using Lim_cmul[of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
- unfolding vector_smult_rzero vector_ssub_ldistrib[of c] by auto
+ unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
}
thus ?thesis using assms unfolding uniformly_continuous_on_sequentially by auto
qed
@@ -3701,13 +3736,11 @@
qed
lemma continuous_open_preimage_univ:
- fixes f :: "real ^ _ \<Rightarrow> real ^ _" (* FIXME: generalize *)
- shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
+ "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
lemma continuous_closed_preimage_univ:
- fixes f :: "real ^ _ \<Rightarrow> real ^ _" (* FIXME: generalize *)
- shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
+ "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
text{* Equality of continuous functions on closure and related results. *}
@@ -3802,20 +3835,20 @@
text{* Some arithmetical combinations (more to prove). *}
lemma open_scaling[intro]:
- fixes s :: "(real ^ _) set"
+ fixes s :: "'a::real_normed_vector set"
assumes "c \<noteq> 0" "open s"
- shows "open((\<lambda>x. c *s x) ` s)"
+ shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
proof-
{ fix x assume "x \<in> s"
then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]] by auto
have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using real_mult_order[OF `e>0`] by auto
moreover
- { fix y assume "dist y (c *s x) < e * \<bar>c\<bar>"
- hence "norm ((1 / c) *s y - x) < e" unfolding dist_norm
- using norm_mul[of c "(1 / c) *s y - x", unfolded vector_ssub_ldistrib, unfolded vector_smult_assoc] assms(1)
+ { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
+ hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
+ using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
- hence "y \<in> op *s c ` s" using rev_image_eqI[of "(1 / c) *s y" s y "op *s c"] e[THEN spec[where x="(1 / c) *s y"]] assms(1) unfolding dist_norm vector_smult_assoc by auto }
- ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *s x) < e \<longrightarrow> x' \<in> op *s c ` s" apply(rule_tac x="e * abs c" in exI) by auto }
+ hence "y \<in> op *\<^sub>R c ` s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"] e[THEN spec[where x="(1 / c) *\<^sub>R y"]] assms(1) unfolding dist_norm scaleR_scaleR by auto }
+ ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s" apply(rule_tac x="e * abs c" in exI) by auto }
thus ?thesis unfolding open_dist by auto
qed
@@ -3825,12 +3858,13 @@
by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
lemma open_negations:
- fixes s :: "(real ^ _) set" (* FIXME: generalize *)
+ fixes s :: "'a::real_normed_vector set"
shows "open s ==> open ((\<lambda> x. -x) ` s)"
- unfolding vector_sneg_minus1 by auto
+ unfolding scaleR_minus1_left [symmetric]
+ by (rule open_scaling, auto)
lemma open_translation:
- fixes s :: "(real ^ _) set" (* FIXME: generalize *)
+ fixes s :: "'a::real_normed_vector set"
assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
proof-
{ fix x have "continuous (at x) (\<lambda>x. x - a)" using continuous_sub[of "at x" "\<lambda>x. x" "\<lambda>x. a"] continuous_at_id[of x] continuous_const[of "at x" a] by auto }
@@ -3841,11 +3875,11 @@
lemma open_affinity:
fixes s :: "(real ^ _) set"
assumes "open s" "c \<noteq> 0"
- shows "open ((\<lambda>x. a + c *s x) ` s)"
+ shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
proof-
- have *:"(\<lambda>x. a + c *s x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *s x)" unfolding o_def ..
- have "op + a ` op *s c ` s = (op + a \<circ> op *s c) ` s" by auto
- thus ?thesis using assms open_translation[of "op *s c ` s" a] unfolding * by auto
+ have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
+ have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
+ thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
qed
lemma interior_translation:
@@ -3875,13 +3909,13 @@
proof-
{ fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
then obtain y where y:"\<forall>n. y n \<in> s \<and> x n = f (y n)" unfolding image_iff Bex_def using choice[of "\<lambda>n xa. xa \<in> s \<and> x n = f xa"] by auto
- then obtain l r where "l\<in>s" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and lr:"((y \<circ> r) ---> l) sequentially" using assms(2)[unfolded compact_def, THEN spec[where x=y]] by auto
+ then obtain l r where "l\<in>s" and r:"subseq r" and lr:"((y \<circ> r) ---> l) sequentially" using assms(2)[unfolded compact_def, THEN spec[where x=y]] by auto
{ fix e::real assume "e>0"
then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' l < d \<longrightarrow> dist (f x') (f l) < e" using assms(1)[unfolded continuous_on_def, THEN bspec[where x=l], OF `l\<in>s`] by auto
then obtain N::nat where N:"\<forall>n\<ge>N. dist ((y \<circ> r) n) l < d" using lr[unfolded Lim_sequentially, THEN spec[where x=d]] by auto
{ fix n::nat assume "n\<ge>N" hence "dist ((x \<circ> r) n) (f l) < e" using N[THEN spec[where x=n]] d[THEN bspec[where x="y (r n)"]] y[THEN spec[where x="r n"]] by auto }
hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
- hence "\<exists>l\<in>f ` s. \<exists>r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((x \<circ> r) ---> l) sequentially" unfolding Lim_sequentially using r lr `l\<in>s` by auto }
+ hence "\<exists>l\<in>f ` s. \<exists>r. subseq r \<and> ((x \<circ> r) ---> l) sequentially" unfolding Lim_sequentially using r lr `l\<in>s` by auto }
thus ?thesis unfolding compact_def by auto
qed
@@ -3938,7 +3972,8 @@
text{* Continuity of inverse function on compact domain. *}
lemma continuous_on_inverse:
- fixes f :: "real ^ _ \<Rightarrow> real ^ _"
+ fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
+ (* TODO: can this be generalized more? *)
assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
shows "continuous_on (f ` s) g"
proof-
@@ -4050,68 +4085,52 @@
subsection{* Topological stuff lifted from and dropped to R *}
-lemma open_vec1:
- fixes s :: "real set" shows
- "open(vec1 ` s) \<longleftrightarrow>
- (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
- unfolding open_dist apply simp unfolding forall_vec1 dist_vec1 vec1_in_image_vec1 by simp
-
-lemma islimpt_approachable_vec1:
+lemma open_real:
fixes s :: "real set" shows
- "(vec1 x) islimpt (vec1 ` s) \<longleftrightarrow>
- (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
- by (auto simp add: islimpt_approachable dist_vec1 vec1_eq)
-
-lemma closed_vec1:
- fixes s :: "real set" shows
- "closed (vec1 ` s) \<longleftrightarrow>
+ "open s \<longleftrightarrow>
+ (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
+ unfolding open_dist dist_norm by simp
+
+lemma islimpt_approachable_real:
+ fixes s :: "real set"
+ shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
+ unfolding islimpt_approachable dist_norm by simp
+
+lemma closed_real:
+ fixes s :: "real set"
+ shows "closed s \<longleftrightarrow>
(\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
--> x \<in> s)"
- unfolding closed_limpt islimpt_approachable forall_vec1 apply simp
- unfolding dist_vec1 vec1_in_image_vec1 abs_minus_commute by auto
-
-lemma continuous_at_vec1_range:
- fixes f :: "real ^ _ \<Rightarrow> real"
- shows "continuous (at x) (vec1 o f) \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
+ unfolding closed_limpt islimpt_approachable dist_norm by simp
+
+lemma continuous_at_real_range:
+ fixes f :: "'a::real_normed_vector \<Rightarrow> real"
+ shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
\<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
- unfolding continuous_at unfolding Lim_at apply simp unfolding dist_vec1 unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
+ unfolding continuous_at unfolding Lim_at
+ unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto
apply(erule_tac x=e in allE) by auto
-lemma continuous_on_vec1_range:
+lemma continuous_on_real_range:
fixes f :: "'a::real_normed_vector \<Rightarrow> real"
- shows "continuous_on s (vec1 o f) \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
- unfolding continuous_on_def apply (simp del: dist_commute) unfolding dist_vec1 unfolding dist_norm ..
-
-lemma continuous_at_vec1_norm:
- fixes x :: "real ^ _"
- shows "continuous (at x) (vec1 o norm)"
- unfolding continuous_at_vec1_range using real_abs_sub_norm order_le_less_trans by blast
-
-lemma continuous_on_vec1_norm:
- fixes s :: "(real ^ _) set"
- shows "continuous_on s (vec1 o norm)"
-unfolding continuous_on_vec1_range norm_vec1[THEN sym] by (metis norm_vec1 order_le_less_trans real_abs_sub_norm)
-
-lemma continuous_at_vec1_component:
- shows "continuous (at (a::real^'a::finite)) (\<lambda> x. vec1(x$i))"
-proof-
- { fix e::real and x assume "0 < dist x a" "dist x a < e" "e>0"
- hence "\<bar>x $ i - a $ i\<bar> < e" using component_le_norm[of "x - a" i] unfolding dist_norm by auto }
- thus ?thesis unfolding continuous_at tendsto_iff eventually_at dist_vec1 by auto
-qed
-
-lemma continuous_on_vec1_component:
- shows "continuous_on s (\<lambda> x::real^'a::finite. vec1(x$i))"
-proof-
- { fix e::real and x xa assume "x\<in>s" "e>0" "xa\<in>s" "0 < norm (xa - x) \<and> norm (xa - x) < e"
- hence "\<bar>xa $ i - x $ i\<bar> < e" using component_le_norm[of "xa - x" i] by auto }
- thus ?thesis unfolding continuous_on Lim_within dist_vec1 unfolding dist_norm by auto
-qed
-
-lemma continuous_at_vec1_infnorm:
- "continuous (at x) (vec1 o infnorm)"
- unfolding continuous_at Lim_at o_def unfolding dist_vec1 unfolding dist_norm
+ shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
+ unfolding continuous_on_def dist_norm by simp
+
+lemma continuous_at_norm: "continuous (at x) norm"
+ unfolding continuous_at by (intro tendsto_intros)
+
+lemma continuous_on_norm: "continuous_on s norm"
+unfolding continuous_on by (intro ballI tendsto_intros)
+
+lemma continuous_at_component: "continuous (at a) (\<lambda>x. x $ i)"
+unfolding continuous_at by (intro tendsto_intros)
+
+lemma continuous_on_component: "continuous_on s (\<lambda>x. x $ i)"
+unfolding continuous_on by (intro ballI tendsto_intros)
+
+lemma continuous_at_infnorm: "continuous (at x) infnorm"
+ unfolding continuous_at Lim_at o_def unfolding dist_norm
apply auto apply (rule_tac x=e in exI) apply auto
using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
@@ -4119,23 +4138,23 @@
lemma compact_attains_sup:
fixes s :: "real set"
- assumes "compact (vec1 ` s)" "s \<noteq> {}"
+ assumes "compact s" "s \<noteq> {}"
shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
proof-
- from assms(1) have a:"bounded (vec1 ` s)" "closed (vec1 ` s)" unfolding compact_eq_bounded_closed by auto
+ from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
{ fix e::real assume as: "\<forall>x\<in>s. x \<le> rsup s" "rsup s \<notin> s" "0 < e" "\<forall>x'\<in>s. x' = rsup s \<or> \<not> rsup s - x' < e"
have "isLub UNIV s (rsup s)" using rsup[OF assms(2)] unfolding setle_def using as(1) by auto
moreover have "isUb UNIV s (rsup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
ultimately have False using isLub_le_isUb[of UNIV s "rsup s" "rsup s - e"] using `e>0` by auto }
- thus ?thesis using bounded_has_rsup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_vec1, THEN spec[where x="rsup s"]]
+ thus ?thesis using bounded_has_rsup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="rsup s"]]
apply(rule_tac x="rsup s" in bexI) by auto
qed
lemma compact_attains_inf:
fixes s :: "real set"
- assumes "compact (vec1 ` s)" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
+ assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
proof-
- from assms(1) have a:"bounded (vec1 ` s)" "closed (vec1 ` s)" unfolding compact_eq_bounded_closed by auto
+ from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
{ fix e::real assume as: "\<forall>x\<in>s. x \<ge> rinf s" "rinf s \<notin> s" "0 < e"
"\<forall>x'\<in>s. x' = rinf s \<or> \<not> abs (x' - rinf s) < e"
have "isGlb UNIV s (rinf s)" using rinf[OF assms(2)] unfolding setge_def using as(1) by auto
@@ -4145,43 +4164,40 @@
have "rinf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
hence "isLb UNIV s (rinf s + e)" unfolding isLb_def and setge_def by auto
ultimately have False using isGlb_le_isLb[of UNIV s "rinf s" "rinf s + e"] using `e>0` by auto }
- thus ?thesis using bounded_has_rinf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_vec1, THEN spec[where x="rinf s"]]
+ thus ?thesis using bounded_has_rinf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="rinf s"]]
apply(rule_tac x="rinf s" in bexI) by auto
qed
lemma continuous_attains_sup:
fixes f :: "'a::metric_space \<Rightarrow> real"
- shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s (vec1 o f)
+ shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
using compact_attains_sup[of "f ` s"]
- using compact_continuous_image[of s "vec1 \<circ> f"] unfolding image_compose by auto
+ using compact_continuous_image[of s f] by auto
lemma continuous_attains_inf:
fixes f :: "'a::metric_space \<Rightarrow> real"
- shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s (vec1 o f)
+ shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
\<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
using compact_attains_inf[of "f ` s"]
- using compact_continuous_image[of s "vec1 \<circ> f"] unfolding image_compose by auto
+ using compact_continuous_image[of s f] by auto
lemma distance_attains_sup:
- fixes s :: "(real ^ _) set"
assumes "compact s" "s \<noteq> {}"
shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
-proof-
- { fix x assume "x\<in>s" fix e::real assume "e>0"
- { fix x' assume "x'\<in>s" and as:"norm (x' - x) < e"
- hence "\<bar>norm (x' - a) - norm (x - a)\<bar> < e"
- using real_abs_sub_norm[of "x' - a" "x - a"] by auto }
- hence "\<exists>d>0. \<forall>x'\<in>s. norm (x' - x) < d \<longrightarrow> \<bar>dist x' a - dist x a\<bar> < e" using `e>0` unfolding dist_norm by auto }
- thus ?thesis using assms
- using continuous_attains_sup[of s "\<lambda>x. dist a x"]
- unfolding continuous_on_vec1_range by (auto simp add: dist_commute)
+proof (rule continuous_attains_sup [OF assms])
+ { fix x assume "x\<in>s"
+ have "(dist a ---> dist a x) (at x within s)"
+ by (intro tendsto_dist tendsto_const Lim_at_within Lim_ident_at)
+ }
+ thus "continuous_on s (dist a)"
+ unfolding continuous_on ..
qed
text{* For *minimal* distance, we only need closure, not compactness. *}
lemma distance_attains_inf:
- fixes a :: "real ^ _" (* FIXME: generalize *)
+ fixes a :: "'a::heine_borel"
assumes "closed s" "s \<noteq> {}"
shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
proof-
@@ -4192,14 +4208,25 @@
moreover
{ fix x assume "x\<in>?B"
fix e::real assume "e>0"
- { fix x' assume "x'\<in>?B" and as:"norm (x' - x) < e"
- hence "\<bar>norm (x' - a) - norm (x - a)\<bar> < e"
- using real_abs_sub_norm[of "x' - a" "x - a"] by auto }
- hence "\<exists>d>0. \<forall>x'\<in>?B. norm (x' - x) < d \<longrightarrow> \<bar>dist x' a - dist x a\<bar> < e" using `e>0` unfolding dist_norm by auto }
- hence "continuous_on (cball a (dist b a) \<inter> s) (vec1 \<circ> dist a)" unfolding continuous_on_vec1_range
- by (auto simp add: dist_commute)
- moreover have "compact ?B" using compact_cball[of a "dist b a"] unfolding compact_eq_bounded_closed using bounded_Int and closed_Int and assms(1) by auto
- ultimately obtain x where "x\<in>cball a (dist b a) \<inter> s" "\<forall>y\<in>cball a (dist b a) \<inter> s. dist a x \<le> dist a y" using continuous_attains_inf[of ?B "dist a"] by fastsimp
+ { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
+ from as have "\<bar>dist a x' - dist a x\<bar> < e"
+ unfolding abs_less_iff minus_diff_eq
+ using dist_triangle2 [of a x' x]
+ using dist_triangle [of a x x']
+ by arith
+ }
+ hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
+ using `e>0` by auto
+ }
+ hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
+ unfolding continuous_on Lim_within dist_norm real_norm_def
+ by fast
+ moreover have "compact ?B"
+ using compact_cball[of a "dist b a"]
+ unfolding compact_eq_bounded_closed
+ using bounded_Int and closed_Int and assms(1) by auto
+ ultimately obtain x where "x\<in>cball a (dist b a) \<inter> s" "\<forall>y\<in>cball a (dist b a) \<inter> s. dist a x \<le> dist a y"
+ using continuous_attains_inf[of ?B "dist a"] by fastsimp
thus ?thesis by fastsimp
qed
@@ -4207,99 +4234,64 @@
lemma Lim_mul:
fixes f :: "'a \<Rightarrow> real ^ _"
- assumes "((vec1 o c) ---> vec1 d) net" "(f ---> l) net"
- shows "((\<lambda>x. c(x) *s f x) ---> (d *s l)) net"
-proof-
- have "bilinear (\<lambda>x. op *s (dest_vec1 (x::real^1)))" unfolding bilinear_def linear_def
- unfolding dest_vec1_add dest_vec1_cmul
- apply vector apply auto unfolding semiring_class.right_distrib semiring_class.left_distrib by auto
- thus ?thesis using Lim_bilinear[OF assms, of "\<lambda>x y. (dest_vec1 x) *s y"] by auto
-qed
+ assumes "(c ---> d) net" "(f ---> l) net"
+ shows "((\<lambda>x. c(x) *\<^sub>R f x) ---> (d *\<^sub>R l)) net"
+ unfolding smult_conv_scaleR using assms by (rule scaleR.tendsto)
lemma Lim_vmul:
- fixes c :: "'a \<Rightarrow> real"
- shows "((vec1 o c) ---> vec1 d) net ==> ((\<lambda>x. c(x) *s v) ---> d *s v) net"
- using Lim_mul[of c d net "\<lambda>x. v" v] using Lim_const[of v] by auto
+ fixes c :: "'a \<Rightarrow> real" and v :: "'b::real_normed_vector"
+ shows "(c ---> d) net ==> ((\<lambda>x. c(x) *\<^sub>R v) ---> d *\<^sub>R v) net"
+ by (intro tendsto_intros)
lemma continuous_vmul:
- fixes c :: "'a::metric_space \<Rightarrow> real"
- shows "continuous net (vec1 o c) ==> continuous net (\<lambda>x. c(x) *s v)"
+ fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
+ shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
unfolding continuous_def using Lim_vmul[of c] by auto
lemma continuous_mul:
fixes c :: "'a::metric_space \<Rightarrow> real"
- shows "continuous net (vec1 o c) \<Longrightarrow> continuous net f
- ==> continuous net (\<lambda>x. c(x) *s f x) "
- unfolding continuous_def using Lim_mul[of c] by auto
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ shows "continuous net c \<Longrightarrow> continuous net f
+ ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
+ unfolding continuous_def by (intro tendsto_intros)
lemma continuous_on_vmul:
- fixes c :: "'a::metric_space \<Rightarrow> real"
- shows "continuous_on s (vec1 o c) ==> continuous_on s (\<lambda>x. c(x) *s v)"
+ fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
+ shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
lemma continuous_on_mul:
fixes c :: "'a::metric_space \<Rightarrow> real"
- shows "continuous_on s (vec1 o c) \<Longrightarrow> continuous_on s f
- ==> continuous_on s (\<lambda>x. c(x) *s f x)"
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ shows "continuous_on s c \<Longrightarrow> continuous_on s f
+ ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
text{* And so we have continuity of inverse. *}
lemma Lim_inv:
fixes f :: "'a \<Rightarrow> real"
- assumes "((vec1 o f) ---> vec1 l) (net::'a net)" "l \<noteq> 0"
- shows "((vec1 o inverse o f) ---> vec1(inverse l)) net"
-proof -
- { fix e::real assume "e>0"
- let ?d = "min (\<bar>l\<bar> / 2) (l\<twosuperior> * e / 2)"
- have "0 < ?d" using `l\<noteq>0` `e>0` mult_pos_pos[of "l^2" "e/2"] by auto
- with assms(1) have "eventually (\<lambda>x. dist ((vec1 o f) x) (vec1 l) < ?d) net"
- by (rule tendstoD)
- moreover
- { fix x assume "dist ((vec1 o f) x) (vec1 l) < ?d"
- hence *:"\<bar>f x - l\<bar> < min (\<bar>l\<bar> / 2) (l\<twosuperior> * e / 2)" unfolding o_def dist_vec1 by auto
- hence fx0:"f x \<noteq> 0" using `l \<noteq> 0` by auto
- hence fxl0: "(f x) * l \<noteq> 0" using `l \<noteq> 0` by auto
- from * have **:"\<bar>f x - l\<bar> < l\<twosuperior> * e / 2" by auto
- have "\<bar>f x\<bar> * 2 \<ge> \<bar>l\<bar>" using * by (auto simp del: less_divide_eq_number_of1)
- hence "\<bar>f x\<bar> * 2 * \<bar>l\<bar> \<ge> \<bar>l\<bar> * \<bar>l\<bar>" unfolding mult_le_cancel_right by auto
- hence "\<bar>f x * l\<bar> * 2 \<ge> \<bar>l\<bar>^2" unfolding real_mult_commute and power2_eq_square by auto
- hence ***:"inverse \<bar>f x * l\<bar> \<le> inverse (l\<twosuperior> / 2)" using fxl0
- using le_imp_inverse_le[of "l^2 / 2" "\<bar>f x * l\<bar>"] by auto
-
- have "dist ((vec1 \<circ> inverse \<circ> f) x) (vec1 (inverse l)) < e" unfolding o_def unfolding dist_vec1
- unfolding inverse_diff_inverse[OF fx0 `l\<noteq>0`] apply simp
- unfolding mult_commute[of "inverse (f x)"]
- unfolding real_divide_def[THEN sym]
- unfolding divide_divide_eq_left
- unfolding nonzero_abs_divide[OF fxl0]
- using mult_less_le_imp_less[OF **, of "inverse \<bar>f x * l\<bar>", of "inverse (l^2 / 2)"] using *** using fx0 `l\<noteq>0`
- unfolding inverse_eq_divide using `e>0` by auto
- }
- ultimately
- have "eventually (\<lambda>x. dist ((vec1 o inverse o f) x) (vec1(inverse l)) < e) net"
- by (auto elim: eventually_rev_mono)
- }
- thus ?thesis unfolding tendsto_iff by auto
-qed
+ assumes "(f ---> l) (net::'a net)" "l \<noteq> 0"
+ shows "((inverse o f) ---> inverse l) net"
+ unfolding o_def using assms by (rule tendsto_inverse)
lemma continuous_inv:
fixes f :: "'a::metric_space \<Rightarrow> real"
- shows "continuous net (vec1 o f) \<Longrightarrow> f(netlimit net) \<noteq> 0
- ==> continuous net (vec1 o inverse o f)"
+ shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
+ ==> continuous net (inverse o f)"
unfolding continuous_def using Lim_inv by auto
lemma continuous_at_within_inv:
- fixes f :: "real ^ _ \<Rightarrow> real"
- assumes "continuous (at a within s) (vec1 o f)" "f a \<noteq> 0"
- shows "continuous (at a within s) (vec1 o inverse o f)"
- using assms unfolding continuous_within o_apply
- by (rule Lim_inv)
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
+ assumes "continuous (at a within s) f" "f a \<noteq> 0"
+ shows "continuous (at a within s) (inverse o f)"
+ using assms unfolding continuous_within o_def
+ by (intro tendsto_intros)
lemma continuous_at_inv:
- fixes f :: "real ^ _ \<Rightarrow> real"
- shows "continuous (at a) (vec1 o f) \<Longrightarrow> f a \<noteq> 0
- ==> continuous (at a) (vec1 o inverse o f) "
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
+ shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
+ ==> continuous (at a) (inverse o f) "
using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
subsection{* Preservation properties for pasted sets. *}
@@ -4316,6 +4308,16 @@
thus ?thesis unfolding bounded_iff by auto
qed
+lemma bounded_Times:
+ assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
+proof-
+ obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
+ using assms [unfolded bounded_def] by auto
+ then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
+ by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
+ thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
+qed
+
lemma closed_pastecart:
fixes s :: "(real ^ 'a::finite) set" (* FIXME: generalize *)
assumes "closed s" "closed t"
@@ -4343,6 +4345,26 @@
shows "compact s \<Longrightarrow> compact t ==> compact {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
unfolding compact_eq_bounded_closed using bounded_pastecart[of s t] closed_pastecart[of s t] by auto
+lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
+by (induct x) simp
+
+lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
+unfolding compact_def
+apply clarify
+apply (drule_tac x="fst \<circ> f" in spec)
+apply (drule mp, simp add: mem_Times_iff)
+apply (clarify, rename_tac l1 r1)
+apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
+apply (drule mp, simp add: mem_Times_iff)
+apply (clarify, rename_tac l2 r2)
+apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
+apply (rule_tac x="r1 \<circ> r2" in exI)
+apply (rule conjI, simp add: subseq_def)
+apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
+apply (drule (1) tendsto_Pair) back
+apply (simp add: o_def)
+done
+
text{* Hence some useful properties follow quite easily. *}
lemma compact_scaleR_image:
@@ -4357,7 +4379,7 @@
lemma compact_scaling:
fixes s :: "(real ^ _) set"
- assumes "compact s" shows "compact ((\<lambda>x. c *s x) ` s)"
+ assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
using assms unfolding smult_conv_scaleR by (rule compact_scaleR_image)
lemma compact_negations:
@@ -4366,30 +4388,27 @@
using compact_scaleR_image [OF assms, of "- 1"] by auto
lemma compact_sums:
- fixes s t :: "(real ^ _) set"
+ fixes s t :: "'a::real_normed_vector set"
assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
proof-
- have *:"{x + y | x y. x \<in> s \<and> y \<in> t} =(\<lambda>z. fstcart z + sndcart z) ` {pastecart x y | x y. x \<in> s \<and> y \<in> t}"
- apply auto unfolding image_iff apply(rule_tac x="pastecart xa y" in bexI) unfolding fstcart_pastecart sndcart_pastecart by auto
- have "linear (\<lambda>z::real^('a + 'a). fstcart z + sndcart z)" unfolding linear_def
- unfolding fstcart_add sndcart_add apply auto
- unfolding vector_add_ldistrib fstcart_cmul[THEN sym] sndcart_cmul[THEN sym] by auto
- hence "continuous_on {pastecart x y |x y. x \<in> s \<and> y \<in> t} (\<lambda>z. fstcart z + sndcart z)"
- using continuous_at_imp_continuous_on linear_continuous_at by auto
- thus ?thesis unfolding * using compact_continuous_image compact_pastecart[OF assms] by auto
+ have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
+ apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
+ have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
+ unfolding continuous_on by (rule ballI) (intro tendsto_intros)
+ thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
qed
lemma compact_differences:
- fixes s t :: "(real ^ 'a::finite) set"
+ fixes s t :: "'a::real_normed_vector set"
assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
proof-
- have "{x - y | x y::real^'a. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
+ have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
qed
lemma compact_translation:
- fixes s :: "(real ^ _) set"
+ fixes s :: "'a::real_normed_vector set"
assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
proof-
have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
@@ -4398,23 +4417,23 @@
lemma compact_affinity:
fixes s :: "(real ^ _) set"
- assumes "compact s" shows "compact ((\<lambda>x. a + c *s x) ` s)"
+ assumes "compact s" shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
proof-
- have "op + a ` op *s c ` s = (\<lambda>x. a + c *s x) ` s" by auto
+ have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
qed
text{* Hence we get the following. *}
lemma compact_sup_maxdistance:
- fixes s :: "(real ^ _) set"
+ fixes s :: "'a::real_normed_vector set"
assumes "compact s" "s \<noteq> {}"
shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
proof-
have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
then obtain x where x:"x\<in>{x - y |x y. x \<in> s \<and> y \<in> s}" "\<forall>y\<in>{x - y |x y. x \<in> s \<and> y \<in> s}. norm y \<le> norm x"
using compact_differences[OF assms(1) assms(1)]
- using distance_attains_sup[unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by(auto simp add: norm_minus_cancel)
+ using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by(auto simp add: norm_minus_cancel)
from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
thus ?thesis using x(2)[unfolded `x = a - b`] by blast
qed
@@ -4458,11 +4477,11 @@
using diameter_bounded by blast
lemma diameter_compact_attained:
- fixes s :: "(real ^ _) set"
+ fixes s :: "'a::real_normed_vector set"
assumes "compact s" "s \<noteq> {}"
shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
proof-
- have b:"bounded s" using assms(1) compact_eq_bounded_closed by auto
+ have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. norm (u - v) \<le> norm (x - y)" using compact_sup_maxdistance[OF assms] by auto
hence "diameter s \<le> norm (x - y)" using rsup_le[of "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}" "norm (x - y)"]
unfolding setle_def and diameter_def by auto
@@ -4495,7 +4514,7 @@
then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
- unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym] norm_scaleR
+ unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
ultimately have "l \<in> scaleR c ` s"
@@ -4507,7 +4526,7 @@
lemma closed_scaling:
fixes s :: "(real ^ _) set"
- assumes "closed s" shows "closed ((\<lambda>x. c *s x) ` s)"
+ assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
using assms unfolding smult_conv_scaleR by (rule closed_scaleR_image)
lemma closed_negations:
@@ -4523,10 +4542,10 @@
{ fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
from as(1) obtain f where f:"\<forall>n. x n = fst (f n) + snd (f n)" "\<forall>n. fst (f n) \<in> s" "\<forall>n. snd (f n) \<in> t"
using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
- obtain l' r where "l'\<in>s" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
+ obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
- using Lim_sub[OF lim_subsequence[OF r as(2)] lr] and f(1) unfolding o_def by auto
+ using Lim_sub[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
hence "l - l' \<in> t"
using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
using f(3) by auto
@@ -4600,7 +4619,7 @@
subsection{* Separation between points and sets. *}
lemma separate_point_closed:
- fixes s :: "(real ^ _) set" (* FIXME: generalize *)
+ fixes s :: "'a::heine_borel set"
shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
proof(cases "s = {}")
case True
@@ -4613,7 +4632,8 @@
qed
lemma separate_compact_closed:
- fixes s t :: "(real ^ _) set"
+ fixes s t :: "'a::{heine_borel, real_normed_vector} set"
+ (* TODO: does this generalize to heine_borel? *)
assumes "compact s" and "closed t" and "s \<inter> t = {}"
shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
proof-
@@ -4629,7 +4649,7 @@
qed
lemma separate_closed_compact:
- fixes s t :: "(real ^ _) set"
+ fixes s t :: "'a::{heine_borel, real_normed_vector} set"
assumes "closed s" and "compact t" and "s \<inter> t = {}"
shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
proof-
@@ -4666,10 +4686,10 @@
hence False using as by auto }
moreover
{ assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
- let ?x = "(1/2) *s (a + b)"
+ let ?x = "(1/2) *\<^sub>R (a + b)"
{ fix i
have "a$i < b$i" using as[THEN spec[where x=i]] by auto
- hence "a$i < ((1/2) *s (a+b)) $ i" "((1/2) *s (a+b)) $ i < b$i"
+ hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
unfolding vector_smult_component and vector_add_component
by (auto simp add: less_divide_eq_number_of1) }
hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
@@ -4681,10 +4701,10 @@
hence False using as by auto }
moreover
{ assume as:"\<forall>i. \<not> (b$i < a$i)"
- let ?x = "(1/2) *s (a + b)"
+ let ?x = "(1/2) *\<^sub>R (a + b)"
{ fix i
have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
- hence "a$i \<le> ((1/2) *s (a+b)) $ i" "((1/2) *s (a+b)) $ i \<le> b$i"
+ hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
unfolding vector_smult_component and vector_add_component
by (auto simp add: less_divide_eq_number_of1) }
hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
@@ -4865,14 +4885,14 @@
then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
{ fix i
- have "dist (x - (e / 2) *s basis i) x < e"
- "dist (x + (e / 2) *s basis i) x < e"
+ have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
+ "dist (x + (e / 2) *\<^sub>R basis i) x < e"
unfolding dist_norm apply auto
- unfolding norm_minus_cancel and norm_mul using norm_basis[of i] and `e>0` by auto
- hence "a $ i \<le> (x - (e / 2) *s basis i) $ i"
- "(x + (e / 2) *s basis i) $ i \<le> b $ i"
- using e[THEN spec[where x="x - (e/2) *s basis i"]]
- and e[THEN spec[where x="x + (e/2) *s basis i"]]
+ unfolding norm_minus_cancel using norm_basis[of i] and `e>0` by auto
+ hence "a $ i \<le> (x - (e / 2) *\<^sub>R basis i) $ i"
+ "(x + (e / 2) *\<^sub>R basis i) $ i \<le> b $ i"
+ using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
+ and e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
unfolding mem_interval by (auto elim!: allE[where x=i])
hence "a $ i < x $ i" and "x $ i < b $ i"
unfolding vector_minus_component and vector_add_component
@@ -4910,10 +4930,10 @@
using bounded_closed_imp_compact using bounded_interval[of a b] using closed_interval[of a b] by auto
lemma open_interval_midpoint: fixes a :: "real^'n::finite"
- assumes "{a<..<b} \<noteq> {}" shows "((1/2) *s (a + b)) \<in> {a<..<b}"
+ assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
proof-
{ fix i
- have "a $ i < ((1 / 2) *s (a + b)) $ i \<and> ((1 / 2) *s (a + b)) $ i < b $ i"
+ have "a $ i < ((1 / 2) *\<^sub>R (a + b)) $ i \<and> ((1 / 2) *\<^sub>R (a + b)) $ i < b $ i"
using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
unfolding vector_smult_component and vector_add_component
by(auto simp add: less_divide_eq_number_of1) }
@@ -4922,7 +4942,7 @@
lemma open_closed_interval_convex: fixes x :: "real^'n::finite"
assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
- shows "(e *s x + (1 - e) *s y) \<in> {a<..<b}"
+ shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
proof-
{ fix i
have "a $ i = e * a$i + (1 - e) * a$i" unfolding left_diff_distrib by simp
@@ -4931,7 +4951,7 @@
using x unfolding mem_interval apply simp
using y unfolding mem_interval apply simp
done
- finally have "a $ i < (e *s x + (1 - e) *s y) $ i" by auto
+ finally have "a $ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i" by auto
moreover {
have "b $ i = e * b$i + (1 - e) * b$i" unfolding left_diff_distrib by simp
also have "\<dots> > e * x $ i + (1 - e) * y $ i" apply(rule add_less_le_mono)
@@ -4939,8 +4959,8 @@
using x unfolding mem_interval apply simp
using y unfolding mem_interval apply simp
done
- finally have "(e *s x + (1 - e) *s y) $ i < b $ i" by auto
- } ultimately have "a $ i < (e *s x + (1 - e) *s y) $ i \<and> (e *s x + (1 - e) *s y) $ i < b $ i" by auto }
+ finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i < b $ i" by auto
+ } ultimately have "a $ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i < b $ i" by auto }
thus ?thesis unfolding mem_interval by auto
qed
@@ -4949,13 +4969,14 @@
shows "closure {a<..<b} = {a .. b}"
proof-
have ab:"a < b" using assms[unfolded interval_ne_empty] unfolding vector_less_def by auto
- let ?c = "(1 / 2) *s (a + b)"
+ let ?c = "(1 / 2) *\<^sub>R (a + b)"
{ fix x assume as:"x \<in> {a .. b}"
- def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *s (?c - x)"
+ def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
{ fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
- have "(inverse (real n + 1)) *s ((1 / 2) *s (a + b)) + (1 - inverse (real n + 1)) *s x =
- x + (inverse (real n + 1)) *s ((1 / 2 *s (a + b)) - x)" by (auto simp add: vector_ssub_ldistrib vector_add_ldistrib field_simps vector_sadd_rdistrib[THEN sym])
+ have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
+ x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
+ by (auto simp add: algebra_simps)
hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto
hence False using fn unfolding f_def using xc by(auto simp add: vector_mul_lcancel vector_ssub_ldistrib) }
moreover
@@ -4965,11 +4986,11 @@
then obtain N::nat where "inverse (real (N + 1)) < e" by auto
hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
- hence "((vec1 \<circ> (\<lambda>n. inverse (real n + 1))) ---> vec1 0) sequentially"
- unfolding Lim_sequentially by(auto simp add: dist_vec1)
+ hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
+ unfolding Lim_sequentially by(auto simp add: dist_norm)
hence "(f ---> x) sequentially" unfolding f_def
- using Lim_add[OF Lim_const, of "\<lambda>n::nat. (inverse (real n + 1)) *s ((1 / 2) *s (a + b) - x)" 0 sequentially x]
- using Lim_vmul[of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *s (a + b) - x)"] by auto }
+ using Lim_add[OF Lim_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
+ using Lim_vmul[of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
ultimately have "x \<in> closure {a<..<b}"
using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
@@ -5128,99 +5149,82 @@
subsection{* Closure of halfspaces and hyperplanes. *}
-lemma Lim_vec1_dot: fixes f :: "real^'m \<Rightarrow> real^'n::finite"
- assumes "(f ---> l) net" shows "((vec1 o (\<lambda>y. a \<bullet> (f y))) ---> vec1(a \<bullet> l)) net"
-proof(cases "a = vec 0")
- case True thus ?thesis using dot_lzero and Lim_const[of 0 net] unfolding vec1_vec and o_def by auto
-next
- case False
- { fix e::real
- assume "0 < e"
- with `a \<noteq> vec 0` have "0 < e / norm a" by (simp add: divide_pos_pos)
- with assms(1) have "eventually (\<lambda>x. dist (f x) l < e / norm a) net"
- by (rule tendstoD)
- moreover
- { fix z assume "dist (f z) l < e / norm a"
- hence "norm a * norm (f z - l) < e" unfolding dist_norm and
- pos_less_divide_eq[OF False[unfolded vec_0 zero_less_norm_iff[of a, THEN sym]]] and real_mult_commute by auto
- hence "\<bar>a \<bullet> f z - a \<bullet> l\<bar> < e"
- using order_le_less_trans[OF norm_cauchy_schwarz_abs[of a "f z - l"], of e]
- unfolding dot_rsub[symmetric] by auto }
- ultimately have "eventually (\<lambda>x. \<bar>a \<bullet> f x - a \<bullet> l\<bar> < e) net"
- by (auto elim: eventually_rev_mono)
- }
- thus ?thesis unfolding tendsto_iff
- by (auto simp add: dist_vec1)
-qed
-
-lemma continuous_at_vec1_dot:
+lemma Lim_dot: fixes f :: "real^'m \<Rightarrow> real^'n::finite"
+ assumes "(f ---> l) net" shows "((\<lambda>y. a \<bullet> (f y)) ---> a \<bullet> l) net"
+ unfolding dot_def by (intro tendsto_intros assms)
+
+lemma continuous_at_dot:
fixes x :: "real ^ _"
- shows "continuous (at x) (vec1 o (\<lambda>y. a \<bullet> y))"
+ shows "continuous (at x) (\<lambda>y. a \<bullet> y)"
proof-
have "((\<lambda>x. x) ---> x) (at x)" unfolding Lim_at by auto
- thus ?thesis unfolding continuous_at and o_def using Lim_vec1_dot[of "\<lambda>x. x" x "at x" a] by auto
-qed
-
-lemma continuous_on_vec1_dot:
+ thus ?thesis unfolding continuous_at and o_def using Lim_dot[of "\<lambda>x. x" x "at x" a] by auto
+qed
+
+lemma continuous_on_dot:
fixes s :: "(real ^ _) set"
- shows "continuous_on s (vec1 o (\<lambda>y. a \<bullet> y)) "
- using continuous_at_imp_continuous_on[of s "vec1 o (\<lambda>y. a \<bullet> y)"]
- using continuous_at_vec1_dot
+ shows "continuous_on s (\<lambda>y. a \<bullet> y)"
+ using continuous_at_imp_continuous_on[of s "\<lambda>y. a \<bullet> y"]
+ using continuous_at_dot
by auto
-lemma closed_halfspace_le: fixes a::"real^'n::finite"
- shows "closed {x. a \<bullet> x \<le> b}"
+lemma continuous_on_inner:
+ fixes s :: "'a::real_inner set"
+ shows "continuous_on s (inner a)"
+ unfolding continuous_on by (rule ballI) (intro tendsto_intros)
+
+lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
proof-
- have *:"{x \<in> UNIV. (vec1 \<circ> op \<bullet> a) x \<in> vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b}} = {x. a \<bullet> x \<le> b}" by auto
- let ?T = "{x::real^1. (\<forall>i. x$i \<le> (vec1 b)$i)}"
- have "closed ?T" using closed_interval_left[of "vec1 b"] by simp
- moreover have "vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b} = range (vec1 \<circ> op \<bullet> a) \<inter> ?T" unfolding all_1
+ have *:"{x \<in> UNIV. inner a x \<in> {r. \<exists>x. inner a x = r \<and> r \<le> b}} = {x. inner a x \<le> b}" by auto
+ let ?T = "{..b}"
+ have "closed ?T" by (rule closed_real_atMost)
+ moreover have "{r. \<exists>x. inner a x = r \<and> r \<le> b} = range (inner a) \<inter> ?T"
unfolding image_def by auto
- ultimately have "\<exists>T. closed T \<and> vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b} = range (vec1 \<circ> op \<bullet> a) \<inter> T" by auto
- hence "closedin euclidean {x \<in> UNIV. (vec1 \<circ> op \<bullet> a) x \<in> vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b}}"
- using continuous_on_vec1_dot[of UNIV a, unfolded continuous_on_closed subtopology_UNIV] unfolding closedin_closed
- by (auto elim!: allE[where x="vec1 ` {r. (\<exists>x. a \<bullet> x = r \<and> r \<le> b)}"])
+ ultimately have "\<exists>T. closed T \<and> {r. \<exists>x. inner a x = r \<and> r \<le> b} = range (inner a) \<inter> T" by fast
+ hence "closedin euclidean {x \<in> UNIV. inner a x \<in> {r. \<exists>x. inner a x = r \<and> r \<le> b}}"
+ using continuous_on_inner[of UNIV a, unfolded continuous_on_closed subtopology_UNIV] unfolding closedin_closed
+ by (fast elim!: allE[where x="{r. (\<exists>x. inner a x = r \<and> r \<le> b)}"])
thus ?thesis unfolding closed_closedin[THEN sym] and * by auto
qed
-lemma closed_halfspace_ge: "closed {x::real^_. a \<bullet> x \<ge> b}"
- using closed_halfspace_le[of "-a" "-b"] unfolding dot_lneg by auto
-
-lemma closed_hyperplane: "closed {x::real^_. a \<bullet> x = b}"
+lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
+ using closed_halfspace_le[of "-a" "-b"] unfolding inner_minus_left by auto
+
+lemma closed_hyperplane: "closed {x. inner a x = b}"
proof-
- have "{x. a \<bullet> x = b} = {x. a \<bullet> x \<ge> b} \<inter> {x. a \<bullet> x \<le> b}" by auto
+ have "{x. inner a x = b} = {x. inner a x \<ge> b} \<inter> {x. inner a x \<le> b}" by auto
thus ?thesis using closed_halfspace_le[of a b] and closed_halfspace_ge[of b a] using closed_Int by auto
qed
lemma closed_halfspace_component_le:
shows "closed {x::real^'n::finite. x$i \<le> a}"
- using closed_halfspace_le[of "(basis i)::real^'n" a] unfolding dot_basis[OF assms] by auto
+ using closed_halfspace_le[of "(basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
lemma closed_halfspace_component_ge:
shows "closed {x::real^'n::finite. x$i \<ge> a}"
- using closed_halfspace_ge[of a "(basis i)::real^'n"] unfolding dot_basis[OF assms] by auto
+ using closed_halfspace_ge[of a "(basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
text{* Openness of halfspaces. *}
-lemma open_halfspace_lt: "open {x::real^_. a \<bullet> x < b}"
+lemma open_halfspace_lt: "open {x. inner a x < b}"
proof-
- have "UNIV - {x. b \<le> a \<bullet> x} = {x. a \<bullet> x < b}" by auto
+ have "UNIV - {x. b \<le> inner a x} = {x. inner a x < b}" by auto
thus ?thesis using closed_halfspace_ge[unfolded closed_def Compl_eq_Diff_UNIV, of b a] by auto
qed
-lemma open_halfspace_gt: "open {x::real^_. a \<bullet> x > b}"
+lemma open_halfspace_gt: "open {x. inner a x > b}"
proof-
- have "UNIV - {x. b \<ge> a \<bullet> x} = {x. a \<bullet> x > b}" by auto
+ have "UNIV - {x. b \<ge> inner a x} = {x. inner a x > b}" by auto
thus ?thesis using closed_halfspace_le[unfolded closed_def Compl_eq_Diff_UNIV, of a b] by auto
qed
lemma open_halfspace_component_lt:
shows "open {x::real^'n::finite. x$i < a}"
- using open_halfspace_lt[of "(basis i)::real^'n" a] unfolding dot_basis[OF assms] by auto
+ using open_halfspace_lt[of "(basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
lemma open_halfspace_component_gt:
shows "open {x::real^'n::finite. x$i > a}"
- using open_halfspace_gt[of a "(basis i)::real^'n"] unfolding dot_basis[OF assms] by auto
+ using open_halfspace_gt[of a "(basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
text{* This gives a simple derivation of limit component bounds. *}
@@ -5228,8 +5232,8 @@
assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$i \<le> b) net"
shows "l$i \<le> b"
proof-
- { fix x have "x \<in> {x::real^'n. basis i \<bullet> x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding dot_basis by auto } note * = this
- show ?thesis using Lim_in_closed_set[of "{x. basis i \<bullet> x \<le> b}" f net l] unfolding *
+ { fix x have "x \<in> {x::real^'n. inner (basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding inner_basis by auto } note * = this
+ show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
using closed_halfspace_le[of "(basis i)::real^'n" b] and assms(1,2,3) by auto
qed
@@ -5237,8 +5241,8 @@
assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$i) net"
shows "b \<le> l$i"
proof-
- { fix x have "x \<in> {x::real^'n. basis i \<bullet> x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding dot_basis by auto } note * = this
- show ?thesis using Lim_in_closed_set[of "{x. basis i \<bullet> x \<ge> b}" f net l] unfolding *
+ { fix x have "x \<in> {x::real^'n. inner (basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding inner_basis by auto } note * = this
+ show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
using closed_halfspace_ge[of b "(basis i)::real^'n"] and assms(1,2,3) by auto
qed
@@ -5292,13 +5296,13 @@
text{* Some more convenient intermediate-value theorem formulations. *}
-lemma connected_ivt_hyperplane: fixes y :: "real^'n::finite"
- assumes "connected s" "x \<in> s" "y \<in> s" "a \<bullet> x \<le> b" "b \<le> a \<bullet> y"
- shows "\<exists>z \<in> s. a \<bullet> z = b"
+lemma connected_ivt_hyperplane:
+ assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
+ shows "\<exists>z \<in> s. inner a z = b"
proof(rule ccontr)
- assume as:"\<not> (\<exists>z\<in>s. a \<bullet> z = b)"
- let ?A = "{x::real^'n. a \<bullet> x < b}"
- let ?B = "{x::real^'n. a \<bullet> x > b}"
+ assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
+ let ?A = "{x. inner a x < b}"
+ let ?B = "{x. inner a x > b}"
have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
moreover have "?A \<inter> ?B = {}" by auto
moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
@@ -5307,7 +5311,7 @@
lemma connected_ivt_component: fixes x::"real^'n::finite" shows
"connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s. z$k = a)"
- using connected_ivt_hyperplane[of s x y "(basis k)::real^'n" a] by (auto simp add: dot_basis)
+ using connected_ivt_hyperplane[of s x y "(basis k)::real^'n" a] by (auto simp add: inner_basis)
text{* Also more convenient formulations of monotone convergence. *}
@@ -5320,7 +5324,7 @@
have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE) by(auto simp add: not_less_eq_eq) }
hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)" by auto
- then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a] by auto
+ then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto
thus ?thesis unfolding Lim_sequentially apply(rule_tac x="vec1 l" in exI)
unfolding dist_norm unfolding abs_dest_vec1 and dest_vec1_sub by auto
qed
@@ -5429,11 +5433,11 @@
text{* Results on translation, scaling etc. *}
lemma homeomorphic_scaling:
- assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *s x) ` s)"
+ assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
unfolding homeomorphic_minimal
- apply(rule_tac x="\<lambda>x. c *s x" in exI)
- apply(rule_tac x="\<lambda>x. (1 / c) *s x" in exI)
- apply auto unfolding vector_smult_assoc using assms apply auto
+ apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
+ apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
+ using assms apply auto
using continuous_on_cmul[OF continuous_on_id] by auto
lemma homeomorphic_translation:
@@ -5444,13 +5448,13 @@
using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
lemma homeomorphic_affinity:
- assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *s x) ` s)"
+ assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
proof-
- have *:"op + a ` op *s c ` s = (\<lambda>x. a + c *s x) ` s" by auto
+ have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
show ?thesis
using homeomorphic_trans
using homeomorphic_scaling[OF assms, of s]
- using homeomorphic_translation[of "(\<lambda>x. c *s x) ` s" a] unfolding * by auto
+ using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
qed
lemma homeomorphic_balls: fixes a b ::"real^'a::finite"
@@ -5460,27 +5464,23 @@
proof-
have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
show ?th unfolding homeomorphic_minimal
- apply(rule_tac x="\<lambda>x. b + (e/d) *s (x - a)" in exI)
- apply(rule_tac x="\<lambda>x. a + (d/e) *s (x - b)" in exI)
- apply (auto simp add: dist_commute) unfolding dist_norm and vector_smult_assoc using assms apply auto
- unfolding norm_minus_cancel and norm_mul
- using continuous_on_add[OF continuous_on_const continuous_on_cmul[OF continuous_on_sub[OF continuous_on_id continuous_on_const]]]
- apply (auto simp add: dist_commute)
- using pos_less_divide_eq[OF *(1), THEN sym] unfolding real_mult_commute[of _ "\<bar>e / d\<bar>"]
- using pos_less_divide_eq[OF *(2), THEN sym] unfolding real_mult_commute[of _ "\<bar>d / e\<bar>"]
- by (auto simp add: dist_commute)
+ apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
+ apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
+ using assms apply (auto simp add: dist_commute)
+ unfolding dist_norm
+ apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
+ unfolding continuous_on
+ by (intro ballI tendsto_intros, simp, assumption)+
next
have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
show ?cth unfolding homeomorphic_minimal
- apply(rule_tac x="\<lambda>x. b + (e/d) *s (x - a)" in exI)
- apply(rule_tac x="\<lambda>x. a + (d/e) *s (x - b)" in exI)
- apply (auto simp add: dist_commute) unfolding dist_norm and vector_smult_assoc using assms apply auto
- unfolding norm_minus_cancel and norm_mul
- using continuous_on_add[OF continuous_on_const continuous_on_cmul[OF continuous_on_sub[OF continuous_on_id continuous_on_const]]]
- apply auto
- using pos_le_divide_eq[OF *(1), THEN sym] unfolding real_mult_commute[of _ "\<bar>e / d\<bar>"]
- using pos_le_divide_eq[OF *(2), THEN sym] unfolding real_mult_commute[of _ "\<bar>d / e\<bar>"]
- by auto
+ apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
+ apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
+ using assms apply (auto simp add: dist_commute)
+ unfolding dist_norm
+ apply (auto simp add: pos_divide_le_eq)
+ unfolding continuous_on
+ by (intro ballI tendsto_intros, simp, assumption)+
qed
text{* "Isometry" (up to constant bounds) of injective linear map etc. *}
@@ -5568,10 +5568,10 @@
next
case False
hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
- have "\<forall>c. \<forall>x\<in>s. c *s x \<in> s" using s[unfolded subspace_def] by auto
- hence "(norm a / norm x) *s x \<in> {x \<in> s. norm x = norm a}" using `x\<in>s` and `x\<noteq>0` by auto
- thus "norm (f b) / norm b * norm x \<le> norm (f x)" using b[THEN bspec[where x="(norm a / norm x) *s x"]]
- unfolding linear_cmul[OF f(1)] and norm_mul and ba using `x\<noteq>0` `a\<noteq>0`
+ have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def smult_conv_scaleR] by auto
+ hence "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}" using `x\<in>s` and `x\<noteq>0` by auto
+ thus "norm (f b) / norm b * norm x \<le> norm (f x)" using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
+ unfolding linear_cmul[OF f(1), unfolded smult_conv_scaleR] and ba using `x\<noteq>0` `a\<noteq>0`
by (auto simp add: real_mult_commute pos_le_divide_eq pos_divide_le_eq)
qed }
ultimately
@@ -5598,16 +5598,16 @@
"closed {x::real^'n::finite. \<forall>i. P i --> x$i = 0}" (is "closed ?A")
proof-
let ?D = "{i. P i}"
- let ?Bs = "{{x::real^'n. basis i \<bullet> x = 0}| i. i \<in> ?D}"
+ let ?Bs = "{{x::real^'n. inner (basis i) x = 0}| i. i \<in> ?D}"
{ fix x
{ assume "x\<in>?A"
hence x:"\<forall>i\<in>?D. x $ i = 0" by auto
- hence "x\<in> \<Inter> ?Bs" by(auto simp add: dot_basis x) }
+ hence "x\<in> \<Inter> ?Bs" by(auto simp add: inner_basis x) }
moreover
{ assume x:"x\<in>\<Inter>?Bs"
{ fix i assume i:"i \<in> ?D"
- then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. basis i \<bullet> x = 0}" by auto
- hence "x $ i = 0" unfolding B using x unfolding dot_basis by auto }
+ then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. inner (basis i) x = 0}" by auto
+ hence "x $ i = 0" unfolding B using x unfolding inner_basis by auto }
hence "x\<in>?A" by auto }
ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" by auto }
hence "?A = \<Inter> ?Bs" by auto
@@ -5635,8 +5635,8 @@
case (insert k F)
hence *:"\<forall>i. i \<notin> insert k F \<longrightarrow> x $ i = 0" by auto
have **:"F \<subseteq> insert k F" by auto
- def y \<equiv> "x - x$k *s basis k"
- have y:"x = y + (x$k) *s basis k" unfolding y_def by auto
+ def y \<equiv> "x - x$k *\<^sub>R basis k"
+ have y:"x = y + (x$k) *\<^sub>R basis k" unfolding y_def by auto
{ fix i assume i':"i \<notin> F"
hence "y $ i = 0" unfolding y_def unfolding vector_minus_component
and vector_smult_component and basis_component
@@ -5646,9 +5646,10 @@
using image_mono[OF **, of basis] by auto
moreover
have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
- hence "x$k *s basis k \<in> span (?bas ` (insert k F))" using span_mul by auto
+ hence "x$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
+ using span_mul [where 'a=real, unfolded smult_conv_scaleR] by auto
ultimately
- have "y + x$k *s basis k \<in> span (?bas ` (insert k F))"
+ have "y + x$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
using span_add by auto
thus ?case using y by auto
qed
@@ -5769,45 +5770,45 @@
lemma image_affinity_interval: fixes m::real
fixes a b c :: "real^'n::finite"
- shows "(\<lambda>x. m *s x + c) ` {a .. b} =
+ shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
(if {a .. b} = {} then {}
- else (if 0 \<le> m then {m *s a + c .. m *s b + c}
- else {m *s b + c .. m *s a + c}))"
+ else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
+ else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
proof(cases "m=0")
{ fix x assume "x \<le> c" "c \<le> x"
hence "x=c" unfolding vector_less_eq_def and Cart_eq by (auto intro: order_antisym) }
moreover case True
- moreover have "c \<in> {m *s a + c..m *s b + c}" unfolding True by(auto simp add: vector_less_eq_def)
+ moreover have "c \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}" unfolding True by(auto simp add: vector_less_eq_def)
ultimately show ?thesis by auto
next
case False
{ fix y assume "a \<le> y" "y \<le> b" "m > 0"
- hence "m *s a + c \<le> m *s y + c" "m *s y + c \<le> m *s b + c"
+ hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
unfolding vector_less_eq_def by(auto simp add: vector_smult_component vector_add_component)
} moreover
{ fix y assume "a \<le> y" "y \<le> b" "m < 0"
- hence "m *s b + c \<le> m *s y + c" "m *s y + c \<le> m *s a + c"
+ hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
unfolding vector_less_eq_def by(auto simp add: vector_smult_component vector_add_component mult_left_mono_neg elim!:ballE)
} moreover
- { fix y assume "m > 0" "m *s a + c \<le> y" "y \<le> m *s b + c"
- hence "y \<in> (\<lambda>x. m *s x + c) ` {a..b}"
+ { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
+ hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
unfolding image_iff Bex_def mem_interval vector_less_eq_def
apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
- intro!: exI[where x="(1 / m) *s (y - c)"])
+ intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
by(auto simp add: pos_le_divide_eq pos_divide_le_eq real_mult_commute diff_le_iff)
} moreover
- { fix y assume "m *s b + c \<le> y" "y \<le> m *s a + c" "m < 0"
- hence "y \<in> (\<lambda>x. m *s x + c) ` {a..b}"
+ { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
+ hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
unfolding image_iff Bex_def mem_interval vector_less_eq_def
apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
- intro!: exI[where x="(1 / m) *s (y - c)"])
+ intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
by(auto simp add: neg_le_divide_eq neg_divide_le_eq real_mult_commute diff_le_iff)
}
ultimately show ?thesis using False by auto
qed
-lemma image_smult_interval:"(\<lambda>x. m *s (x::real^'n::finite)) ` {a..b} =
- (if {a..b} = {} then {} else if 0 \<le> m then {m *s a..m *s b} else {m *s b..m *s a})"
+lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::real^'n::finite)) ` {a..b} =
+ (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
using image_affinity_interval[of m 0 a b] by auto
subsection{* Banach fixed point theorem (not really topological...) *}
@@ -5940,9 +5941,10 @@
subsection{* Edelstein fixed point theorem. *}
lemma edelstein_fix:
+ fixes s :: "'a::real_normed_vector set"
assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
- shows "\<exists>! x::real^'a::finite\<in>s. g x = x"
+ shows "\<exists>! x\<in>s. g x = x"
proof(cases "\<exists>x\<in>s. g x \<noteq> x")
obtain x where "x\<in>s" using s(2) by auto
case False hence g:"\<forall>x\<in>s. g x = x" by auto
@@ -5985,26 +5987,23 @@
qed
qed } note distf = this
- def h \<equiv> "\<lambda>n. pastecart (f n x) (f n y)"
- let ?s2 = "{pastecart x y |x y. x \<in> s \<and> y \<in> s}"
- obtain l r where "l\<in>?s2" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and lr:"((h \<circ> r) ---> l) sequentially"
- using compact_pastecart[OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
+ def h \<equiv> "\<lambda>n. (f n x, f n y)"
+ let ?s2 = "s \<times> s"
+ obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
+ using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
- def a \<equiv> "fstcart l" def b \<equiv> "sndcart l"
- have lab:"l = pastecart a b" unfolding a_def b_def and pastecart_fst_snd by simp
+ def a \<equiv> "fst l" def b \<equiv> "snd l"
+ have lab:"l = (a, b)" unfolding a_def b_def by simp
have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
- have "continuous_on (UNIV :: (real ^ _) set) fstcart"
- and "continuous_on (UNIV :: (real ^ _) set) sndcart"
- using linear_continuous_on using linear_fstcart and linear_sndcart by auto
- hence lima:"((fstcart \<circ> (h \<circ> r)) ---> a) sequentially" and limb:"((sndcart \<circ> (h \<circ> r)) ---> b) sequentially"
- unfolding atomize_conj unfolding continuous_on_sequentially
- apply(erule_tac x="h \<circ> r" in allE) apply(erule_tac x="h \<circ> r" in allE) using lr
- unfolding o_def and h_def a_def b_def by auto
+ have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
+ and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
+ using lr
+ unfolding o_def a_def b_def by (simp_all add: tendsto_intros)
{ fix n::nat
- have *:"\<And>fx fy (x::real^_) y. dist fx fy \<le> dist x y \<Longrightarrow> \<not> (dist (fx - fy) (a - b) < dist a b - dist x y)" unfolding dist_norm by norm
- { fix x y ::"real^'a"
+ have *:"\<And>fx fy (x::'a) y. dist fx fy \<le> dist x y \<Longrightarrow> \<not> (dist (fx - fy) (a - b) < dist a b - dist x y)" unfolding dist_norm by norm
+ { fix x y :: 'a
have "dist (-x) (-y) = dist x y" unfolding dist_norm
using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
@@ -6021,7 +6020,7 @@
moreover
have "dist (f (r (Na + Nb + n)) x - f (r (Na + Nb + n)) y) (a - b) \<ge> dist a b - dist (f n x) (f n y)"
using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
- using monotone_bigger[OF r, of "Na+Nb+n"]
+ using subseq_bigger[OF r, of "Na+Nb+n"]
using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
ultimately have False by simp
}
@@ -6049,8 +6048,8 @@
have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
hence "continuous_on s g" unfolding continuous_on_def by auto
- hence "((sndcart \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
- apply (rule allE[where x="\<lambda>n. (fstcart \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
+ hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
+ apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
hence "g a = a" using Lim_unique[OF trivial_limit_sequentially limb, of "g a"]
unfolding `a=b` and o_assoc by auto
--- a/src/HOL/Limits.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/Limits.thy Sat Jun 13 16:32:38 2009 +0200
@@ -358,6 +358,14 @@
where [code del]:
"(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
+ML{*
+structure TendstoIntros =
+ NamedThmsFun(val name = "tendsto_intros"
+ val description = "introduction rules for tendsto");
+*}
+
+setup TendstoIntros.setup
+
lemma topological_tendstoI:
"(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
\<Longrightarrow> (f ---> l) net"
@@ -395,12 +403,38 @@
lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
by (simp only: tendsto_iff Zfun_def dist_norm)
-lemma tendsto_const: "((\<lambda>x. k) ---> k) net"
+lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
+unfolding tendsto_def eventually_at_topological by auto
+
+lemma tendsto_ident_at_within [tendsto_intros]:
+ "a \<in> S \<Longrightarrow> ((\<lambda>x. x) ---> a) (at a within S)"
+unfolding tendsto_def eventually_within eventually_at_topological by auto
+
+lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
by (simp add: tendsto_def)
-lemma tendsto_norm:
- fixes a :: "'a::real_normed_vector"
- shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
+lemma tendsto_dist [tendsto_intros]:
+ assumes f: "(f ---> l) net" and g: "(g ---> m) net"
+ shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
+proof (rule tendstoI)
+ fix e :: real assume "0 < e"
+ hence e2: "0 < e/2" by simp
+ from tendstoD [OF f e2] tendstoD [OF g e2]
+ show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
+ proof (rule eventually_elim2)
+ fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
+ then show "dist (dist (f x) (g x)) (dist l m) < e"
+ unfolding dist_real_def
+ using dist_triangle2 [of "f x" "g x" "l"]
+ using dist_triangle2 [of "g x" "l" "m"]
+ using dist_triangle3 [of "l" "m" "f x"]
+ using dist_triangle [of "f x" "m" "g x"]
+ by arith
+ qed
+qed
+
+lemma tendsto_norm [tendsto_intros]:
+ "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
apply (simp add: tendsto_iff dist_norm, safe)
apply (drule_tac x="e" in spec, safe)
apply (erule eventually_elim1)
@@ -417,12 +451,12 @@
shows "(- a) - (- b) = - (a - b)"
by simp
-lemma tendsto_add:
+lemma tendsto_add [tendsto_intros]:
fixes a b :: "'a::real_normed_vector"
shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
-lemma tendsto_minus:
+lemma tendsto_minus [tendsto_intros]:
fixes a :: "'a::real_normed_vector"
shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
@@ -432,16 +466,34 @@
shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
by (drule tendsto_minus, simp)
-lemma tendsto_diff:
+lemma tendsto_diff [tendsto_intros]:
fixes a b :: "'a::real_normed_vector"
shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
by (simp add: diff_minus tendsto_add tendsto_minus)
-lemma (in bounded_linear) tendsto:
+lemma tendsto_setsum [tendsto_intros]:
+ fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
+ assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
+ shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
+proof (cases "finite S")
+ assume "finite S" thus ?thesis using assms
+ proof (induct set: finite)
+ case empty show ?case
+ by (simp add: tendsto_const)
+ next
+ case (insert i F) thus ?case
+ by (simp add: tendsto_add)
+ qed
+next
+ assume "\<not> finite S" thus ?thesis
+ by (simp add: tendsto_const)
+qed
+
+lemma (in bounded_linear) tendsto [tendsto_intros]:
"(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
-lemma (in bounded_bilinear) tendsto:
+lemma (in bounded_bilinear) tendsto [tendsto_intros]:
"\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
by (simp only: tendsto_Zfun_iff prod_diff_prod
Zfun_add Zfun Zfun_left Zfun_right)
@@ -556,7 +608,7 @@
apply (simp add: tendsto_Zfun_iff)
done
-lemma tendsto_inverse:
+lemma tendsto_inverse [tendsto_intros]:
fixes a :: "'a::real_normed_div_algebra"
assumes f: "(f ---> a) net"
assumes a: "a \<noteq> 0"
@@ -571,7 +623,7 @@
by (rule tendsto_inverse_lemma)
qed
-lemma tendsto_divide:
+lemma tendsto_divide [tendsto_intros]:
fixes a b :: "'a::real_normed_field"
shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
\<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
--- a/src/HOL/List.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/List.thy Sat Jun 13 16:32:38 2009 +0200
@@ -2931,7 +2931,7 @@
done
-subsubsection {* Infiniteness *}
+subsubsection {* (In)finiteness *}
lemma finite_maxlen:
"finite (M::'a list set) ==> EX n. ALL s:M. size s < n"
@@ -2944,6 +2944,27 @@
thus ?case ..
qed
+lemma finite_lists_length_eq:
+assumes "finite A"
+shows "finite {xs. set xs \<subseteq> A \<and> length xs = n}" (is "finite (?S n)")
+proof(induct n)
+ case 0 show ?case by simp
+next
+ case (Suc n)
+ have "?S (Suc n) = (\<Union>x\<in>A. (\<lambda>xs. x#xs) ` ?S n)"
+ by (auto simp:length_Suc_conv)
+ then show ?case using `finite A`
+ by (auto intro: finite_imageI Suc) (* FIXME metis? *)
+qed
+
+lemma finite_lists_length_le:
+ assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
+ (is "finite ?S")
+proof-
+ have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto
+ thus ?thesis by (auto intro: finite_lists_length_eq[OF `finite A`])
+qed
+
lemma infinite_UNIV_listI: "~ finite(UNIV::'a list set)"
apply(rule notI)
apply(drule finite_maxlen)
--- a/src/HOL/RealVector.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/RealVector.thy Sat Jun 13 16:32:38 2009 +0200
@@ -116,11 +116,11 @@
thus "a = b" by (simp only: right_minus_eq)
qed
-lemma scale_cancel_left:
+lemma scale_cancel_left [simp]:
"scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
by (auto intro: scale_left_imp_eq)
-lemma scale_cancel_right:
+lemma scale_cancel_right [simp]:
"scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
by (auto intro: scale_right_imp_eq)
@@ -530,6 +530,9 @@
lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
using dist_triangle2 [of x z y] by (simp add: dist_commute)
+lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
+using dist_triangle2 [of x y a] by (simp add: dist_commute)
+
subclass topological_space
proof
have "\<exists>e::real. 0 < e"
@@ -568,7 +571,7 @@
assumes norm_ge_zero [simp]: "0 \<le> norm x"
and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
- and norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
+ and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
class real_normed_algebra = real_algebra + real_normed_vector +
assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
@@ -592,32 +595,6 @@
thus "norm (1::'a) = 1" by simp
qed
-instantiation real :: real_normed_field
-begin
-
-definition real_norm_def [simp]:
- "norm r = \<bar>r\<bar>"
-
-definition dist_real_def:
- "dist x y = \<bar>x - y\<bar>"
-
-definition open_real_def [code del]:
- "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-
-instance
-apply (intro_classes, unfold real_norm_def real_scaleR_def)
-apply (rule dist_real_def)
-apply (rule open_real_def)
-apply (simp add: real_sgn_def)
-apply (rule abs_ge_zero)
-apply (rule abs_eq_0)
-apply (rule abs_triangle_ineq)
-apply (rule abs_mult)
-apply (rule abs_mult)
-done
-
-end
-
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
by simp
@@ -724,7 +701,7 @@
lemma norm_of_real [simp]:
"norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
-unfolding of_real_def by (simp add: norm_scaleR)
+unfolding of_real_def by simp
lemma norm_number_of [simp]:
"norm (number_of w::'a::{number_ring,real_normed_algebra_1})
@@ -797,6 +774,76 @@
using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
qed
+
+subsection {* Class instances for real numbers *}
+
+instantiation real :: real_normed_field
+begin
+
+definition real_norm_def [simp]:
+ "norm r = \<bar>r\<bar>"
+
+definition dist_real_def:
+ "dist x y = \<bar>x - y\<bar>"
+
+definition open_real_def [code del]:
+ "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
+
+instance
+apply (intro_classes, unfold real_norm_def real_scaleR_def)
+apply (rule dist_real_def)
+apply (rule open_real_def)
+apply (simp add: real_sgn_def)
+apply (rule abs_ge_zero)
+apply (rule abs_eq_0)
+apply (rule abs_triangle_ineq)
+apply (rule abs_mult)
+apply (rule abs_mult)
+done
+
+end
+
+lemma open_real_lessThan [simp]:
+ fixes a :: real shows "open {..<a}"
+unfolding open_real_def dist_real_def
+proof (clarify)
+ fix x assume "x < a"
+ hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
+ thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
+qed
+
+lemma open_real_greaterThan [simp]:
+ fixes a :: real shows "open {a<..}"
+unfolding open_real_def dist_real_def
+proof (clarify)
+ fix x assume "a < x"
+ hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
+ thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
+qed
+
+lemma open_real_greaterThanLessThan [simp]:
+ fixes a b :: real shows "open {a<..<b}"
+proof -
+ have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
+ thus "open {a<..<b}" by (simp add: open_Int)
+qed
+
+lemma closed_real_atMost [simp]:
+ fixes a :: real shows "closed {..a}"
+unfolding closed_open by simp
+
+lemma closed_real_atLeast [simp]:
+ fixes a :: real shows "closed {a..}"
+unfolding closed_open by simp
+
+lemma closed_real_atLeastAtMost [simp]:
+ fixes a b :: real shows "closed {a..b}"
+proof -
+ have "{a..b} = {a..} \<inter> {..b}" by auto
+ thus "closed {a..b}" by (simp add: closed_Int)
+qed
+
+
subsection {* Extra type constraints *}
text {* Only allow @{term "open"} in class @{text topological_space}. *}
@@ -819,7 +866,7 @@
lemma norm_sgn:
"norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
-by (simp add: sgn_div_norm norm_scaleR)
+by (simp add: sgn_div_norm)
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
by (simp add: sgn_div_norm)
@@ -832,7 +879,7 @@
lemma sgn_scaleR:
"sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
-by (simp add: sgn_div_norm norm_scaleR mult_ac)
+by (simp add: sgn_div_norm mult_ac)
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
by (simp add: sgn_div_norm)
@@ -1000,8 +1047,7 @@
apply (rule scaleR_right_distrib)
apply simp
apply (rule scaleR_left_commute)
-apply (rule_tac x="1" in exI)
-apply (simp add: norm_scaleR)
+apply (rule_tac x="1" in exI, simp)
done
interpretation scaleR_left: bounded_linear "\<lambda>r. scaleR r x"
--- a/src/HOL/SEQ.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/SEQ.thy Sat Jun 13 16:32:38 2009 +0200
@@ -348,23 +348,7 @@
fixes L :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
-proof (cases "finite S")
- case True
- thus ?thesis using n
- proof (induct)
- case empty
- show ?case
- by (simp add: LIMSEQ_const)
- next
- case insert
- thus ?case
- by (simp add: LIMSEQ_add)
- qed
-next
- case False
- thus ?thesis
- by (simp add: LIMSEQ_const)
-qed
+using n unfolding LIMSEQ_conv_tendsto by (rule tendsto_setsum)
lemma LIMSEQ_setprod:
fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
--- a/src/HOL/Transitive_Closure.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/Transitive_Closure.thy Sat Jun 13 16:32:38 2009 +0200
@@ -486,6 +486,28 @@
lemmas tranclD = tranclpD [to_set]
+lemma converse_tranclpE:
+ assumes major: "tranclp r x z"
+ assumes base: "r x z ==> P"
+ assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
+ shows P
+proof -
+ from tranclpD[OF major]
+ obtain y where "r x y" and "rtranclp r y z" by iprover
+ from this(2) show P
+ proof (cases rule: rtranclp.cases)
+ case rtrancl_refl
+ with `r x y` base show P by iprover
+ next
+ case rtrancl_into_rtrancl
+ from this have "tranclp r y z"
+ by (iprover intro: rtranclp_into_tranclp1)
+ with `r x y` step show P by iprover
+ qed
+qed
+
+lemmas converse_tranclE = converse_tranclpE [to_set]
+
lemma tranclD2:
"(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
by (blast elim: tranclE intro: trancl_into_rtrancl)
--- a/src/HOL/ex/Predicate_Compile_ex.thy Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/ex/Predicate_Compile_ex.thy Sat Jun 13 16:32:38 2009 +0200
@@ -7,12 +7,8 @@
| "even n \<Longrightarrow> odd (Suc n)"
| "odd n \<Longrightarrow> even (Suc n)"
+code_pred even .
-(*
-code_pred even
- using assms by (rule even.cases)
-*)
-setup {* Predicate_Compile.add_equations @{const_name even} *}
thm odd.equation
thm even.equation
@@ -31,15 +27,7 @@
"rev [] []"
| "rev xs xs' ==> append xs' [x] ys ==> rev (x#xs) ys"
-setup {* Predicate_Compile.add_equations @{const_name rev} *}
-
-thm rev.equation
-thm append.equation
-
-(*
-code_pred append
- using assms by (rule append.cases)
-*)
+code_pred rev .
thm append.equation
@@ -54,36 +42,44 @@
| "f x \<Longrightarrow> partition f xs ys zs \<Longrightarrow> partition f (x # xs) (x # ys) zs"
| "\<not> f x \<Longrightarrow> partition f xs ys zs \<Longrightarrow> partition f (x # xs) ys (x # zs)"
-setup {* Predicate_Compile.add_equations @{const_name partition} *}
+(* FIXME: correct handling of parameters *)
(*
-code_pred partition
- using assms by (rule partition.cases)
-*)
+ML {* reset Predicate_Compile.do_proofs *}
+code_pred partition .
thm partition.equation
+ML {* set Predicate_Compile.do_proofs *}
+*)
+(* TODO: requires to handle abstractions in parameter positions correctly *)
(*FIXME values 10 "{(ys, zs). partition (\<lambda>n. n mod 2 = 0)
- [0, Suc 0, 2, 3, 4, 5, 6, 7] ys zs}"*)
+ [0, Suc 0, 2, 3, 4, 5, 6, 7] ys zs}" *)
+
-setup {* Predicate_Compile.add_equations @{const_name tranclp} *}
+lemma [code_pred_intros]:
+"r a b ==> tranclp r a b"
+"r a b ==> tranclp r b c ==> tranclp r a c"
+by auto
+
+(* Setup requires quick and dirty proof *)
(*
code_pred tranclp
- using assms by (rule tranclp.cases)
-*)
+proof -
+ case tranclp
+ from this converse_tranclpE[OF this(1)] show thesis by metis
+qed
thm tranclp.equation
+*)
inductive succ :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
"succ 0 1"
| "succ m n \<Longrightarrow> succ (Suc m) (Suc n)"
-setup {* Predicate_Compile.add_equations @{const_name succ} *}
-(*
-code_pred succ
- using assms by (rule succ.cases)
-*)
+code_pred succ .
+
thm succ.equation
-
+(* FIXME: why does this not terminate? *)
(*
values 20 "{n. tranclp succ 10 n}"
values "{n. tranclp succ n 10}"
--- a/src/HOL/ex/predicate_compile.ML Sat Jun 13 10:01:01 2009 +0200
+++ b/src/HOL/ex/predicate_compile.ML Sat Jun 13 16:32:38 2009 +0200
@@ -37,13 +37,17 @@
fun tracing s = (if ! Toplevel.debug then Output.tracing s else ());
fun print_tac s = (if ! Toplevel.debug then Tactical.print_tac s else Seq.single);
-fun debug_tac msg = (fn st => (tracing msg; Seq.single st));
+fun new_print_tac s = Tactical.print_tac s
+fun debug_tac msg = (fn st => (Output.tracing msg; Seq.single st));
val do_proofs = ref true;
fun mycheat_tac thy i st =
(Tactic.rtac (SkipProof.make_thm thy (Var (("A", 0), propT))) i) st
+fun remove_last_goal thy st =
+ (Tactic.rtac (SkipProof.make_thm thy (Var (("A", 0), propT))) (nprems_of st)) st
+
(* reference to preprocessing of InductiveSet package *)
val ind_set_codegen_preproc = InductiveSetPackage.codegen_preproc;
@@ -131,6 +135,7 @@
cat_lines (map (fn (s, ms) => s ^ ": " ^ commas (map
string_of_mode ms)) modes));
+
datatype predfun_data = PredfunData of {
name : string,
definition : thm,
@@ -176,9 +181,7 @@
(* queries *)
fun lookup_pred_data thy name =
- case try (Graph.get_node (PredData.get thy)) name of
- SOME pred_data => SOME (rep_pred_data pred_data)
- | NONE => NONE
+ Option.map rep_pred_data (try (Graph.get_node (PredData.get thy)) name)
fun the_pred_data thy name = case lookup_pred_data thy name
of NONE => error ("No such predicate " ^ quote name)
@@ -239,16 +242,73 @@
in thy end;
*)
+
+fun imp_prems_conv cv ct =
+ case Thm.term_of ct of
+ Const ("==>", _) $ _ $ _ => Conv.combination_conv (Conv.arg_conv cv) (imp_prems_conv cv) ct
+ | _ => Conv.all_conv ct
+
+fun Trueprop_conv cv ct =
+ case Thm.term_of ct of
+ Const ("Trueprop", _) $ _ => Conv.arg_conv cv ct
+ | _ => error "Trueprop_conv"
+
+fun preprocess_intro thy rule =
+ Conv.fconv_rule
+ (imp_prems_conv
+ (Trueprop_conv (Conv.try_conv (Conv.rewr_conv (Thm.symmetric @{thm Predicate.eq_is_eq})))))
+ (Thm.transfer thy rule)
+
+fun preprocess_elim thy nargs elimrule = let
+ fun replace_eqs (Const ("Trueprop", _) $ (Const ("op =", T) $ lhs $ rhs)) =
+ HOLogic.mk_Trueprop (Const (@{const_name Predicate.eq}, T) $ lhs $ rhs)
+ | replace_eqs t = t
+ fun preprocess_case t = let
+ val params = Logic.strip_params t
+ val (assums1, assums2) = chop nargs (Logic.strip_assums_hyp t)
+ val assums_hyp' = assums1 @ (map replace_eqs assums2)
+ in list_all (params, Logic.list_implies (assums_hyp', Logic.strip_assums_concl t)) end
+ val prems = Thm.prems_of elimrule
+ val cases' = map preprocess_case (tl prems)
+ val elimrule' = Logic.list_implies ((hd prems) :: cases', Thm.concl_of elimrule)
+ in
+ Thm.equal_elim
+ (Thm.symmetric (Conv.implies_concl_conv (MetaSimplifier.rewrite true [@{thm eq_is_eq}])
+ (cterm_of thy elimrule')))
+ elimrule
+ end;
+
+fun fetch_pred_data thy name =
+ case try (InductivePackage.the_inductive (ProofContext.init thy)) name of
+ SOME (info as (_, result)) =>
+ let
+ fun is_intro_of intro =
+ let
+ val (const, _) = strip_comb (HOLogic.dest_Trueprop (concl_of intro))
+ in (fst (dest_Const const) = name) end;
+ val intros = map (preprocess_intro thy) (filter is_intro_of (#intrs result))
+ val elim = nth (#elims result) (find_index (fn s => s = name) (#names (fst info)))
+ val nparams = length (InductivePackage.params_of (#raw_induct result))
+ in (intros, elim, nparams) end
+ | NONE => error ("No such predicate: " ^ quote name)
+
(* updaters *)
fun add_predfun name mode data = let
val add = apsnd (cons (mode, mk_predfun_data data))
in PredData.map (Graph.map_node name (map_pred_data add)) end
-fun add_intro thm = let
+fun add_intro thm thy = let
val (name, _) = dest_Const (fst (strip_intro_concl 0 (prop_of thm)))
- fun set (intros, elim, nparams) = (thm::intros, elim, nparams)
- in PredData.map (Graph.map_node name (map_pred_data (apfst set))) end
+ fun cons_intro gr =
+ case try (Graph.get_node gr) name of
+ SOME pred_data => Graph.map_node name (map_pred_data
+ (apfst (fn (intro, elim, nparams) => (thm::intro, elim, nparams)))) gr
+ | NONE =>
+ let
+ val nparams = the_default 0 (try (#3 o fetch_pred_data thy) name)
+ in Graph.new_node (name, mk_pred_data (([thm], NONE, nparams), [])) gr end;
+ in PredData.map cons_intro thy end
fun set_elim thm = let
val (name, _) = dest_Const (fst
@@ -733,21 +793,27 @@
val args = map Free (argnames ~~ (Ts1' @ Ts2))
val (params, io_args) = chop nparams args
val (inargs, outargs) = get_args (snd mode) io_args
+ val param_names = Name.variant_list argnames
+ (map (fn i => "p" ^ string_of_int i) (1 upto nparams))
+ val param_vs = map Free (param_names ~~ Ts1)
val (params', names) = fold_map mk_Eval_of ((params ~~ Ts1) ~~ (fst mode)) []
- val predprop = HOLogic.mk_Trueprop (list_comb (pred, params' @ io_args))
+ val predpropI = HOLogic.mk_Trueprop (list_comb (pred, param_vs @ io_args))
+ val predpropE = HOLogic.mk_Trueprop (list_comb (pred, params' @ io_args))
+ val param_eqs = map (HOLogic.mk_Trueprop o HOLogic.mk_eq) (param_vs ~~ params')
val funargs = params @ inargs
val funpropE = HOLogic.mk_Trueprop (mk_Eval (list_comb (funtrm, funargs),
if null outargs then Free("y", HOLogic.unitT) else mk_tuple outargs))
val funpropI = HOLogic.mk_Trueprop (mk_Eval (list_comb (funtrm, funargs),
mk_tuple outargs))
- val introtrm = Logic.mk_implies (predprop, funpropI)
+ val introtrm = Logic.list_implies (predpropI :: param_eqs, funpropI)
+ val _ = Output.tracing (Syntax.string_of_term_global thy introtrm)
val simprules = [defthm, @{thm eval_pred},
@{thm "split_beta"}, @{thm "fst_conv"}, @{thm "snd_conv"}]
val unfolddef_tac = (Simplifier.asm_full_simp_tac (HOL_basic_ss addsimps simprules) 1)
- val introthm = Goal.prove (ProofContext.init thy) (argnames @ ["y"]) [] introtrm (fn {...} => unfolddef_tac)
+ val introthm = Goal.prove (ProofContext.init thy) (argnames @ param_names @ ["y"]) [] introtrm (fn {...} => unfolddef_tac)
val P = HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT));
- val elimtrm = Logic.list_implies ([funpropE, Logic.mk_implies (predprop, P)], P)
- val elimthm = Goal.prove (ProofContext.init thy) (argnames @ ["y", "P"]) [] elimtrm (fn {...} => unfolddef_tac)
+ val elimtrm = Logic.list_implies ([funpropE, Logic.mk_implies (predpropE, P)], P)
+ val elimthm = Goal.prove (ProofContext.init thy) (argnames @ param_names @ ["y", "P"]) [] elimtrm (fn {...} => unfolddef_tac)
in
(introthm, elimthm)
end;
@@ -792,6 +858,7 @@
in thy' |> add_predfun name mode (mode_id, definition, intro, elim)
|> PureThy.store_thm (Binding.name (Long_Name.base_name mode_id ^ "I"), intro) |> snd
|> PureThy.store_thm (Binding.name (Long_Name.base_name mode_id ^ "E"), elim) |> snd
+ |> Theory.checkpoint
end;
in
fold create_definition modes thy
@@ -803,42 +870,6 @@
fun is_Type (Type _) = true
| is_Type _ = false
-fun imp_prems_conv cv ct =
- case Thm.term_of ct of
- Const ("==>", _) $ _ $ _ => Conv.combination_conv (Conv.arg_conv cv) (imp_prems_conv cv) ct
- | _ => Conv.all_conv ct
-
-fun Trueprop_conv cv ct =
- case Thm.term_of ct of
- Const ("Trueprop", _) $ _ => Conv.arg_conv cv ct
- | _ => error "Trueprop_conv"
-
-fun preprocess_intro thy rule =
- Conv.fconv_rule
- (imp_prems_conv
- (Trueprop_conv (Conv.try_conv (Conv.rewr_conv (Thm.symmetric @{thm Predicate.eq_is_eq})))))
- (Thm.transfer thy rule)
-
-fun preprocess_elim thy nargs elimrule = let
- fun replace_eqs (Const ("Trueprop", _) $ (Const ("op =", T) $ lhs $ rhs)) =
- HOLogic.mk_Trueprop (Const (@{const_name Predicate.eq}, T) $ lhs $ rhs)
- | replace_eqs t = t
- fun preprocess_case t = let
- val params = Logic.strip_params t
- val (assums1, assums2) = chop nargs (Logic.strip_assums_hyp t)
- val assums_hyp' = assums1 @ (map replace_eqs assums2)
- in list_all (params, Logic.list_implies (assums_hyp', Logic.strip_assums_concl t)) end
- val prems = Thm.prems_of elimrule
- val cases' = map preprocess_case (tl prems)
- val elimrule' = Logic.list_implies ((hd prems) :: cases', Thm.concl_of elimrule)
- in
- Thm.equal_elim
- (Thm.symmetric (Conv.implies_concl_conv (MetaSimplifier.rewrite true [@{thm eq_is_eq}])
- (cterm_of thy elimrule')))
- elimrule
- end;
-
-
(* returns true if t is an application of an datatype constructor *)
(* which then consequently would be splitted *)
(* else false *)
@@ -859,7 +890,16 @@
fun prove_param thy modes (NONE, t) =
all_tac
| prove_param thy modes (m as SOME (Mode (mode, is, ms)), t) =
- let
+ REPEAT_DETERM (etac @{thm thin_rl} 1)
+ THEN REPEAT_DETERM (rtac @{thm ext} 1)
+ THEN (rtac @{thm iffI} 1)
+ THEN new_print_tac "prove_param"
+ (* proof in one direction *)
+ THEN (atac 1)
+ (* proof in the other direction *)
+ THEN (atac 1)
+ THEN new_print_tac "after prove_param"
+(* let
val (f, args) = strip_comb t
val (params, _) = chop (length ms) args
val f_tac = case f of
@@ -872,11 +912,10 @@
print_tac "before simplification in prove_args:"
THEN f_tac
THEN print_tac "after simplification in prove_args"
- (* work with parameter arguments *)
THEN (EVERY (map (prove_param thy modes) (ms ~~ params)))
THEN (REPEAT_DETERM (atac 1))
end
-
+*)
fun prove_expr thy modes (SOME (Mode (mode, is, ms)), t, us) (premposition : int) =
(case strip_comb t of
(Const (name, T), args) =>
@@ -897,8 +936,10 @@
(* for the right assumption in first position *)
THEN rotate_tac premposition 1
THEN rtac introrule 1
- THEN print_tac "after intro rule"
+ THEN new_print_tac "after intro rule"
(* work with parameter arguments *)
+ THEN (atac 1)
+ THEN (new_print_tac "parameter goal")
THEN (EVERY (map (prove_param thy modes) (ms ~~ args1)))
THEN (REPEAT_DETERM (atac 1)) end)
else error "Prove expr if case not implemented"
@@ -1032,7 +1073,7 @@
val nargs = length (binder_types T) - nparams_of thy pred
val pred_case_rule = singleton (ind_set_codegen_preproc thy)
(preprocess_elim thy nargs (the_elim_of thy pred))
- (* FIXME preprocessor |> Simplifier.full_simplify (HOL_basic_ss addsimps [@ {thm Predicate.memb_code}])*)
+ (* FIXME preprocessor |> Simplifier.full_simplify (HOL_basic_ss addsimps [@{thm Predicate.memb_code}])*)
in
REPEAT_DETERM (CHANGED (rewtac @{thm "split_paired_all"}))
THEN etac (predfun_elim_of thy pred mode) 1
@@ -1041,6 +1082,7 @@
(fn i => EVERY' (select_sup (length clauses) i) i)
(1 upto (length clauses))))
THEN (EVERY (map (prove_clause thy nargs all_vs param_vs modes mode) clauses))
+ THEN new_print_tac "proved one direction"
end;
(*******************************************************************************************************)
@@ -1073,7 +1115,8 @@
(* VERY LARGE SIMILIRATIY to function prove_param
-- join both functions
-*)
+*)
+(*
fun prove_param2 thy modes (NONE, t) = all_tac
| prove_param2 thy modes (m as SOME (Mode (mode, is, ms)), t) = let
val (f, args) = strip_comb t
@@ -1087,9 +1130,9 @@
print_tac "before simplification in prove_args:"
THEN f_tac
THEN print_tac "after simplification in prove_args"
- (* work with parameter arguments *)
THEN (EVERY (map (prove_param2 thy modes) (ms ~~ params)))
end
+*)
fun prove_expr2 thy modes (SOME (Mode (mode, is, ms)), t) =
(case strip_comb t of
@@ -1097,8 +1140,14 @@
if AList.defined op = modes name then
etac @{thm bindE} 1
THEN (REPEAT_DETERM (CHANGED (rewtac @{thm "split_paired_all"})))
+ THEN (debug_tac (Syntax.string_of_term_global thy
+ (prop_of (predfun_elim_of thy name mode))))
THEN (etac (predfun_elim_of thy name mode) 1)
- THEN (EVERY (map (prove_param2 thy modes) (ms ~~ args)))
+ THEN new_print_tac "prove_expr2"
+ (* TODO -- FIXME: replace remove_last_goal*)
+ THEN (EVERY (replicate (length args) (remove_last_goal thy)))
+ THEN new_print_tac "finished prove_expr2"
+ (* THEN (EVERY (map (prove_param thy modes) (ms ~~ args))) *)
else error "Prove expr2 if case not implemented"
| _ => etac @{thm bindE} 1)
| prove_expr2 _ _ _ = error "Prove expr2 not implemented"
@@ -1177,7 +1226,7 @@
THEN (if is_some name then
full_simp_tac (HOL_basic_ss addsimps [predfun_definition_of thy (the name) (iss, js)]) 1
THEN etac @{thm not_predE} 1
- THEN (EVERY (map (prove_param2 thy modes) (param_modes ~~ params)))
+ THEN (EVERY (map (prove_param thy modes) (param_modes ~~ params)))
else
etac @{thm not_predE'} 1)
THEN rec_tac
@@ -1252,6 +1301,7 @@
fun prepare_intrs thy prednames =
let
+ (* FIXME: preprocessing moved to fetch_pred_data *)
val intrs = map (preprocess_intro thy) (maps (intros_of thy) prednames)
|> ind_set_codegen_preproc thy (*FIXME preprocessor
|> map (Simplifier.full_simplify (HOL_basic_ss addsimps [@ {thm Predicate.memb_code}]))*)
@@ -1316,7 +1366,7 @@
val modes = infer_modes thy extra_modes arities param_vs clauses
val _ = print_modes modes
val _ = tracing "Defining executable functions..."
- val thy' = fold (create_definitions preds nparams) modes thy
+ val thy' = fold (create_definitions preds nparams) modes thy |> Theory.checkpoint
val clauses' = map (fn (s, cls) => (s, (the (AList.lookup (op =) preds s), cls))) clauses
val _ = tracing "Compiling equations..."
val ts = compile_preds thy' all_vs param_vs (extra_modes @ modes) clauses'
@@ -1330,15 +1380,16 @@
[((Binding.qualify true (Long_Name.base_name name) (Binding.name "equation"), result_thms),
[Attrib.attribute_i thy Code.add_default_eqn_attrib])] thy))
(arrange ((map (fn ((name, _), _) => name) pred_mode) ~~ result_thms)) thy'
+ |> Theory.checkpoint
in
thy''
end
(* generation of case rules from user-given introduction rules *)
-fun mk_casesrule introrules nparams ctxt =
+fun mk_casesrule ctxt nparams introrules =
let
- val intros = map prop_of introrules
+ val intros = map (Logic.unvarify o prop_of) introrules
val (pred, (params, args)) = strip_intro_concl nparams (hd intros)
val ([propname], ctxt1) = Variable.variant_fixes ["thesis"] ctxt
val prop = HOLogic.mk_Trueprop (Free (propname, HOLogic.boolT))
@@ -1356,32 +1407,16 @@
in fold Logic.all frees (Logic.list_implies (eqprems @ prems, prop)) end
val assm = HOLogic.mk_Trueprop (list_comb (pred, params @ argvs))
val cases = map mk_case intros
- val (_, ctxt3) = ProofContext.add_assms_i Assumption.assume_export
- [((Binding.name AutoBind.assmsN, []), map (fn t => (t, [])) (assm :: cases))]
- ctxt2
- in (pred, prop, ctxt3) end;
+ in Logic.list_implies (assm :: cases, prop) end;
(* code dependency graph *)
-
-fun fetch_pred_data thy name =
- case try (InductivePackage.the_inductive (ProofContext.init thy)) name of
- SOME (info as (_, result)) =>
- let
- fun is_intro_of intro =
- let
- val (const, _) = strip_comb (HOLogic.dest_Trueprop (concl_of intro))
- in (fst (dest_Const const) = name) end;
- val intros = map (preprocess_intro thy) (filter is_intro_of (#intrs result))
- val elim = nth (#elims result) (find_index (fn s => s = name) (#names (fst info)))
- val nparams = length (InductivePackage.params_of (#raw_induct result))
- in mk_pred_data ((intros, SOME elim, nparams), []) end
- | NONE => error ("No such predicate: " ^ quote name)
-fun dependencies_of (thy : theory) name =
+fun dependencies_of thy name =
let
fun is_inductive_predicate thy name =
is_some (try (InductivePackage.the_inductive (ProofContext.init thy)) name)
- val data = fetch_pred_data thy name
+ val (intro, elim, nparams) = fetch_pred_data thy name
+ val data = mk_pred_data ((intro, SOME elim, nparams), [])
val intros = map Thm.prop_of (#intros (rep_pred_data data))
val keys = fold Term.add_consts intros [] |> map fst
|> filter (is_inductive_predicate thy)
@@ -1391,7 +1426,7 @@
fun add_equations name thy =
let
- val thy' = PredData.map (Graph.extend (dependencies_of thy) name) thy;
+ val thy' = PredData.map (Graph.extend (dependencies_of thy) name) thy |> Theory.checkpoint;
(*val preds = Graph.all_preds (PredData.get thy') [name] |> filter_out (has_elim thy') *)
fun strong_conn_of gr keys =
Graph.strong_conn (Graph.subgraph (member (op =) (Graph.all_succs gr keys)) gr)
@@ -1399,7 +1434,7 @@
val thy'' = fold_rev
(fn preds => fn thy =>
if forall (null o modes_of thy) preds then add_equations_of preds thy else thy)
- scc thy'
+ scc thy' |> Theory.checkpoint
in thy'' end
(** user interface **)
@@ -1417,38 +1452,33 @@
(* TODO: must create state to prove multiple cases *)
fun generic_code_pred prep_const raw_const lthy =
let
- val thy = (ProofContext.theory_of lthy)
+
+ val thy = ProofContext.theory_of lthy
val const = prep_const thy raw_const
- val lthy' = lthy
- val thy' = PredData.map (Graph.extend (dependencies_of thy) const) thy
+
+ val lthy' = LocalTheory.theory (PredData.map (Graph.extend (dependencies_of thy) const)) lthy
+ |> LocalTheory.checkpoint
+ val thy' = ProofContext.theory_of lthy'
val preds = Graph.all_preds (PredData.get thy') [const] |> filter_out (has_elim thy')
- val _ = Output.tracing ("preds: " ^ commas preds)
- (*
- fun mk_elim pred =
+
+ fun mk_cases const =
let
- val nparams = nparams_of thy pred
- val intros = intros_of thy pred
- val (((tfrees, frees), fact), lthy'') =
- Variable.import_thms true intros lthy';
- val (pred, prop, lthy''') = mk_casesrule fact nparams lthy''
- in (pred, prop, lthy''') end;
-
- val (predname, _) = dest_Const pred
- *)
- val nparams = nparams_of thy' const
- val intros = intros_of thy' const
- val (((tfrees, frees), fact), lthy'') =
- Variable.import_thms true intros lthy';
- val (pred, prop, lthy''') = mk_casesrule fact nparams lthy''
- val (predname, _) = dest_Const pred
- fun after_qed [[th]] lthy''' =
- lthy'''
- |> LocalTheory.note Thm.generatedK
- ((Binding.empty, []), [th])
- |> snd
- |> LocalTheory.theory (add_equations_of [predname])
+ val nparams = nparams_of thy' const
+ val intros = intros_of thy' const
+ in mk_casesrule lthy' nparams intros end
+ val cases_rules = map mk_cases preds
+ val cases =
+ map (fn case_rule => RuleCases.Case {fixes = [],
+ assumes = [("", Logic.strip_imp_prems case_rule)],
+ binds = [], cases = []}) cases_rules
+ val case_env = map2 (fn p => fn c => (Long_Name.base_name p, SOME c)) preds cases
+ val _ = Output.tracing (commas (map fst case_env))
+ val lthy'' = ProofContext.add_cases true case_env lthy'
+
+ fun after_qed thms =
+ LocalTheory.theory (fold set_elim (map the_single thms) #> add_equations const)
in
- Proof.theorem_i NONE after_qed [[(prop, [])]] lthy'''
+ Proof.theorem_i NONE after_qed (map (single o (rpair [])) cases_rules) lthy''
end;
structure P = OuterParse
--- a/src/Pure/General/symbol.ML Sat Jun 13 10:01:01 2009 +0200
+++ b/src/Pure/General/symbol.ML Sat Jun 13 16:32:38 2009 +0200
@@ -433,7 +433,7 @@
val scan_encoded_newline =
$$ "\^M" -- $$ "\n" >> K "\n" ||
$$ "\^M" >> K "\n" ||
- $$ "\\" -- Scan.optional ($$ "\\") "" -- Scan.this_string "<^newline>" >> K "\n";
+ Scan.this_string "\\<^newline>" >> K "\n";
val scan_raw =
Scan.this_string "raw:" ^^ (Scan.many raw_chr >> implode) ||
@@ -442,7 +442,7 @@
val scan =
Scan.one is_plain ||
scan_encoded_newline ||
- (($$ "\\" --| Scan.optional ($$ "\\") "") ^^ $$ "<" ^^
+ ($$ "\\" ^^ $$ "<" ^^
!! (fn (cs, _) => malformed_msg (beginning 10 ("\\" :: "<" :: cs)))
(($$ "^" ^^ (scan_raw || scan_id) || scan_id) ^^ $$ ">")) ||
Scan.one not_eof;
@@ -453,7 +453,7 @@
Scan.this_string "{*" || Scan.this_string "*}";
val recover =
- (Scan.this (explode "\\\\<") || Scan.this (explode "\\<")) @@@
+ Scan.this ["\\", "<"] @@@
Scan.repeat (Scan.unless scan_resync (Scan.one not_eof))
>> (fn ss => malformed :: ss @ [end_malformed]);
--- a/src/Pure/General/symbol.scala Sat Jun 13 10:01:01 2009 +0200
+++ b/src/Pure/General/symbol.scala Sat Jun 13 16:32:38 2009 +0200
@@ -20,12 +20,12 @@
[^\\ \ud800-\udfff] | [\ud800-\udbff][\udc00-\udfff] """)
private val symbol = new Regex("""(?xs)
- \\ \\? < (?:
+ \\ < (?:
\^? [A-Za-z][A-Za-z0-9_']* |
\^raw: [\x20-\x7e\u0100-\uffff && [^.>]]* ) >""")
private val bad_symbol = new Regex("(?xs) (?!" + symbol + ")" +
- """ \\ \\? < (?: (?! \s | [\"`\\] | \(\* | \*\) | \{\* | \*\} ) . )*""")
+ """ \\ < (?: (?! \s | [\"`\\] | \(\* | \*\) | \{\* | \*\} ) . )*""")
// total pattern
val regex = new Regex(plain + "|" + symbol + "|" + bad_symbol + "| .")
@@ -133,8 +133,8 @@
}
val ch = new String(Character.toChars(code))
} yield (sym, ch)
- (new Recoder(mapping ++ (for ((x, y) <- mapping) yield ("\\" + x, y))),
- new Recoder(for ((x, y) <- mapping) yield (y, x)))
+ (new Recoder(mapping),
+ new Recoder(mapping map { case (x, y) => (y, x) }))
}
def decode(text: String) = decoder.recode(text)
--- a/src/Pure/ML/ml_lex.ML Sat Jun 13 10:01:01 2009 +0200
+++ b/src/Pure/ML/ml_lex.ML Sat Jun 13 16:32:38 2009 +0200
@@ -247,7 +247,11 @@
Symbol_Pos.source (Position.line 1) src
|> Source.source Symbol_Pos.stopper (Scan.bulk (!!! "bad input" scan_ml)) (SOME (false, recover));
-val tokenize = Source.of_string #> source #> Source.exhaust;
+val tokenize =
+ Source.of_string #>
+ Symbol.source {do_recover = true} #>
+ source #>
+ Source.exhaust;
fun read_antiq (syms, pos) =
(Source.of_list syms
--- a/src/Pure/ML/ml_syntax.ML Sat Jun 13 10:01:01 2009 +0200
+++ b/src/Pure/ML/ml_syntax.ML Sat Jun 13 16:32:38 2009 +0200
@@ -58,7 +58,7 @@
| print_option f (SOME x) = "SOME (" ^ f x ^ ")";
fun print_char s =
- if not (Symbol.is_char s) then raise Fail ("Bad character: " ^ quote s)
+ if not (Symbol.is_char s) then s
else if s = "\"" then "\\\""
else if s = "\\" then "\\\\"
else
@@ -68,7 +68,7 @@
else "\\" ^ string_of_int c
end;
-val print_string = quote o translate_string print_char;
+val print_string = quote o implode o map print_char o Symbol.explode;
val print_strings = print_list print_string;
val print_properties = print_list (print_pair print_string print_string);
--- a/src/Pure/Syntax/syn_trans.ML Sat Jun 13 10:01:01 2009 +0200
+++ b/src/Pure/Syntax/syn_trans.ML Sat Jun 13 16:32:38 2009 +0200
@@ -223,7 +223,7 @@
fun the_struct structs i =
if 1 <= i andalso i <= length structs then nth structs (i - 1)
- else raise error ("Illegal reference to implicit structure #" ^ string_of_int i);
+ else error ("Illegal reference to implicit structure #" ^ string_of_int i);
fun struct_tr structs (*"_struct"*) [Const ("_indexdefault", _)] =
Lexicon.free (the_struct structs 1)
@@ -503,7 +503,7 @@
val exn_results = map (Exn.capture ast_of) pts;
val exns = map_filter Exn.get_exn exn_results;
val results = map_filter Exn.get_result exn_results
- in (case (results, exns) of ([], exn :: _) => raise exn | _ => results) end;
+ in (case (results, exns) of ([], exn :: _) => reraise exn | _ => results) end;
@@ -534,6 +534,6 @@
val exn_results = map (Exn.capture (term_of #> free_fixed)) asts;
val exns = map_filter Exn.get_exn exn_results;
val results = map_filter Exn.get_result exn_results
- in (case (results, exns) of ([], exn :: _) => raise exn | _ => results) end;
+ in (case (results, exns) of ([], exn :: _) => reraise exn | _ => results) end;
end;
--- a/src/Pure/Thy/thy_info.ML Sat Jun 13 10:01:01 2009 +0200
+++ b/src/Pure/Thy/thy_info.ML Sat Jun 13 16:32:38 2009 +0200
@@ -387,7 +387,7 @@
(case Graph.get_node tasks name of
Task body =>
let val after_load = body ()
- in after_load () handle exn => (kill_thy name; raise exn) end
+ in after_load () handle exn => (kill_thy name; reraise exn) end
| _ => ()));
in